subject:(Splitting methods)

Baylor University

1. Padgett, Josh Lee, 1990-. Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.

Degree: PhD, Baylor University. Dept. of Mathematics., 2017, Baylor University

URL: http://hdl.handle.net/2104/10099

► In this dissertation, we explore and analyze highly effective and efficient computational procedures for solving a class of nonlinear and stochastic partial differential equations. We… (more)

▼ In this dissertation, we explore and analyze highly effective and efficient computational procedures for solving a class of nonlinear and stochastic partial differential equations. We are particularly interested in degenerate Kawarada equations arising from numerous multiphysics applications including solid fuel combustion and oil pipeline decay. These differential equations are characterized by their strong degeneracies on the boundary, quenching blow-up singularities in the spatial domain, and vibrant stochastic influences in the evolution processes. In addition, physically relevant numerical solutions to Kawarada-type equations must preserve their positivity and monotonicity in time throughout computations, for all valid initial datum. In light of these concerns and challenges, the numerical methods and analysis developed in this dissertation incorporate nonuniform finite difference approximations of the underlying singular differential equation on arbitrary spatial grids. These approximations are then advanced in time via proper adaptive mechanisms and splitting procedures, when multiple dimensions are involved. Degenerate stochastic Kawarada equations in one-, two-, and three-dimensions are explored. The numerical procedures developed in each case are rigorously shown to not only be unconditionally stable, but also preserve the anticipated solution positivity and monotonicity throughout computations. While the numerical stability is proven in the local linearized sense, the stability analysis has been successfully extended to a novel setting that significantly considers all nonlinear influences. Nonlinear convergence of the quenching numerical solutions, as well as their temporal derivatives, are discussed and assessed through generalized Milne devices for the two-dimensional problem. These explorations offer new insights into the computations of similar singular and stochastic differential equation problems. Intensive simulation experiments and validations are provided. Advisors/Committee Members: Sheng, Qin. (advisor).

Subjects/Keywords: Partial differential equations. Operator splitting methods.

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APA (6^th Edition):

Padgett, Josh Lee, 1. (2017). Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods. (Doctoral Dissertation). Baylor University. Retrieved from http://hdl.handle.net/2104/10099

Chicago Manual of Style (16^th Edition):

Padgett, Josh Lee, 1990-. “Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.” 2017. Doctoral Dissertation, Baylor University. Accessed January 16, 2021. http://hdl.handle.net/2104/10099.

MLA Handbook (7^th Edition):

Padgett, Josh Lee, 1990-. “Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.” 2017. Web. 16 Jan 2021.

Vancouver:

Padgett, Josh Lee 1. Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods. [Internet] [Doctoral dissertation]. Baylor University; 2017. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/2104/10099.

Council of Science Editors:

Padgett, Josh Lee 1. Solving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods. [Doctoral Dissertation]. Baylor University; 2017. Available from: http://hdl.handle.net/2104/10099

King Abdullah University of Science and Technology

2. Felício dos Reis, João Miguel. Stability of BDF-ADI Discretizations.

Degree: 2017, King Abdullah University of Science and Technology

URL: http://hdl.handle.net/10754/625356

► We present new results on absolute stability for BDF-ADI (Backward differentiation formula Alternating Direction Implicit) methods applied to a linear advection and diffusion equations. Unconditional… (more)

Subjects/Keywords: Numerical analysis; Splitting methods; BDF-ADI; absolute stability; scour-cohn algorithm

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APA (6^th Edition):

Felício dos Reis, J. M. (2017). Stability of BDF-ADI Discretizations. (Thesis). King Abdullah University of Science and Technology. Retrieved from http://hdl.handle.net/10754/625356

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Felício dos Reis, João Miguel. “Stability of BDF-ADI Discretizations.” 2017. Thesis, King Abdullah University of Science and Technology. Accessed January 16, 2021. http://hdl.handle.net/10754/625356.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Felício dos Reis, João Miguel. “Stability of BDF-ADI Discretizations.” 2017. Web. 16 Jan 2021.

Vancouver:

Felício dos Reis JM. Stability of BDF-ADI Discretizations. [Internet] [Thesis]. King Abdullah University of Science and Technology; 2017. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/10754/625356.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Felício dos Reis JM. Stability of BDF-ADI Discretizations. [Thesis]. King Abdullah University of Science and Technology; 2017. Available from: http://hdl.handle.net/10754/625356

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

3. Silva lopes, Laura. Méthodes numériques pour la simulation d'évènements rares en dynamique moléculaire : Numerical methods for simulating rare events in molecular dynamics.

Degree: Docteur es, Mathématiques, 2019, Université Paris-Est

URL: http://www.theses.fr/2019PESC1045

► Dans les systèmes dynamiques aléatoires, tels ceux rencontrés en dynamique moléculaire, les événements rares apparaissent naturellement, comme étant liés à des fluctuations de probabilité faible.… (more)

▼ Dans les systèmes dynamiques aléatoires, tels ceux rencontrés en dynamique moléculaire, les événements rares apparaissent naturellement, comme étant liés à des fluctuations de probabilité faible. En dynamique moléculaire, le repliement des protéines, la dissociation protéine-ligand, et la fermeture ou l’ouverture des canaux ioniques dans les membranes, sont des exemples d’événements rares. La simulation d’événements rares est un domaine de recherche important en biophysique depuis presque trois décennies.En dynamique moléculaire, on est particulièrement intéressé par la simulation de la transition entre les états métastables, qui sont des régions de l’espace des phases dans lesquelles le système reste piégé sur des longues périodes de temps. Ces transitions sont rares, leurs simulations sont donc assez coûteuses et parfois même impossibles. Pour contourner ces difficultés, des méthodes d’échantillonnage ont été développées pour simuler efficacement ces événement rares. Parmi celles-ci les méthodes de splitting consistent à diviser l’événement rare en sous-événements successifs plus probables. Par exemple, la trajectoire réactive est divisée en morceaux qui progressent graduellement de l’état initial vers l’état final.Le Adaptive Multilevel Splitting (AMS) est une méthode de splitting où les positions des interfaces intermédiaires sont obtenues de façon naturelle au cours de l’algorithme. Les surfaces sont définies de telle sorte que les probabilités de transition entre elles soient constantes et ceci minimise la variance de l’estimateur de la probabilité de l’événement rare. AMS est une méthode avec peu de paramètres numériques à choisir par l’utilisateur, tout en garantissant une grande robustesse par rapport au choix de ces paramètres.Cette thèse porte sur l’application de la méthode adaptive multilevel splitting en dynamique moléculaire. Deux types de systèmes ont été étudiés. La première famille est constituée de modèles simples, qui nous ont permis d’améliorer la méthode. La seconde famille est faite de systèmes plus réalistes qui représentent des vrai défis, où AMS est utilisé pour avancer nos connaissances sur les mécanismes moléculaires. Cette thèse contient donc à la fois des contributions de nature méthodologique et numérique.Dans un premier temps, une étude conduite sur le changement conformationnel d’une biomolécule simple a permis de valider l’algorithme. Nous avons ensuite proposé une nouvelle technique utilisant une combinaison d’AMS avec une méthode d’échantillonnage préférentiel de l’ensemble des conditions initiales pour estimer plus efficacement le temps de transition. Celle-ci a été validée sur un problème simple et nos résultats ouvrent des perspectives prometteuses pour des applications à des systèmes plus complexes. Une nouvelle approche pour extraire les mécanismes réactionnels liés aux transitions est aussi proposée dans cette thèse. Elle consiste à appliquer des méthodes de clustering sur les trajectoires réactives générées par AMS. Pendant ce travail de thèse, l’implémentation de la méthode… Advisors/Committee Members: Lelièvre, Tony (thesis director), Hénin, Jérôme (thesis director).

Subjects/Keywords: Dynamique moléculaire; Événements rares; Adaptive multilevel splitting; Méthodes numériques; Molecular dynamics; Rare events; Adaptive multilevel splitting; Numerical methods; 510

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Silva lopes, L. (2019). Méthodes numériques pour la simulation d'évènements rares en dynamique moléculaire : Numerical methods for simulating rare events in molecular dynamics. (Doctoral Dissertation). Université Paris-Est. Retrieved from http://www.theses.fr/2019PESC1045

Chicago Manual of Style (16^th Edition):

Silva lopes, Laura. “Méthodes numériques pour la simulation d'évènements rares en dynamique moléculaire : Numerical methods for simulating rare events in molecular dynamics.” 2019. Doctoral Dissertation, Université Paris-Est. Accessed January 16, 2021. http://www.theses.fr/2019PESC1045.

MLA Handbook (7^th Edition):

Silva lopes, Laura. “Méthodes numériques pour la simulation d'évènements rares en dynamique moléculaire : Numerical methods for simulating rare events in molecular dynamics.” 2019. Web. 16 Jan 2021.

Vancouver:

Silva lopes L. Méthodes numériques pour la simulation d'évènements rares en dynamique moléculaire : Numerical methods for simulating rare events in molecular dynamics. [Internet] [Doctoral dissertation]. Université Paris-Est; 2019. [cited 2021 Jan 16]. Available from: http://www.theses.fr/2019PESC1045.

Council of Science Editors:

Silva lopes L. Méthodes numériques pour la simulation d'évènements rares en dynamique moléculaire : Numerical methods for simulating rare events in molecular dynamics. [Doctoral Dissertation]. Université Paris-Est; 2019. Available from: http://www.theses.fr/2019PESC1045

University of Michigan

4. Le, Mai. Reconstruction Methods for Free-Breathing Dynamic Contrast-Enhanced MRI.

Degree: PhD, Electrical Engineering: Systems, 2017, University of Michigan

URL: http://hdl.handle.net/2027.42/138498

► Dynamic Contrast-Enhanced Magnetic Resonance Imaging (DCE-MRI) is a valuable diagnostic tool due to the combination of anatomical and physiological information it provides. However, the sequential… (more)

▼ Dynamic Contrast-Enhanced Magnetic Resonance Imaging (DCE-MRI) is a valuable diagnostic tool due to the combination of anatomical and physiological information it provides. However, the sequential sampling of MRI presents an inherent tradeoff between spatial and temporal resolution. Compressed Sensing (CS) methods have been applied to undersampled MRI to reconstruct full-resolution images at sub-Nyquist sampling rates. In exchange for shorter data acquisition times, CS-MRI requires more computationally intensive iterative reconstruction methods. We present several model-based image reconstruction (MBIR) methods to improve the spatial and temporal resolution of MR images and/or the computational time for multi-coil MRI reconstruction. We propose efficient variable splitting (VS) methods for support-constrained MRI reconstruction, image reconstruction and denoising with non-circulant boundary conditions, and improved temporal regularization for breast DCE-MRI. These proposed VS algorithms decouple the system model and sparsity terms of the convex optimization problem. By leveraging matrix structures in the system model and sparsifying operator, we perform alternating minimization over a list of auxiliary variables, each of which can be performed efficiently. We demonstrate the computational benefits of our proposed VS algorithms compared to similar proposed methods. We also demonstrate convergence guarantees for two proposed methods, ADMM-tridiag and ADMM-FP-tridiag. With simulation experiments, we demonstrate lower error in spatial and temporal dimensions for these VS methods compared to other object models. We also propose a method for indirect motion compensation in 5D liver DCE-MRI. 5D MRI separates temporal changes due to contrast from anatomical changes due to respiratory motion into two distinct dimensions. This work applies a pre-computed motion model to perform motion-compensated regularization across the respiratory dimension and improve the conditioning of this highly sparse 5D reconstruction problem. We demonstrate a proof of concept using a digital phantom with contrast and respiratory changes, and we show preliminary results for motion model-informed regularization on in vivo patient data. Advisors/Committee Members: Fessler, Jeffrey A (committee member), Balter, James M (committee member), Balzano, Laura Kathryn (committee member), Epelman, Marina A (committee member).

Subjects/Keywords: MRI reconstruction; Dynamic Contrast-Enhanced MRI; Variable Splitting Methods for Image Reconstruction; Electrical Engineering; Engineering

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Le, M. (2017). Reconstruction Methods for Free-Breathing Dynamic Contrast-Enhanced MRI. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/138498

Chicago Manual of Style (16^th Edition):

Le, Mai. “Reconstruction Methods for Free-Breathing Dynamic Contrast-Enhanced MRI.” 2017. Doctoral Dissertation, University of Michigan. Accessed January 16, 2021. http://hdl.handle.net/2027.42/138498.

MLA Handbook (7^th Edition):

Le, Mai. “Reconstruction Methods for Free-Breathing Dynamic Contrast-Enhanced MRI.” 2017. Web. 16 Jan 2021.

Vancouver:

Le M. Reconstruction Methods for Free-Breathing Dynamic Contrast-Enhanced MRI. [Internet] [Doctoral dissertation]. University of Michigan; 2017. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/2027.42/138498.

Council of Science Editors:

Le M. Reconstruction Methods for Free-Breathing Dynamic Contrast-Enhanced MRI. [Doctoral Dissertation]. University of Michigan; 2017. Available from: http://hdl.handle.net/2027.42/138498

University of Oxford

5. Banjac, Goran. Operator splitting methods for convex optimization : analysis and implementation.

Degree: PhD, 2018, University of Oxford

URL: https://ora.ox.ac.uk/objects/uuid:17ac73af-9fdf-4cf6-a946-3048da3fc9c2 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740972

► Convex optimization problems are a class of mathematical problems which arise in numerous applications. Although interior-point methods can in principle solve these problems efficiently, they… (more)

▼ Convex optimization problems are a class of mathematical problems which arise in numerous applications. Although interior-point methods can in principle solve these problems efficiently, they may become intractable for solving large-scale problems or be unsuitable for real-time embedded applications. Iterations of operator splitting methods are relatively simple and computationally inexpensive, which makes them suitable for these applications. However, some of their known limitations are slow asymptotic convergence, sensitivity to ill-conditioning, and inability to detect infeasible problems. The aim of this thesis is to better understand operator splitting methods and to develop reliable software tools for convex optimization. The main analytical tool in our investigation of these methods is their characterization as the fixed-point iteration of a nonexpansive operator. The fixed-point theory of nonexpansive operators has been studied for several decades. By exploiting the properties of such an operator, it is possible to show that the alternating direction method of multipliers (ADMM) can detect infeasible problems. Although ADMM iterates diverge when the problem at hand is unsolvable, the differences between subsequent iterates converge to a constant vector which is also a certificate of primal and/or dual infeasibility. Reliable termination criteria for detecting infeasibility are proposed based on this result. Similar ideas are used to derive necessary and sufficient conditions for linear (geometric) convergence of an operator splitting method and a bound on the achievable convergence rate. The new bound turns out to be tight for the class of averaged operators. Next, the OSQP solver is presented. OSQP is a novel general-purpose solver for quadratic programs (QPs) based on ADMM. The solver is very robust, is able to detect infeasible problems, and has been extensively tested on many problem instances from a wide variety of application areas. Finally, operator splitting methods can also be effective in nonconvex optimization. The developed algorithm significantly outperforms a common approach based on convex relaxation of the original nonconvex problem.

Subjects/Keywords: 629.8; Mathematical optimization; Convex optimization; Operator splitting methods; Infeasibility detection; Linear convergence

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APA (6^th Edition):

Banjac, G. (2018). Operator splitting methods for convex optimization : analysis and implementation. (Doctoral Dissertation). University of Oxford. Retrieved from https://ora.ox.ac.uk/objects/uuid:17ac73af-9fdf-4cf6-a946-3048da3fc9c2 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740972

Chicago Manual of Style (16^th Edition):

Banjac, Goran. “Operator splitting methods for convex optimization : analysis and implementation.” 2018. Doctoral Dissertation, University of Oxford. Accessed January 16, 2021. https://ora.ox.ac.uk/objects/uuid:17ac73af-9fdf-4cf6-a946-3048da3fc9c2 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740972.

MLA Handbook (7^th Edition):

Banjac, Goran. “Operator splitting methods for convex optimization : analysis and implementation.” 2018. Web. 16 Jan 2021.

Vancouver:

Banjac G. Operator splitting methods for convex optimization : analysis and implementation. [Internet] [Doctoral dissertation]. University of Oxford; 2018. [cited 2021 Jan 16]. Available from: https://ora.ox.ac.uk/objects/uuid:17ac73af-9fdf-4cf6-a946-3048da3fc9c2 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740972.

Council of Science Editors:

Banjac G. Operator splitting methods for convex optimization : analysis and implementation. [Doctoral Dissertation]. University of Oxford; 2018. Available from: https://ora.ox.ac.uk/objects/uuid:17ac73af-9fdf-4cf6-a946-3048da3fc9c2 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.740972

Universitat Autònoma de Barcelona

6. Vecil, Francesco. A contribution to the simulation of Vlasov-based models.

Degree: Departament de Matemàtiques, 2007, Universitat Autònoma de Barcelona

URL: http://hdl.handle.net/10803/3100

► This thesis is dedicated to the development, application and test of numerical methods for the numerical simulation of problems arising from physics and electronic engineering.… (more)

▼ This thesis is dedicated to the development, application and test of numerical methods for the numerical simulation of problems arising from physics and electronic engineering. The main tool which is used all along the work is the Vlasov (transport) equation in the form of the Boltzmann Transport Equation (BTE) for the description of the transport and collisions of charged particles in plasmas and electronic devices: charge carriers are driven by a force field and scattered by other carriers or phonons (pseudo-particles giving an effective representation of the oscillating field produced by the vibrating ions). The BTE must be coupled to an equation or a system of equations for the computation of the force field: for simple structures the Poisson equation is used; for plasmas, where the magnetic phenomena cannot be neglected due to the high velocities of the particles, the Lorentz force is used, so the Maxwell equations have to be solved; for nanostructures, e.g. transistors with confined dimensions, the Poisson equation needs coupling with Schrödinger equation for the description of the quantum dimensions and the decomposition into subbands, or energy levels. Collisions mean the scattering the carriers suffer due to the interactions with other carriers or the fixed lattice, in form of phonons. All along the thesis several scattering operator are used: the simplest ones are linear relaxation-time operators; a model for the simulation of a semiconductor is studied in which collisions are taken into account with acoustic phonons, in the elastic approximation, and optical phonons. After the introduction, in the first chapter the most important numerical methods are developed: first of all a pointwise non-oscillatory interpolation method (PWENO) needed to avoid the simple Lagrange polynomial reconstruction, which increases the total variation when shocks appear: oscillations are part of the physics of the problem, but if the method adds spurious, non-physical oscillations, then the numerical result is meaningless, or it simply blows up. The second fundamental numerical method is the splitting technique: when solving a complicated problem, if we are able to subdivide it into sub-problem and solve them for separate, then we can reconstruct an approximation for the complete problem; this technique is used for both time splitting (separate transport from collisions) and dimensional splitting (split the phase space into either dimensions). The third fundamental instrument is the solver for linear advections: two methods are used, one based on pointwise following backwards the characteristics and another one based on reconstructing integral values along segments instead of point values; the first one controls better oscillations, the second one forces mass conservation. These methods are applied in chapter 2 to some well-known benchmark tests to control their robustness. In chapter 3 these methods are applied to the simulation of a diode, and the results compared to previous results obtained by Runge-Kutta… Advisors/Committee Members: [email protected] (authoremail), true (authoremailshow), Carrillo de la Plata, José Antonio (director), Abdallah, Naoufel Ben (director).

Subjects/Keywords: Weno(methods); Kinetic equations; Time Splitting; Ciències Experimentals; 51

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Vecil, F. (2007). A contribution to the simulation of Vlasov-based models. (Thesis). Universitat Autònoma de Barcelona. Retrieved from http://hdl.handle.net/10803/3100

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Vecil, Francesco. “A contribution to the simulation of Vlasov-based models.” 2007. Thesis, Universitat Autònoma de Barcelona. Accessed January 16, 2021. http://hdl.handle.net/10803/3100.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Vecil, Francesco. “A contribution to the simulation of Vlasov-based models.” 2007. Web. 16 Jan 2021.

Vancouver:

Vecil F. A contribution to the simulation of Vlasov-based models. [Internet] [Thesis]. Universitat Autònoma de Barcelona; 2007. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/10803/3100.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Vecil F. A contribution to the simulation of Vlasov-based models. [Thesis]. Universitat Autònoma de Barcelona; 2007. Available from: http://hdl.handle.net/10803/3100

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Universitat Politècnica de València

7. Bader, Philipp Karl-Heinz. Geometric Integrators for Schrödinger Equations .

Degree: 2014, Universitat Politècnica de València

URL: http://hdl.handle.net/10251/38716

► The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution… (more)

▼ The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special… Advisors/Committee Members: Blanes Zamora, Sergio (advisor).

Subjects/Keywords: Numerical analysis; Geometric integrators; Splitting methods; Magnus expansion; Algebraic techniques; Schrödinger equation; Gross-Piatevskii equation; Semiclassical limit; Imaginary time

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Bader, P. K. (2014). Geometric Integrators for Schrödinger Equations . (Doctoral Dissertation). Universitat Politècnica de València. Retrieved from http://hdl.handle.net/10251/38716

Chicago Manual of Style (16^th Edition):

Bader, Philipp Karl-Heinz. “Geometric Integrators for Schrödinger Equations .” 2014. Doctoral Dissertation, Universitat Politècnica de València. Accessed January 16, 2021. http://hdl.handle.net/10251/38716.

MLA Handbook (7^th Edition):

Bader, Philipp Karl-Heinz. “Geometric Integrators for Schrödinger Equations .” 2014. Web. 16 Jan 2021.

Vancouver:

Bader PK. Geometric Integrators for Schrödinger Equations . [Internet] [Doctoral dissertation]. Universitat Politècnica de València; 2014. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/10251/38716.

Council of Science Editors:

Bader PK. Geometric Integrators for Schrödinger Equations . [Doctoral Dissertation]. Universitat Politècnica de València; 2014. Available from: http://hdl.handle.net/10251/38716

Universitat Politècnica de València

8. Kopylov, Nikita. Magnus-based geometric integrators for dynamical systems with time-dependent potentials .

Degree: 2019, Universitat Politècnica de València

URL: http://hdl.handle.net/10251/118798

► [ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en… (more)

▼ [ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en la física matemática, porque provienen de la mecánica cuántica, clásica y celestial. La meta de la tesis es construir integradores para unos problemas relevantes no autónomos: la ecuación de Schrödinger, que es el fundamento de la mecánica cuántica; las ecuaciones de Hill y de onda, que describen sistemas oscilatorios; el problema de Kepler con la masa variante en el tiempo. El Capítulo 1 describe la motivación y los objetivos de la obra en el contexto histórico de la integración numérica. En el Capítulo 2 se introducen los conceptos esenciales y unas herramientas fundamentales utilizadas a lo largo de la tesis. El diseño de los integradores propuestos se basa en los métodos de composición y escisión y en el desarrollo de Magnus. En el Capítulo 3 se describe el primero. Su idea principal consta de una recombinación de unos integradores sencillos para obtener la solución del problema. El concepto importante de las condiciones de orden se describe en ese capítulo. En el Capítulo 4 se hace un resumen de las álgebras de Lie y del desarrollo de Magnus que son las herramientas algebraicas que permiten expresar la solución de ecuaciones diferenciales dependientes del tiempo. La ecuación lineal de Schrödinger con potencial dependiente del tiempo está examinada en el Capítulo 5. Dado su estructura particular, nuevos métodos casi sin conmutadores, basados en el desarrollo de Magnus, son construidos. Su eficiencia es demostrada en unos experimentos numéricos con el modelo de Walker-Preston de una molécula dentro de un campo electromagnético. En el Capítulo 6, se diseñan los métodos de Magnus-escisión para las ecuaciones de onda y de Hill. Su eficiencia está demostrada en los experimentos numéricos con varios sistemas oscilatorios: con la ecuación de Mathieu, la ec. de Hill matricial, las ecuaciones de onda y de Klein-Gordon-Fock. El Capítulo 7 explica cómo el enfoque algebraico y el desarrollo de Magnus pueden generalizarse a los problemas no lineales. El ejemplo utilizado es el problema de Kepler con masa decreciente. El Capítulo 8 concluye la tesis, reseña los resultados y traza las posibles direcciones de la investigación futura.; [CAT] Aquesta tesi tracta de la integració numèrica de sistemes hamiltonians amb potencials explícitament dependents del temps. Els problemes d'aquest tipus són comuns en la física matemàtica, perquè provenen de la mecànica quàntica, clàssica i celest. L'objectiu de la tesi és construir integradors per a uns problemes rellevants no autònoms: l'equació de Schrödinger, que és el fonament de la mecànica quàntica; les equacions de Hill i d'ona, que descriuen sistemes oscil·latoris; el problema de Kepler amb la massa variant en el temps. El Capítol 1 descriu la motivació i els objectius de l'obra en el context històric de la integració numèrica. En Capítol 2 s'introdueixen els conceptes essencials i unes ferramentes… Advisors/Committee Members: Bader, Philipp Karl Heinz (advisor), Blanes Zamora, Sergio (advisor).

Subjects/Keywords: Numerical analysis; Geometric numerical integration; Symplectic integrator; Structure preservation; Differential equations; Time-dependent; Non-autonomous; Magnus expansion; Splitting methods; Composition methods; Schrödinger equation; Wave equation; Hill equation; Mathieu equation; Kepler problem; Quasi-commutator-free; Quasi-Magnus; Magnus-splitting

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Kopylov, N. (2019). Magnus-based geometric integrators for dynamical systems with time-dependent potentials . (Doctoral Dissertation). Universitat Politècnica de València. Retrieved from http://hdl.handle.net/10251/118798

Chicago Manual of Style (16^th Edition):

Kopylov, Nikita. “Magnus-based geometric integrators for dynamical systems with time-dependent potentials .” 2019. Doctoral Dissertation, Universitat Politècnica de València. Accessed January 16, 2021. http://hdl.handle.net/10251/118798.

MLA Handbook (7^th Edition):

Kopylov, Nikita. “Magnus-based geometric integrators for dynamical systems with time-dependent potentials .” 2019. Web. 16 Jan 2021.

Vancouver:

Kopylov N. Magnus-based geometric integrators for dynamical systems with time-dependent potentials . [Internet] [Doctoral dissertation]. Universitat Politècnica de València; 2019. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/10251/118798.

Council of Science Editors:

Kopylov N. Magnus-based geometric integrators for dynamical systems with time-dependent potentials . [Doctoral Dissertation]. Universitat Politècnica de València; 2019. Available from: http://hdl.handle.net/10251/118798

9. Zou, Qinmeng. Iterative methods with retards for the solution of large-scale linear systems : Méthodes itératives à retards pour la résolution des systèmes linéaires à grande échelle.

Degree: Docteur es, Mathématiques appliquées, 2019, Université Paris-Saclay (ComUE)

URL: http://www.theses.fr/2019SACLC042

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Toute perturbation dans les systèmes linéaires peut gravement dégrader la performance des méthodes itératives lorsque les directions conjuguées sont constituées. Ce problème peut être partiellement… (more)

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Toute perturbation dans les systèmes linéaires peut gravement dégrader la performance des méthodes itératives lorsque les directions conjuguées sont constituées. Ce problème peut être partiellement résolu par les méthodes du gradient à retards, qui ne garantissent pas la descente de la fonction quadratique, mais peuvent améliorer la convergence par rapport aux méthodes traditionnelles. Les travaux ultérieurs se sont concentrés sur les méthodes du gradient alternées avec deux ou plusieurs types de pas afin d'interrompre le zigzag. Des papiers récents ont suggéré que la révélation d'information de second ordre avec des pas à retards pourrait réduire de manière asymptotique les espaces de recherche dans des dimensions de plus en plus petites. Ceci a conduit aux méthodes du gradient avec alignement dans lesquelles l'étape essentielle et l'étape auxiliaire sont effectuées en alternance. Des expériences numériques ont démontré leur efficacité. Cette thèse considère d'abord des méthodes du gradient efficaces pour résoudre les systèmes linéaires symétriques définis positifs. Nous commençons par étudier une méthode alternée avec la propriété de terminaison finie à deux dimensions. Ensuite, nous déduisons davantage de propriétés spectrales pour les méthodes du gradient traditionnelles. Ces propriétés nous permettent d’élargir la famille de méthodes du gradient avec alignement et d’établir la convergence de nouvelles méthodes. Nous traitons également les itérations de gradient comme un processus peu coûteux intégré aux méthodes de splitting. En particulier, nous abordons le problème de l’estimation de paramètre et suggérons d’utiliser les méthodes du gradient rapide comme solveurs internes à faible précision. Dans le cas parallèle, nous nous concentrons sur les formulations avec retards pour lesquelles il est possible de réduire les coûts de communication. Nous présentons également de nouvelles propriétés et méthodes pour les itérations de gradient s-dimensionnelles. En résumé, cette thèse s'intéresse aux trois sujets interreliés dans lesquelles les itérations de gradient peuvent être utilisées en tant que solveurs efficaces, qu’outils intégrés pour les méthodes de splitting et que solveurs parallèles pour réduire la communication. Des exemples numériques sont présentés à la fin de chaque sujet pour appuyer nos résultats théoriques.

Any perturbation in linear systems may severely degrade the performance of iterative methods when conjugate directions are constructed. This issue can be partially remedied by lagged gradient methods, which does not guarantee descent in the quadratic function but can improve the convergence compared with traditional gradient methods. Later work focused on alternate gradient methods with two or more steplengths in order to break the zigzag pattern. Recent papers suggested that revealing of second-order information along with lagged steps could reduce asymptotically the search spaces in smaller and smaller dimensions. This led to gradient methods with alignment in which essential and auxiliary…

Advisors/Committee Members: Magoulès, Frédéric (thesis director).

Subjects/Keywords: Méthodes du gradient avec alignement; Méthodes du gradient à retards; Propriétés spectrales; Splitting hermitien et anti-Hermitien; Calcul parallèle; Gradient methods with alignment; Lagged gradient methods; Spectral properties; Hermitian and skew-Hermitian splitting; Parallel computing

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Zou, Q. (2019). Iterative methods with retards for the solution of large-scale linear systems : Méthodes itératives à retards pour la résolution des systèmes linéaires à grande échelle. (Doctoral Dissertation). Université Paris-Saclay (ComUE). Retrieved from http://www.theses.fr/2019SACLC042

Chicago Manual of Style (16^th Edition):

Zou, Qinmeng. “Iterative methods with retards for the solution of large-scale linear systems : Méthodes itératives à retards pour la résolution des systèmes linéaires à grande échelle.” 2019. Doctoral Dissertation, Université Paris-Saclay (ComUE). Accessed January 16, 2021. http://www.theses.fr/2019SACLC042.

MLA Handbook (7^th Edition):

Zou, Qinmeng. “Iterative methods with retards for the solution of large-scale linear systems : Méthodes itératives à retards pour la résolution des systèmes linéaires à grande échelle.” 2019. Web. 16 Jan 2021.

Vancouver:

Zou Q. Iterative methods with retards for the solution of large-scale linear systems : Méthodes itératives à retards pour la résolution des systèmes linéaires à grande échelle. [Internet] [Doctoral dissertation]. Université Paris-Saclay (ComUE); 2019. [cited 2021 Jan 16]. Available from: http://www.theses.fr/2019SACLC042.

Council of Science Editors:

Zou Q. Iterative methods with retards for the solution of large-scale linear systems : Méthodes itératives à retards pour la résolution des systèmes linéaires à grande échelle. [Doctoral Dissertation]. Université Paris-Saclay (ComUE); 2019. Available from: http://www.theses.fr/2019SACLC042

Université de Bordeaux I

10. Poux, Alexandre. Conditions limites de sortie pour les méthodes de time-splitting appliquées aux équations Navier-Stokes : Outflow boundary conditions for time-splitting methods applied to Navier-Stokes equations.

Degree: Docteur es, Mécanique, 2012, Université de Bordeaux I

URL: http://www.theses.fr/2012BOR14656

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La simulation d’écoulements incompressibles pose de nombreuses difficultés. Une première est la question de savoir comment traiter la contrainte d’incompressibilité et le couplage vitesse/pression afin… (more)

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La simulation d’écoulements incompressibles pose de nombreuses difficultés. Une première est la question de savoir comment traiter la contrainte d’incompressibilité et le couplage vitesse/pression afin d’obtenir une solution précise à moindre coût. Pour cela, nous nous intéressons en particulier à deux méthodes de time splitting : la correction de pression et la correction de vitesse. Une seconde difficulté porte sur des conditions limites de sortie. Nous nous intéressons ici à deux d’entre elles : la condition limite de traction et la condition limite d’Orlanski. Après avoir détaillé les difficultés pouvant apparaître lors de l’implémentation des méthodes de time-splitting, nous proposons une nouvelle implémentation de la condition limite de traction qui permet d’améliorer les ordres de convergence obtenus. Nous nous intéressons ensuite à la condition limite d’Orlanski qui nécessite une certaine vitesse d’advection C dans la direction normale à la limite dont nous proposons ici une nouvelle définition. Nos propositions sont confrontées à de multiples écoulements physiques afin de valider leurs comportements : l’écoulement en aval d’une marche descendante, l’écoulement au niveau d’une bifurcation,l’écoulement autour d’un obstacle et des écoulements de Poiseuille-Rayleigh-Bénard.

One of the understudied difficulties in the simulation of incompressible flows is how to treat the incompressibilityconstraint and the velocity/pressure coupling in order to obtain an accurate solution at low computationnalcost. In this context, we develop two methods: pressure-correction and velocity-correction. An anotherdifficulty is due to the boundary conditions. We study here two of them : the traction boundary condition andthe Orlanski boundary condition. After having developed the difficulties that appears when implementing timesplittingmethods, we propose a new way to enforce the traction boundary condition which improves the orderof convergence. Then we propose a new definition of the advective velocity C which is needed for the Orlanskiboundary condition. Our propositions are validated against multiple physical flows: flow over a backward facingstep, flow around a biffurcation, flow around an obstacle and several Poiseuille-Rayleigh-Bénard flows.

Advisors/Committee Members: Azaïez, Mejdi (thesis director), Glockner, Stéphane (thesis director).

Subjects/Keywords: Navier-Stokes; Méthodes de time-splitting; Condition limite de sortie; Condition limite de traction; Condition limite d’Orlanski; Navier-Stokes; Time-splitting methods; Outflow boundary condition; Traction boundary condition; Orlanski boundary condition

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Poux, A. (2012). Conditions limites de sortie pour les méthodes de time-splitting appliquées aux équations Navier-Stokes : Outflow boundary conditions for time-splitting methods applied to Navier-Stokes equations. (Doctoral Dissertation). Université de Bordeaux I. Retrieved from http://www.theses.fr/2012BOR14656

Chicago Manual of Style (16^th Edition):

Poux, Alexandre. “Conditions limites de sortie pour les méthodes de time-splitting appliquées aux équations Navier-Stokes : Outflow boundary conditions for time-splitting methods applied to Navier-Stokes equations.” 2012. Doctoral Dissertation, Université de Bordeaux I. Accessed January 16, 2021. http://www.theses.fr/2012BOR14656.

MLA Handbook (7^th Edition):

Poux, Alexandre. “Conditions limites de sortie pour les méthodes de time-splitting appliquées aux équations Navier-Stokes : Outflow boundary conditions for time-splitting methods applied to Navier-Stokes equations.” 2012. Web. 16 Jan 2021.

Vancouver:

Poux A. Conditions limites de sortie pour les méthodes de time-splitting appliquées aux équations Navier-Stokes : Outflow boundary conditions for time-splitting methods applied to Navier-Stokes equations. [Internet] [Doctoral dissertation]. Université de Bordeaux I; 2012. [cited 2021 Jan 16]. Available from: http://www.theses.fr/2012BOR14656.

Council of Science Editors:

Poux A. Conditions limites de sortie pour les méthodes de time-splitting appliquées aux équations Navier-Stokes : Outflow boundary conditions for time-splitting methods applied to Navier-Stokes equations. [Doctoral Dissertation]. Université de Bordeaux I; 2012. Available from: http://www.theses.fr/2012BOR14656

11. Blanc, Emilie. Time-domain numerical modeling of poroelastic waves : the Biot-JKD model with fractional derivatives : Groundwater resources vulnerability : origins of salinity in coastal karst groundwater, contamination by heavy metals in post closure mine : multiple tracers, geochemical approach.

Degree: Docteur es, Acoustique, 2013, Aix Marseille Université

URL: http://www.theses.fr/2013AIXM4774

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Une modélisation numérique des ondes poroélastiques, décrites par le modèle de Biot, est proposée dans le domaine temporel. La dissipation visqueuse à l'intérieur des pores… (more)

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Une modélisation numérique des ondes poroélastiques, décrites par le modèle de Biot, est proposée dans le domaine temporel. La dissipation visqueuse à l'intérieur des pores est décrite par le modèle de perméabilité dynamique de Johnson-Koplik-Dashen (JKD). Certains coefficients du modèle de Biot-JKD sont proportionnels à la racine carrée de la fréquence, introduisant dans le domaine temporel des dérivées fractionnaires décalées d'ordre 1/2, revenant à un produit de convolution. Basé sur une représentation diffusive, le produit de convolution est remplacé par un nombre fini de variables de mémoire satisfaisant une équation différentielle ordinaire locale en temps, menant au modèle de Biot-DA (diffusive approximation). Les propriétés des deux modèles sont analysées : hyperbolicité, décroissance de l'énergie, dispersion. On montre que la meilleure méthode de détermination des coefficients de l'approximation diffusive - quadratures de Gauss, optimisation linéaire ou non-linéaire sur la plage de fréquence d'intérêt - est l'optimisation non-linéaire. Une méthode de splitting est utilisée numériquement : la partie propagative est discrétisée par un schéma aux différences finies ADER d'ordre 4, et la partie diffusive est intégrée exactement. Les conditions de saut aux interfaces sont discrétisées avec une méthode d'interface immergée. Des simulations numériques sont présentées pour des milieux isotropes et isotropes transverses. Des comparaisons avec des solutions analytiques montrent l'efficacité et la précision de cette approche. Des simulations numériques en milieux complexes sont réalisées : influence de la porosité d'os spongieux, diffusion multiple en milieu aléatoire.

A time-domain numerical modeling of Biot poroelastic waves is proposed. The viscous dissipation in the pores is described using the dynamic permeability model of Johnson-Koplik-Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency: in the time-domain, these coefficients introduce shifted fractional derivatives of order 1/2, involving a convolution product. Based on a diffusive representation, the convolution product is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations, resulting in the Biot-DA (diffusive approximation). The properties of the two models are analyzed: hyperbolicity, decrease of energy, dispersion. To determine the coefficients of the diffusive approximation, different methods of quadrature are analyzed: Gaussian quadratures, linear or nonlinear optimization procedures in the frequency range of interest. The nonlinear optimization is shown to be the best way of determination. A splitting strategy is applied numerically: the propagative part is discretized using a fourth-order ADER scheme on a Cartesian grid, and the diffusive part is solved exactly. An immersed interface method is implemented to discretize the jump conditions at interfaces. Numerical experiments are presented for isotropic and transversely isotropic media.…

Advisors/Committee Members: Lombard, Bruno (thesis director), Chiavassa, Guillaume (thesis director).

Subjects/Keywords: Milieux poreux; Ondes élastiques; Modèle de Biot-JKD; Dérivées fractionnaires; Méthode de splitting; Méthodes de différences finies; Méthode d'interface immergée; Porous media; Elastic waves; Biot-JKD model; Fractional derivatives; Time splitting method; Finite difference methods; Immersed interface method; 534

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Blanc, E. (2013). Time-domain numerical modeling of poroelastic waves : the Biot-JKD model with fractional derivatives : Groundwater resources vulnerability : origins of salinity in coastal karst groundwater, contamination by heavy metals in post closure mine : multiple tracers, geochemical approach. (Doctoral Dissertation). Aix Marseille Université. Retrieved from http://www.theses.fr/2013AIXM4774

Chicago Manual of Style (16^th Edition):

Blanc, Emilie. “Time-domain numerical modeling of poroelastic waves : the Biot-JKD model with fractional derivatives : Groundwater resources vulnerability : origins of salinity in coastal karst groundwater, contamination by heavy metals in post closure mine : multiple tracers, geochemical approach.” 2013. Doctoral Dissertation, Aix Marseille Université. Accessed January 16, 2021. http://www.theses.fr/2013AIXM4774.

MLA Handbook (7^th Edition):

Blanc, Emilie. “Time-domain numerical modeling of poroelastic waves : the Biot-JKD model with fractional derivatives : Groundwater resources vulnerability : origins of salinity in coastal karst groundwater, contamination by heavy metals in post closure mine : multiple tracers, geochemical approach.” 2013. Web. 16 Jan 2021.

Vancouver:

Blanc E. Time-domain numerical modeling of poroelastic waves : the Biot-JKD model with fractional derivatives : Groundwater resources vulnerability : origins of salinity in coastal karst groundwater, contamination by heavy metals in post closure mine : multiple tracers, geochemical approach. [Internet] [Doctoral dissertation]. Aix Marseille Université 2013. [cited 2021 Jan 16]. Available from: http://www.theses.fr/2013AIXM4774.

Council of Science Editors:

Blanc E. Time-domain numerical modeling of poroelastic waves : the Biot-JKD model with fractional derivatives : Groundwater resources vulnerability : origins of salinity in coastal karst groundwater, contamination by heavy metals in post closure mine : multiple tracers, geochemical approach. [Doctoral Dissertation]. Aix Marseille Université 2013. Available from: http://www.theses.fr/2013AIXM4774

12. Diegel, Amanda Emily. Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn-Hilliard-Fluid-Flow Equations.

Degree: 2015, University of Tennessee – Knoxville

URL: https://trace.tennessee.edu/utk_graddiss/3332

► This dissertation investigates numerical schemes for the Cahn-Hilliard equation and the Cahn-Hilliard equation coupled with a Darcy-Stokes flow. Considered independently, the Cahn-Hilliard equation is a… (more)

▼ This dissertation investigates numerical schemes for the Cahn-Hilliard equation and the Cahn-Hilliard equation coupled with a Darcy-Stokes flow. Considered independently, the Cahn-Hilliard equation is a model for spinodal decomposition and domain coarsening. When coupled with a Darcy-Stokes flow, the resulting system describes the flow of a very viscous block copolymer fluid. Challenges in creating numerical schemes for these equations arise due to the nonlinear nature and high derivative order of the Cahn-Hilliard equation. Further challenges arise during the coupling process as the coupling terms tend to be nonlinear as well. The numerical schemes presented herein preserve the energy dissipative structure of the Cahn- Hilliard equation while maintaining unique solvability and optimal error bounds. Specifically, we devise and analyze two mixed finite element schemes: a first order in time numerical scheme for a modified Cahn-Hilliard equation coupled with a non- steady Darcy-Stokes flow and a second order in time numerical scheme for the Cahn- Hilliard equation in two and three dimensions. The time discretizations are based on a convex splitting of the energy of the systems. We prove that our schemes are unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energies and unconditionally uniquely solvable. For each system, we prove that the discrete phase variable is essentially bounded in both time and space with respect to the Lebesque integral and the discrete chemical potential is Lesbegue square integrable in space and essentially bounded in time. We show these bounds are completely independent of the time and space step sizes in two and three dimensions. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms. The analyses included in this dissertation will provide a bridge to the development of stable, efficient, and optimally convergent numerical schemes for more robust and descriptive coupled Cahn-Hilliard-Fluid-Flow systems.

Subjects/Keywords: Cahn Hilliard; mixed finite element methods; convex splitting; fluid flow and phase separation; energy stability; second order accuracy; Numerical Analysis and Computation; Partial Differential Equations

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Diegel, A. E. (2015). Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn-Hilliard-Fluid-Flow Equations. (Doctoral Dissertation). University of Tennessee – Knoxville. Retrieved from https://trace.tennessee.edu/utk_graddiss/3332

Chicago Manual of Style (16^th Edition):

Diegel, Amanda Emily. “Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn-Hilliard-Fluid-Flow Equations.” 2015. Doctoral Dissertation, University of Tennessee – Knoxville. Accessed January 16, 2021. https://trace.tennessee.edu/utk_graddiss/3332.

MLA Handbook (7^th Edition):

Diegel, Amanda Emily. “Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn-Hilliard-Fluid-Flow Equations.” 2015. Web. 16 Jan 2021.

Vancouver:

Diegel AE. Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn-Hilliard-Fluid-Flow Equations. [Internet] [Doctoral dissertation]. University of Tennessee – Knoxville; 2015. [cited 2021 Jan 16]. Available from: https://trace.tennessee.edu/utk_graddiss/3332.

Council of Science Editors:

Diegel AE. Numerical Analysis of Convex Splitting Schemes for Cahn-Hilliard and Coupled Cahn-Hilliard-Fluid-Flow Equations. [Doctoral Dissertation]. University of Tennessee – Knoxville; 2015. Available from: https://trace.tennessee.edu/utk_graddiss/3332

13. Horsin, Romain. Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics.

Degree: Docteur es, Mathématiques et Applications, 2017, Rennes 1

URL: http://www.theses.fr/2017REN1S062

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Cette thèse porte sur le comportement en temps long de solutions d’équations de type Vlasov, principalement le modèle Vlasov-HMF. On s’intéresse en particulier au phénomène… (more)

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Cette thèse porte sur le comportement en temps long de solutions d’équations de type Vlasov, principalement le modèle Vlasov-HMF. On s’intéresse en particulier au phénomène d’amortissement Landau, prouvé mathématiquement dans divers cadres, pour plusieurs équations de type Vlasov, comme l’équation de Vlasov-Poisson ou le modèle Vlasov-HMF, et présentant certaines analogies avec le phénomène d’amortissement non visqueux pour l’équation d’Euler 2D. Les résultats qui y sont décrits sont les suivants. Le premier est un théorème d’amortissement Landau pour des solutions numériques du modèle Vlasov-HMF, obtenues par discrétisation en temps de ce dernier via des méthodes de splitting. Nous prouvons en outre la convergence des schémas numériques. Le second est un théorème d’amortissment Landau pour des solutions du modéle Vlasov-HMF linéarisé autour d’états stationnaires inhomogènes. Ce théorème est accompagné de nombreuses simulations numériques destinées à étudier numériquement le cas non-linéaire, et semblant mettre en lumière de nouveaux phénomènes. Enfin, le dernier résultat porte sur la discrétisation en temps de l’équation d’Euler 2D par un intégrateur de Crouch-Grossman symplectique. Nous prouvons la convergence du schéma.

This thesis concerns the long time behavior of certain Vlasov equations, mainly the Vlasov- HMF model. We are in particular interested in the celebrated phenomenon of Landau damp- ing, proved mathematically in various frameworks, foar several Vlasov equations, such as the Vlasov-Poisson equation or the Vlasov-HMF model, and exhibiting certain analogies with the inviscid damping phenomenon for the 2D Euler equation. The results described in the document are the following.The first one is a Landau damping theorem for numerical solutions of the Vlasov-HMF model, constructed by means of time-discretizations by splitting methods. We prove more- over the convergence of the schemes. The second result is a Landau damping theorem for solutions of the Vlasov-HMF model linearized around inhomogeneous stationary states. We provide moreover a quite large amount of numerical simulations, which are designed to study numerically the nonlinear case, and which seem to show new phenomenons. The last result is the convergence of a scheme that discretizes in time the 2D Euler equation by means of a symplectic Crouch-Grossmann integrator.

Advisors/Committee Members: Faou, Erwan (thesis director), Rousset, Frédéric (thesis director).

Subjects/Keywords: Équations de type Vlasov; Équations d’Euler; Équations de transport; Amortissement Landau; État stationnaire; Méthodes de splitting; Méthodes semi-Lagrangiennes; Intégrateur symplectique; Intégrateur de Crouch-Grossman; Analyse d’erreur rétrograde; Systèmes hamiltoniens; Coordonnées action-angle; Vlasov equations; Euler equations; Transport equations; Landau damping; Stationary state; Splitting methods; Semi-Lagrangian methods; Symplectic integrator; Crouch-Grossman integrator; Backward error analysis; Hamiltonian systems; Angle-action variables

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APA (6^th Edition):

Horsin, R. (2017). Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics. (Doctoral Dissertation). Rennes 1. Retrieved from http://www.theses.fr/2017REN1S062

Chicago Manual of Style (16^th Edition):

Horsin, Romain. “Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics.” 2017. Doctoral Dissertation, Rennes 1. Accessed January 16, 2021. http://www.theses.fr/2017REN1S062.

MLA Handbook (7^th Edition):

Horsin, Romain. “Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics.” 2017. Web. 16 Jan 2021.

Vancouver:

Horsin R. Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics. [Internet] [Doctoral dissertation]. Rennes 1; 2017. [cited 2021 Jan 16]. Available from: http://www.theses.fr/2017REN1S062.

Council of Science Editors:

Horsin R. Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics. [Doctoral Dissertation]. Rennes 1; 2017. Available from: http://www.theses.fr/2017REN1S062

Indian Institute of Science

14. Shrivathsa, B. Time Splitting Methods Applied To A Nonlinear Advective Equation.

Degree: MSc Engg, Faculty of Engineering, 2009, Indian Institute of Science

URL: http://etd.iisc.ac.in/handle/2005/417

► Time splitting is a numerical procedure used in solution of partial diﬀerential equations whose solutions allow multiple time scales. Numerical schemes are split for handling… (more)

▼ Time splitting is a numerical procedure used in solution of partial diﬀerential equations whose solutions allow multiple time scales. Numerical schemes are split for handling the stiﬀness in equations, i.e. when there are multiple time scales with a few time scales being smaller than the others. When there are such terms with smaller time scales, due to the Courant number restriction, the computational cost becomes high if these terms are treated explicitly. In the present work a nonlinear advective equation is solved numerically using diﬀerent techniques based on a generalised framework for splitting methods. The nonlinear advective equation was chosen because it has an analytical solution making comparisons with numerical schemes amenable and also because its nonlinearity mimics the equations encountered in atmospheric modelling. Using the nonlinear advective equation as a test bed, an analysis of the splitting methods and their inﬂuence on the split solutions has been made. An understanding of inﬂuence of splitting schemes requires knowledge of behaviour of unsplit schemes beforehand. Hence a study on unsplit methods has also been made. In the present work, using the nonlinear advective equation, it shown that the three time level schemes have high phase errors and underestimate energy (even though they have a higher order of accuracy in time). It is also found that the leap-frog method, which is used widely in atmospheric modelling, is the worst among examined unsplit methods. The semi implicit method, again a popular splitting method with atmospheric modellers is the worst among examined split methods. Three time-level schemes also need explicit ﬁltering to remove the computational mode. This ﬁltering can have a signiﬁcant impact on the obtained numerical solutions, and hence three-time level schemes appear to be unattractive in the context of the nonlinear convective equation. Based on this experience, splitting methods for the two-time level schemes is proposed. These schemes realistically capture the phase and energy of the nonlinear advective equation. Advisors/Committee Members: Nanjundiah, R S (advisor).

Subjects/Keywords: Nonlinear Differential Equations; MultipleTime Scales; Atmosphere (Geophysics); Meterology - Time Splitting Methods; Atmospheric Modellling; Nonlinear Advective Equation - Splitting Methods; Split Schemes; Unsplit Schemes; Geophysics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Shrivathsa, B. (2009). Time Splitting Methods Applied To A Nonlinear Advective Equation. (Masters Thesis). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/417

Chicago Manual of Style (16^th Edition):

Shrivathsa, B. “Time Splitting Methods Applied To A Nonlinear Advective Equation.” 2009. Masters Thesis, Indian Institute of Science. Accessed January 16, 2021. http://etd.iisc.ac.in/handle/2005/417.

MLA Handbook (7^th Edition):

Shrivathsa, B. “Time Splitting Methods Applied To A Nonlinear Advective Equation.” 2009. Web. 16 Jan 2021.

Vancouver:

Shrivathsa B. Time Splitting Methods Applied To A Nonlinear Advective Equation. [Internet] [Masters thesis]. Indian Institute of Science; 2009. [cited 2021 Jan 16]. Available from: http://etd.iisc.ac.in/handle/2005/417.

Council of Science Editors:

Shrivathsa B. Time Splitting Methods Applied To A Nonlinear Advective Equation. [Masters Thesis]. Indian Institute of Science; 2009. Available from: http://etd.iisc.ac.in/handle/2005/417

15. Garrigos, Guillaume. Descent dynamical systems and algorithms for tame optimization, and multi-objective problems : Systèmes dynamiques de descente et algorithmes pour l'optimisation modérée, et les problèmes multi-objectif.

Degree: Docteur es, Mathématiques et modélisation, 2015, Montpellier; Universidad técnica Federico Santa María (Valparaiso, Chili)

URL: http://www.theses.fr/2015MONTS191

►

Dans une première partie, nous nous intéressons aux systèmes dynamiques gradients gouvernés par des fonctions non lisses, mais aussi non convexes, satisfaisant l'inégalité de Kurdyka-Lojasiewicz.… (more)

▼

Dans une première partie, nous nous intéressons aux systèmes dynamiques gradients gouvernés par des fonctions non lisses, mais aussi non convexes, satisfaisant l'inégalité de Kurdyka-Lojasiewicz. Après avoir obtenu quelques résultats préliminaires pour la dynamique de la plus grande pente continue, nous étudions un algorithme de descente général. Nous prouvons, sous une hypothèse de compacité, que tout suite générée par ce schéma général converge vers un point critique de la fonction. Nous obtenons aussi de nouveaux résultats sur la vitesse de convergence, tant pour les valeurs que pour les itérés. Ce schéma général couvre en particulier des versions parallélisées de la méthode forward-backward, autorisant une métrique variable et des erreurs relatives. Cela nous permet par exemple de proposer une version non convexe non lisse de l'algorithme Levenberg-Marquardt. Enfin, nous proposons quelques applications de ces algorithmes aux problèmes de faisabilité, et aux problèmes inverses. Dans une seconde partie, cette thèse développe une dynamique de descente associée à des problèmes d'optimisation vectoriels sous contrainte. Pour cela, nous adaptons la dynamique de la plus grande pente usuelle aux fonctions à valeurs dans un espace ordonné par un cône convexe fermé solide. Cette dynamique peut être vue comme l'analogue continu de nombreux algorithmes développés ces dernières années. Nous avons un intérêt particulier pour les problèmes de décision multi-objectifs, pour lesquels cette dynamique de descente fait décroitre toutes les fonctions objectif au cours du temps. Nous prouvons l'existence de trajectoires pour cette dynamique continue, ainsi que leur convergence vers des points faiblement efficients. Finalement, nous explorons une nouvelle dynamique inertielle pour les problèmes multi-objectif, avec l'ambition de développer des méthodes rapides convergeant vers des équilibres de Pareto.

In a first part, we focus on gradient dynamical systems governed by non-smooth but also non-convex functions, satisfying the so-called Kurdyka-Lojasiewicz inequality.After obtaining preliminary results for a continuous steepest descent dynamic, we study a general descent algorithm. We prove, under a compactness assumption, that any sequence generated by this general scheme converges to a critical point of the function.We also obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward-backward method, with variable metric and relative errors. As an example, a non-smooth and non-convex version of the Levenberg-Marquardt algorithm is detailed.Applications to non-convex feasibility problems, and to sparse inverse problems are discussed.In a second part, the thesis explores descent dynamics associated to constrained vector optimization problems. For this, we adapt the classic steepest descent dynamic to functions with values in a vector space ordered by a solid closed convex cone. It can be seen as the continuous analogue of various descent algorithms developed in the last…

Advisors/Committee Members: Attouch, Hedy (thesis director), Peypouquet, Juan (thesis director).

Subjects/Keywords: Optimisation; Optimisation non-Lisse non-Convexe; Méthodes de descente; Optimisation multi-Objectif; Méthodes d'éclatement; Système dynamique continus; Optimization; Nonsmooth nonconvex optimization; Descent methods; Multi-Objective optimization; Splitting methods; Continuous dynamical systems

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Garrigos, G. (2015). Descent dynamical systems and algorithms for tame optimization, and multi-objective problems : Systèmes dynamiques de descente et algorithmes pour l'optimisation modérée, et les problèmes multi-objectif. (Doctoral Dissertation). Montpellier; Universidad técnica Federico Santa María (Valparaiso, Chili). Retrieved from http://www.theses.fr/2015MONTS191

Chicago Manual of Style (16^th Edition):

Garrigos, Guillaume. “Descent dynamical systems and algorithms for tame optimization, and multi-objective problems : Systèmes dynamiques de descente et algorithmes pour l'optimisation modérée, et les problèmes multi-objectif.” 2015. Doctoral Dissertation, Montpellier; Universidad técnica Federico Santa María (Valparaiso, Chili). Accessed January 16, 2021. http://www.theses.fr/2015MONTS191.

MLA Handbook (7^th Edition):

Garrigos, Guillaume. “Descent dynamical systems and algorithms for tame optimization, and multi-objective problems : Systèmes dynamiques de descente et algorithmes pour l'optimisation modérée, et les problèmes multi-objectif.” 2015. Web. 16 Jan 2021.

Vancouver:

Garrigos G. Descent dynamical systems and algorithms for tame optimization, and multi-objective problems : Systèmes dynamiques de descente et algorithmes pour l'optimisation modérée, et les problèmes multi-objectif. [Internet] [Doctoral dissertation]. Montpellier; Universidad técnica Federico Santa María (Valparaiso, Chili); 2015. [cited 2021 Jan 16]. Available from: http://www.theses.fr/2015MONTS191.

Council of Science Editors:

Garrigos G. Descent dynamical systems and algorithms for tame optimization, and multi-objective problems : Systèmes dynamiques de descente et algorithmes pour l'optimisation modérée, et les problèmes multi-objectif. [Doctoral Dissertation]. Montpellier; Universidad técnica Federico Santa María (Valparaiso, Chili); 2015. Available from: http://www.theses.fr/2015MONTS191

16. Calatroni, Luca. New PDE models for imaging problems and applications.

Degree: PhD, 2016, University of Cambridge

URL: https://www.repository.cam.ac.uk/handle/1810/256139https://www.repository.cam.ac.uk/bitstream/1810/256139/4/Calatroni-2015-PhD.pdf.txt ; https://www.repository.cam.ac.uk/bitstream/1810/256139/5/Calatroni-2015-PhD.pdf.jpg

► Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as… (more)

▼ Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed.

Subjects/Keywords: Research Subject Categories::MATHEMATICS::Algebra, geometry and mathematical analysis; total variation; higher-order PDEs; directional splitting; quasi-Newton methods; image denoising; image inpainting; mixed noise distribution; bilevel optimisation; parameter learning; SemiSmooth Newton methods; image segmentation; graph clustering; matrix completion; Hough transform

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Calatroni, L. (2016). New PDE models for imaging problems and applications. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/256139https://www.repository.cam.ac.uk/bitstream/1810/256139/4/Calatroni-2015-PhD.pdf.txt ; https://www.repository.cam.ac.uk/bitstream/1810/256139/5/Calatroni-2015-PhD.pdf.jpg

Chicago Manual of Style (16^th Edition):

Calatroni, Luca. “New PDE models for imaging problems and applications.” 2016. Doctoral Dissertation, University of Cambridge. Accessed January 16, 2021. https://www.repository.cam.ac.uk/handle/1810/256139https://www.repository.cam.ac.uk/bitstream/1810/256139/4/Calatroni-2015-PhD.pdf.txt ; https://www.repository.cam.ac.uk/bitstream/1810/256139/5/Calatroni-2015-PhD.pdf.jpg.

MLA Handbook (7^th Edition):

Calatroni, Luca. “New PDE models for imaging problems and applications.” 2016. Web. 16 Jan 2021.

Vancouver:

Calatroni L. New PDE models for imaging problems and applications. [Internet] [Doctoral dissertation]. University of Cambridge; 2016. [cited 2021 Jan 16]. Available from: https://www.repository.cam.ac.uk/handle/1810/256139https://www.repository.cam.ac.uk/bitstream/1810/256139/4/Calatroni-2015-PhD.pdf.txt ; https://www.repository.cam.ac.uk/bitstream/1810/256139/5/Calatroni-2015-PhD.pdf.jpg.

Council of Science Editors:

Calatroni L. New PDE models for imaging problems and applications. [Doctoral Dissertation]. University of Cambridge; 2016. Available from: https://www.repository.cam.ac.uk/handle/1810/256139https://www.repository.cam.ac.uk/bitstream/1810/256139/4/Calatroni-2015-PhD.pdf.txt ; https://www.repository.cam.ac.uk/bitstream/1810/256139/5/Calatroni-2015-PhD.pdf.jpg

17. YIN JIA. MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC/NONLINEAR DIRAC EQUATION.

Degree: 2019, National University of Singapore

URL: https://scholarbank.nus.edu.sg/handle/10635/162434

Subjects/Keywords: Dirac equation; super-resolution; time-splitting methods; nonrelativistic regime; semiclassical regime; finite difference methods

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

JIA, Y. (2019). MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC/NONLINEAR DIRAC EQUATION. (Thesis). National University of Singapore. Retrieved from https://scholarbank.nus.edu.sg/handle/10635/162434

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

JIA, YIN. “MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC/NONLINEAR DIRAC EQUATION.” 2019. Thesis, National University of Singapore. Accessed January 16, 2021. https://scholarbank.nus.edu.sg/handle/10635/162434.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

JIA, YIN. “MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC/NONLINEAR DIRAC EQUATION.” 2019. Web. 16 Jan 2021.

Vancouver:

JIA Y. MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC/NONLINEAR DIRAC EQUATION. [Internet] [Thesis]. National University of Singapore; 2019. [cited 2021 Jan 16]. Available from: https://scholarbank.nus.edu.sg/handle/10635/162434.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

JIA Y. MULTISCALE METHODS AND ANALYSIS FOR THE DIRAC/NONLINEAR DIRAC EQUATION. [Thesis]. National University of Singapore; 2019. Available from: https://scholarbank.nus.edu.sg/handle/10635/162434

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

University of Cambridge

18. Calatroni, Luca. New PDE models for imaging problems and applications.

Degree: PhD, 2016, University of Cambridge

URL: https://doi.org/10.17863/CAM.79 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.693417

► Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as… (more)

▼ Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed.

Subjects/Keywords: 515; Algebra; geometry and mathematical analysis; total variation; higher-order PDEs; directional splitting; quasi-Newton methods; image denoising; image inpainting; mixed noise distribution; bilevel optimisation; parameter learning; SemiSmooth Newton methods; image segmentation; graph clustering; matrix completion; Hough transform

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Calatroni, L. (2016). New PDE models for imaging problems and applications. (Doctoral Dissertation). University of Cambridge. Retrieved from https://doi.org/10.17863/CAM.79 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.693417

Chicago Manual of Style (16^th Edition):

Calatroni, Luca. “New PDE models for imaging problems and applications.” 2016. Doctoral Dissertation, University of Cambridge. Accessed January 16, 2021. https://doi.org/10.17863/CAM.79 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.693417.

MLA Handbook (7^th Edition):

Calatroni, Luca. “New PDE models for imaging problems and applications.” 2016. Web. 16 Jan 2021.

Vancouver:

Calatroni L. New PDE models for imaging problems and applications. [Internet] [Doctoral dissertation]. University of Cambridge; 2016. [cited 2021 Jan 16]. Available from: https://doi.org/10.17863/CAM.79 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.693417.

Council of Science Editors:

Calatroni L. New PDE models for imaging problems and applications. [Doctoral Dissertation]. University of Cambridge; 2016. Available from: https://doi.org/10.17863/CAM.79 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.693417

19. Mahadevan, Vijay Subramaniam. High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis.

Degree: PhD, Nuclear Engineering, 2011, Texas A&M University

URL: http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8579

► The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a… (more)

▼ The modeling of nuclear reactors involves the solution of a multi-physics problem with widely varying time and length scales. This translates mathematically to solving a system of coupled, non-linear, and stiff partial differential equations (PDEs). Multi-physics applications possess the added complexity that most of the solution fields participate in various physics components, potentially yielding spatial and/or temporal coupling errors. This dissertation deals with the verification aspects associated with such a multi-physics code, i.e., the substantiation that the mathematical description of the multi-physics equations are solved correctly (both in time and space). Conventional paradigms used in reactor analysis problems employed to couple various physics components are often non-iterative and can be inconsistent in their treatment of the non-linear terms. This leads to the usage of smaller time steps to maintain stability and accuracy requirements, thereby increasing the overall computational time for simulation. The inconsistencies of these weakly coupled solution methods can be overcome using tighter coupling strategies and yield a better approximation to the coupled non-linear operator, by resolving the dominant spatial and temporal scales involved in the multi-physics simulation. A multi-physics framework, KARMA (K(c)ode for Analysis of Reactor and other Multi-physics Applications), is presented. KARMA uses tight coupling strategies for various physical models based on a Matrix-free Nonlinear-Krylov (MFNK) framework in order to attain high-order spatio-temporal accuracy for all solution fields in amenable wall clock times, for various test problems. The framework also utilizes traditional loosely coupled methods as lower-order solvers, which serve as efficient preconditioners for the tightly coupled solution. Since the software platform employs both lower and higher-order coupling strategies, it can easily be used to test and evaluate different coupling strategies and numerical methods and to compare their efficiency for problems of interest. Multi-physics code verification efforts pertaining to reactor applications are described and associated numerical results obtained using the developed multi-physics framework are provided. The versatility of numerical methods used here for coupled problems and feasibility of general non-linear solvers with appropriate physics-based preconditioners in the KARMA framework offer significantly efficient techniques to solve multi-physics problems in reactor analysis. Advisors/Committee Members: Ragusa, Jean C. (advisor), Adams, Marvin L. (committee member), Morel, Jim E. (committee member), Guermond, Jean L. (committee member).

Subjects/Keywords: Multi-physics; Operator splitting; Coupling methods; Jacobian-Free; Newton iteration; Picard iteration; Reactor analysis

…69 3.4 4.1 4.2 4.3 3.3.1 Nonlinear Iteration Methods . . . . . . . . . 3.3.2 Krylov… …Methods for Solving Linear Systems 3.3.3 Preconditioners for the Linear Iteration . . . 3.3.4… …structural deformation and vice versa [8] [9]. Solution methods for non… …combining consistent numerical methods with principles in software engineering to create a coupled… …development and usage of existing analysis and numerical methods for creating a unified and verified…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Mahadevan, V. S. (2011). High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8579

Chicago Manual of Style (16^th Edition):

Mahadevan, Vijay Subramaniam. “High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis.” 2011. Doctoral Dissertation, Texas A&M University. Accessed January 16, 2021. http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8579.

MLA Handbook (7^th Edition):

Mahadevan, Vijay Subramaniam. “High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis.” 2011. Web. 16 Jan 2021.

Vancouver:

Mahadevan VS. High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis. [Internet] [Doctoral dissertation]. Texas A&M University; 2011. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8579.

Council of Science Editors:

Mahadevan VS. High Resolution Numerical Methods for Coupled Non-linear Multi-physics Simulations with Applications in Reactor Analysis. [Doctoral Dissertation]. Texas A&M University; 2011. Available from: http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8579

20. Jegourel, Cyrille. Rare event simulation for statistical model checking : Simulation d'événements rares pour le model checking statistique.

Degree: Docteur es, Informatique, 2014, Rennes 1

URL: http://www.theses.fr/2014REN1S084

►

Dans cette thèse, nous considérons deux problèmes auxquels le model checking statistique doit faire face. Le premier concerne les systèmes hétérogènes qui introduisent complexité et… (more)

▼

Dans cette thèse, nous considérons deux problèmes auxquels le model checking statistique doit faire face. Le premier concerne les systèmes hétérogènes qui introduisent complexité et non-déterminisme dans l'analyse. Le second problème est celui des propriétés rares, difficiles à observer et donc à quantifier. Pour le premier point, nous présentons des contributions originales pour le formalisme des systèmes composites dans le langage BIP. Nous en proposons une extension stochastique, SBIP, qui permet le recours à l'abstraction stochastique de composants et d'éliminer le non-déterminisme. Ce double effet a pour avantage de réduire la taille du système initial en le remplaçant par un système dont la sémantique est purement stochastique sur lequel les algorithmes de model checking statistique sont définis. La deuxième partie de cette thèse est consacrée à la vérification de propriétés rares. Nous avons proposé le recours à un algorithme original d'échantillonnage préférentiel pour les modèles dont le comportement est décrit à travers un ensemble de commandes. Nous avons également introduit les méthodes multi-niveaux pour la vérification de propriétés rares et nous avons justifié et mis en place l'utilisation d'un algorithme multi-niveau optimal. Ces deux méthodes poursuivent le même objectif de réduire la variance de l'estimateur et le nombre de simulations. Néanmoins, elles sont fondamentalement différentes, la première attaquant le problème au travers du modèle et la seconde au travers des propriétés.

In this thesis, we consider two problems that statistical model checking must cope. The first problem concerns heterogeneous systems, that naturally introduce complexity and non-determinism into the analysis. The second problem concerns rare properties, difficult to observe, and so to quantify. About the first point, we present original contributions for the formalism of composite systems in BIP language. We propose SBIP, a stochastic extension and define its semantics. SBIP allows the recourse to the stochastic abstraction of components and eliminate the non-determinism. This double effect has the advantage of reducing the size of the initial system by replacing it by a system whose semantics is purely stochastic, a necessary requirement for standard statistical model checking algorithms to be applicable. The second part of this thesis is devoted to the verification of rare properties in statistical model checking. We present a state-of-the-art algorithm for models described by a set of guarded commands. Lastly, we motivate the use of importance splitting for statistical model checking and set up an optimal splitting algorithm. Both methods pursue a common goal to reduce the variance of the estimator and the number of simulations. Nevertheless, they are fundamentally different, the first tackling the problem through the model and the second through the properties.

Advisors/Committee Members: Legay, Axel (thesis director).

Subjects/Keywords: Ingénierie des systèmes; Vérification formelle; Model Checking statistique; Méthodes de Monte-Carlo; Simulation d'événements rares; Échantillonnage statistique; Méthodes de Monte-Carlo multi-Niveaux; System Engineering; Formal Verification; Statistical Model Checking; Monte-Carlo methods; Rare Event Simulation; Important Sampling; Important Splitting

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Jegourel, C. (2014). Rare event simulation for statistical model checking : Simulation d'événements rares pour le model checking statistique. (Doctoral Dissertation). Rennes 1. Retrieved from http://www.theses.fr/2014REN1S084

Chicago Manual of Style (16^th Edition):

Jegourel, Cyrille. “Rare event simulation for statistical model checking : Simulation d'événements rares pour le model checking statistique.” 2014. Doctoral Dissertation, Rennes 1. Accessed January 16, 2021. http://www.theses.fr/2014REN1S084.

MLA Handbook (7^th Edition):

Jegourel, Cyrille. “Rare event simulation for statistical model checking : Simulation d'événements rares pour le model checking statistique.” 2014. Web. 16 Jan 2021.

Vancouver:

Jegourel C. Rare event simulation for statistical model checking : Simulation d'événements rares pour le model checking statistique. [Internet] [Doctoral dissertation]. Rennes 1; 2014. [cited 2021 Jan 16]. Available from: http://www.theses.fr/2014REN1S084.

Council of Science Editors:

Jegourel C. Rare event simulation for statistical model checking : Simulation d'événements rares pour le model checking statistique. [Doctoral Dissertation]. Rennes 1; 2014. Available from: http://www.theses.fr/2014REN1S084

University of Cambridge

21. Singh, Pranav. High accuracy computational methods for the semiclassical Schrödinger equation.

Degree: PhD, 2018, University of Cambridge

URL: https://doi.org/10.17863/CAM.22064 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691

► The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for… (more)

▼ The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy{-}we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the ℤ₂-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of it{asymptotic splitting:} exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus – Zassenhaus schemes{-}one where the integrals are discretised using Gauss – Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus – Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

Subjects/Keywords: 515; Semiclassical Schro¨dinger equations; time-dependent potentials; exponential splittings; Zassenhaus splitting; Magnus expansions; Lanczos iterations; Magnus – Zassenhaus schemes; commutator free; high-order methods; asymptotic analysis; Lie algebras; Jordan polynomials; symmetrised differential operators; spectral collocation

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Singh, P. (2018). High accuracy computational methods for the semiclassical Schrödinger equation. (Doctoral Dissertation). University of Cambridge. Retrieved from https://doi.org/10.17863/CAM.22064 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691

Chicago Manual of Style (16^th Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Doctoral Dissertation, University of Cambridge. Accessed January 16, 2021. https://doi.org/10.17863/CAM.22064 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691.

MLA Handbook (7^th Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Web. 16 Jan 2021.

Vancouver:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Internet] [Doctoral dissertation]. University of Cambridge; 2018. [cited 2021 Jan 16]. Available from: https://doi.org/10.17863/CAM.22064 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691.

Council of Science Editors:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Doctoral Dissertation]. University of Cambridge; 2018. Available from: https://doi.org/10.17863/CAM.22064 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.744691

University of Cambridge

22. Jackson, Haran. A unified framework for simulating impact-induced detonation of a combustible material in an elasto-plastic confiner.

Degree: PhD, 2020, University of Cambridge

URL: https://doi.org/10.17863/CAM.52131 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.805879

► A new framework for the computational simulation of problems arising in continuum me- chanics is presented. It is unified in the sense that it can… (more)

▼ A new framework for the computational simulation of problems arising in continuum me- chanics is presented. It is unified in the sense that it can describe all three major phases of matter within the same set of equations. It is able to represent inviscid fluids, Newtonian and non-Newtonian viscous fluids, elastic and plastic solids, and reactive species. These materials are presented with a variety of equations of state, and there is a clear methodology for extending the framework to more exotic materials using other constitutive equations. It is capable of accurately modeling interfaces between regions occupied by different phases, and by the vacuum. The problem of impact-induced detonation in an elastoplastic confiner is one that incorpo- rates the whole range of aforementioned material types, representing a challenge to existing frameworks. This new framework is shown to accurately and efficiently solve this problem. The framework comprises a modification and extension of the Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, along with a new set of operator-splitting-based numerical solvers to allow for the efficient solution of the problems that it is put to, and a new Riemann ghost fluid method for accurate simulation of material interfaces. In addition to this work, novel mathematical analyses of the structure of the GPR equations - and the numerical methods currently used to solve them - are presented in this study. This new framework presents a range of benefits: the conceptual work required to implement a computational simulation involving many different components is greatly reduced, saving time and allowing for greater specialization of computational techniques. This has the po- tential to streamline development of simulation software by reducing the number of different systems of equations that require solvers, and cutting down on the amount of theoretical work required, for example in the treatment of interfaces in multimaterial problems.

Subjects/Keywords: Godunov-Peshkov-Romenski; GPR; CFD; multiphysics; multiphase; multimaterial; solid mechanics; ADER; WENO; finite volume; discontinuous galerkin; continuous galerkin; ghost fluid methods; detonation; elastoplastic; operator splitting; non-newtonian fluids; hyperbolic PDEs; unified framework; Eulerian

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Jackson, H. (2020). A unified framework for simulating impact-induced detonation of a combustible material in an elasto-plastic confiner. (Doctoral Dissertation). University of Cambridge. Retrieved from https://doi.org/10.17863/CAM.52131 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.805879

Chicago Manual of Style (16^th Edition):

Jackson, Haran. “A unified framework for simulating impact-induced detonation of a combustible material in an elasto-plastic confiner.” 2020. Doctoral Dissertation, University of Cambridge. Accessed January 16, 2021. https://doi.org/10.17863/CAM.52131 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.805879.

MLA Handbook (7^th Edition):

Jackson, Haran. “A unified framework for simulating impact-induced detonation of a combustible material in an elasto-plastic confiner.” 2020. Web. 16 Jan 2021.

Vancouver:

Jackson H. A unified framework for simulating impact-induced detonation of a combustible material in an elasto-plastic confiner. [Internet] [Doctoral dissertation]. University of Cambridge; 2020. [cited 2021 Jan 16]. Available from: https://doi.org/10.17863/CAM.52131 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.805879.

Council of Science Editors:

Jackson H. A unified framework for simulating impact-induced detonation of a combustible material in an elasto-plastic confiner. [Doctoral Dissertation]. University of Cambridge; 2020. Available from: https://doi.org/10.17863/CAM.52131 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.805879

23. Singh, Pranav. High accuracy computational methods for the semiclassical Schrödinger equation.

Degree: PhD, 2018, University of Cambridge

URL: https://www.repository.cam.ac.uk/handle/1810/274913

► The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for… (more)

▼ The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy{-}we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the ℤ₂-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of it{asymptotic splitting:} exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus – Zassenhaus schemes{-}one where the integrals are discretised using Gauss – Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus – Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

Subjects/Keywords: Semiclassical Schrödinger equations; time-dependent potentials; exponential splittings; Zassenhaus splitting; Magnus expansions; Lanczos iterations; Magnus – Zassenhaus schemes; commutator free; high-order methods; asymptotic analysis; Lie algebras; Jordan polynomials; symmetrised differential operators; spectral collocation

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Singh, P. (2018). High accuracy computational methods for the semiclassical Schrödinger equation. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/274913

Chicago Manual of Style (16^th Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Doctoral Dissertation, University of Cambridge. Accessed January 16, 2021. https://www.repository.cam.ac.uk/handle/1810/274913.

MLA Handbook (7^th Edition):

Singh, Pranav. “High accuracy computational methods for the semiclassical Schrödinger equation.” 2018. Web. 16 Jan 2021.

Vancouver:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Internet] [Doctoral dissertation]. University of Cambridge; 2018. [cited 2021 Jan 16]. Available from: https://www.repository.cam.ac.uk/handle/1810/274913.

Council of Science Editors:

Singh P. High accuracy computational methods for the semiclassical Schrödinger equation. [Doctoral Dissertation]. University of Cambridge; 2018. Available from: https://www.repository.cam.ac.uk/handle/1810/274913

24. Uhliarik, Marek. Operator Splitting Methods and Artificial Boundary Conditions for a nonlinear Black-Scholes equation.

Degree: MPE-lab, 2010, Halmstad UniversityHalmstad University

URL: http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-6111

► There are some nonlinear models for pricing financial derivatives which can improve the linear Black-Scholes model introduced by Black, Scholes and Merton. In these… (more)

Subjects/Keywords: finacial Mathematics; nonlinear Black-Scholes equation; volatility models; splitting methods; MATHEMATICS; MATEMATIK; Numerical analysis; Numerisk analys

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Uhliarik, M. (2010). Operator Splitting Methods and Artificial Boundary Conditions for a nonlinear Black-Scholes equation. (Thesis). Halmstad UniversityHalmstad University. Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-6111

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Uhliarik, Marek. “Operator Splitting Methods and Artificial Boundary Conditions for a nonlinear Black-Scholes equation.” 2010. Thesis, Halmstad UniversityHalmstad University. Accessed January 16, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-6111.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Uhliarik, Marek. “Operator Splitting Methods and Artificial Boundary Conditions for a nonlinear Black-Scholes equation.” 2010. Web. 16 Jan 2021.

Vancouver:

Uhliarik M. Operator Splitting Methods and Artificial Boundary Conditions for a nonlinear Black-Scholes equation. [Internet] [Thesis]. Halmstad UniversityHalmstad University; 2010. [cited 2021 Jan 16]. Available from: http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-6111.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Uhliarik M. Operator Splitting Methods and Artificial Boundary Conditions for a nonlinear Black-Scholes equation. [Thesis]. Halmstad UniversityHalmstad University; 2010. Available from: http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-6111

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

University of Cambridge

25. Jackson, Haran. A Unified Framework for Simulating Impact-Induced Detonation of a Combustible Material in an Elasto-Plastic Confiner.

Degree: PhD, 2020, University of Cambridge

URL: https://www.repository.cam.ac.uk/handle/1810/305047

► A new framework for the computational simulation of problems arising in continuum me- chanics is presented. It is unified in the sense that it can… (more)

▼ A new framework for the computational simulation of problems arising in continuum me- chanics is presented. It is unified in the sense that it can describe all three major phases of matter within the same set of equations. It is able to represent inviscid fluids, Newtonian and non-Newtonian viscous fluids, elastic and plastic solids, and reactive species. These materials are presented with a variety of equations of state, and there is a clear methodology for extending the framework to more exotic materials using other constitutive equations. It is capable of accurately modeling interfaces between regions occupied by different phases, and by the vacuum. The problem of impact-induced detonation in an elastoplastic confiner is one that incorpo- rates the whole range of aforementioned material types, representing a challenge to existing frameworks. This new framework is shown to accurately and efficiently solve this problem. The framework comprises a modification and extension of the Godunov-Peshkov-Romenski (GPR) model of continuum mechanics, along with a new set of operator-splitting-based numerical solvers to allow for the efficient solution of the problems that it is put to, and a new Riemann ghost fluid method for accurate simulation of material interfaces. In addition to this work, novel mathematical analyses of the structure of the GPR equations - and the numerical methods currently used to solve them - are presented in this study. This new framework presents a range of benefits: the conceptual work required to implement a computational simulation involving many different components is greatly reduced, saving time and allowing for greater specialization of computational techniques. This has the po- tential to streamline development of simulation software by reducing the number of different systems of equations that require solvers, and cutting down on the amount of theoretical work required, for example in the treatment of interfaces in multimaterial problems.

Subjects/Keywords: Godunov-Peshkov-Romenski; GPR; CFD; multiphysics; multiphase; multimaterial; solid mechanics; ADER; WENO; finite volume; discontinuous galerkin; continuous galerkin; ghost fluid methods; detonation; elastoplastic; operator splitting; non-newtonian fluids; hyperbolic PDEs; unified framework; Eulerian

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Jackson, H. (2020). A Unified Framework for Simulating Impact-Induced Detonation of a Combustible Material in an Elasto-Plastic Confiner. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/305047

Chicago Manual of Style (16^th Edition):

Jackson, Haran. “A Unified Framework for Simulating Impact-Induced Detonation of a Combustible Material in an Elasto-Plastic Confiner.” 2020. Doctoral Dissertation, University of Cambridge. Accessed January 16, 2021. https://www.repository.cam.ac.uk/handle/1810/305047.

MLA Handbook (7^th Edition):

Jackson, Haran. “A Unified Framework for Simulating Impact-Induced Detonation of a Combustible Material in an Elasto-Plastic Confiner.” 2020. Web. 16 Jan 2021.

Vancouver:

Jackson H. A Unified Framework for Simulating Impact-Induced Detonation of a Combustible Material in an Elasto-Plastic Confiner. [Internet] [Doctoral dissertation]. University of Cambridge; 2020. [cited 2021 Jan 16]. Available from: https://www.repository.cam.ac.uk/handle/1810/305047.

Council of Science Editors:

Jackson H. A Unified Framework for Simulating Impact-Induced Detonation of a Combustible Material in an Elasto-Plastic Confiner. [Doctoral Dissertation]. University of Cambridge; 2020. Available from: https://www.repository.cam.ac.uk/handle/1810/305047

26. XU WEIBIAO. Analysis and Computation for Coupling Bose-Einstein Condensates in Optical Resonators.

Degree: 2010, National University of Singapore

URL: http://scholarbank.nus.edu.sg/handle/10635/20438

Subjects/Keywords: Coupling Bose-Einstein condensate; optical resonators; gradient flow with discrete normalization; time-splitting Sine spectral methods

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

WEIBIAO, X. (2010). Analysis and Computation for Coupling Bose-Einstein Condensates in Optical Resonators. (Thesis). National University of Singapore. Retrieved from http://scholarbank.nus.edu.sg/handle/10635/20438

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

WEIBIAO, XU. “Analysis and Computation for Coupling Bose-Einstein Condensates in Optical Resonators.” 2010. Thesis, National University of Singapore. Accessed January 16, 2021. http://scholarbank.nus.edu.sg/handle/10635/20438.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

WEIBIAO, XU. “Analysis and Computation for Coupling Bose-Einstein Condensates in Optical Resonators.” 2010. Web. 16 Jan 2021.

Vancouver:

WEIBIAO X. Analysis and Computation for Coupling Bose-Einstein Condensates in Optical Resonators. [Internet] [Thesis]. National University of Singapore; 2010. [cited 2021 Jan 16]. Available from: http://scholarbank.nus.edu.sg/handle/10635/20438.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

WEIBIAO X. Analysis and Computation for Coupling Bose-Einstein Condensates in Optical Resonators. [Thesis]. National University of Singapore; 2010. Available from: http://scholarbank.nus.edu.sg/handle/10635/20438

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

University of Notre Dame

27. Jianfeng Zhu. Application of discontinuous Galerkin finite element methods for vertebrate limb pattern formation</h1>.

Degree: Mathematics, 2010, University of Notre Dame

URL: https://curate.nd.edu/show/h415p843g3v

► Major outstanding questions regarding vertebrate limb development are how the numbers of skeletal elements along the proximodistal (P-D) and anteroposterior (A-P) axes are determined… (more)

Subjects/Keywords: moving domain; triangular meshes; complex geometry; operator splitting; reaction-diffusion equations; limb development; discontinuous Galerkin finite element methods

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Zhu, J. (2010). Application of discontinuous Galerkin finite element methods for vertebrate limb pattern formation</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/h415p843g3v

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Zhu, Jianfeng. “Application of discontinuous Galerkin finite element methods for vertebrate limb pattern formation</h1>.” 2010. Thesis, University of Notre Dame. Accessed January 16, 2021. https://curate.nd.edu/show/h415p843g3v.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Zhu, Jianfeng. “Application of discontinuous Galerkin finite element methods for vertebrate limb pattern formation</h1>.” 2010. Web. 16 Jan 2021.

Vancouver:

Zhu J. Application of discontinuous Galerkin finite element methods for vertebrate limb pattern formation</h1>. [Internet] [Thesis]. University of Notre Dame; 2010. [cited 2021 Jan 16]. Available from: https://curate.nd.edu/show/h415p843g3v.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Zhu J. Application of discontinuous Galerkin finite element methods for vertebrate limb pattern formation</h1>. [Thesis]. University of Notre Dame; 2010. Available from: https://curate.nd.edu/show/h415p843g3v

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

28. Preuss, Adam. A Study of 2-Additive Splitting for Solving Advection-Diffusion-Reaction Equations.

Degree: 2013, University of Saskatchewan

URL: http://hdl.handle.net/10388/ETD-2013-12-1358

► An initial-value problem consists of an ordinary differential equation subject to an initial condition. The right-hand side of the differential equation can be interpreted as… (more)

▼ An initial-value problem consists of an ordinary differential equation subject to an initial condition. The right-hand side of the differential equation can be interpreted as additively split when it is comprised of the sum of two or more contributing factors. For instance, the right-hand sides of initial-value problems derived from advection-diffusion-reaction equations are comprised of the sum of terms emanating from three distinct physical processes: advection, diffusion, and reaction. In some cases, solutions to initial-value problems can be calculated analytically, but when an analytic solution is unknown or nonexistent, methods of numerical integration are used to calculate solutions. The runtime performance of numerical methods is problem dependent; therefore, one must choose an appropriate numerical method to achieve favourable performance, according to characteristics of the problem. Additive methods of numerical integration apply distinct methods to the distinct contributing factors of an additively split problem. Treating the contributing factors with methods that are known to perform well on them individually has the potential to yield an additive method that outperforms single methods applied to the entire (unsplit) problem. Splittings of the right-hand side can be physics-based, i.e., based on physical characteristics of the problem, such as advection, diffusion, or reaction terms. Splittings can also be based on linearization, called Jacobian splitting in this thesis, where the linearized part of the problem is treated with one method and the rest of the problem is treated with another. A comparison of these splitting techniques is performed by applying a set of additive methods to a test suite of problems. Many common non-additive methods are also included to serve as a performance baseline. To perform this numerical study, a problem-solving environment was developed to evaluate permutations of problems, methods, and their associated parameters. The test suite is comprised of several distinct advection-diffusion-reaction equations that have been chosen to represent a wide range of common problem characteristics. When solving split problems in the test suite, it is found that additive Runge–Kutta methods of orders three, four, and five using Jacobian splitting generally outperform those same methods using physics-based splitting. These results provide evidence that Jacobian splitting is an effective approach when solving such initial-value problems in practice. Advisors/Committee Members: Spiteri, Raymond J., Stavness, Ian, Jamali, Nadeem, Sandu, Adrian.

Subjects/Keywords: Performance of Numerical Methods; Problem-Solving Environments; Splitting Strategies; Additive Runge-Kutta Methods; Advection-Diffusion-Reaction Equations

…treated with separate numerical methods [44]. Jacobian splitting for the problem… …splitting versus physics-based splitting for additive methods. Jacobian splitting has not been… …extensively used as an alternative to physics-based splitting. The suite of methods includes three 2… …methods applied with Jacobian splitting outperform those applied with physics-based splitting… …physics-based splitting. Evaluation of multiple numerical methods requires extensive…

14 more images…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Preuss, A. (2013). A Study of 2-Additive Splitting for Solving Advection-Diffusion-Reaction Equations. (Thesis). University of Saskatchewan. Retrieved from http://hdl.handle.net/10388/ETD-2013-12-1358

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Preuss, Adam. “A Study of 2-Additive Splitting for Solving Advection-Diffusion-Reaction Equations.” 2013. Thesis, University of Saskatchewan. Accessed January 16, 2021. http://hdl.handle.net/10388/ETD-2013-12-1358.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Preuss, Adam. “A Study of 2-Additive Splitting for Solving Advection-Diffusion-Reaction Equations.” 2013. Web. 16 Jan 2021.

Vancouver:

Preuss A. A Study of 2-Additive Splitting for Solving Advection-Diffusion-Reaction Equations. [Internet] [Thesis]. University of Saskatchewan; 2013. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/10388/ETD-2013-12-1358.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Preuss A. A Study of 2-Additive Splitting for Solving Advection-Diffusion-Reaction Equations. [Thesis]. University of Saskatchewan; 2013. Available from: http://hdl.handle.net/10388/ETD-2013-12-1358

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

29. Stary, Tomas. Mathematical and computational study of Markovian models of ion channels in cardiac excitation.

Degree: PhD, 2016, University of Exeter

URL: http://hdl.handle.net/10871/24166

► This thesis studies numerical methods for integrating the master equations describing Markov chain models of cardiac ion channels. Such models describe the time evolution of… (more)

▼ This thesis studies numerical methods for integrating the master equations describing Markov chain models of cardiac ion channels. Such models describe the time evolution of the probability that ion channels are in a particular state. Numerical simulations of such models are often computationally demanding because many solvers require relatively small time steps to ensure numerical stability. The aim of this project is to analyse selected Markov chains and develop more efficient and accurate solvers. We separate a Markov chain model into fast and slow time-scales based on the speed of transitions between states. Eliminating the fast transitions, we find an asymptotic reduction of zeroth-order and first-order in a small parameter describing the time-scales separation. We apply the theory to a Markov chain model of the fast sodium channel INa. We consider several variants for classifying some transitions as fast in order to find reduced systems that yield a good accuracy. However, the time step size is still restricted by numerical instabilities. We adapt the Rush-Larsen technique originally developed for gate models. Assuming that a transition matrix can be considered constant during each time step, we solve the Markov chain model analytically. The solution provides a recipe for a stable exponential solver, which we call "Matrix Rush-Larsen" (MRL). Using operator splitting we design an even more flexible "hybrid" method that combines the MRL with other solvers. The resulting improvement in stability allows a large increase in the time step size. In some models, we obtain reasonably accurate results 27 times faster using a hybrid method than with the forward Euler method, even with the maximal time step allowed by the stability constraint. Finally, we extend the cardiac simulation package BeatBox by the developed exponential solvers. We upgrade a format of "ionic" modules which describe a cardiac cell, in order to allow for a specific definition of Markov chain models. We also modify a particular integrator for ionic modules to include the MRL and the hybrid method. To test the functionality of the code, we have converted a number of cellular models into the ionic format. The documented code is available in the official BeatBox package distribution.

Subjects/Keywords: 519.2; Terms-Markov chain; ion channel; numerical methods; Rush-Larsen method; exponential time-differentiation; operator splitting; asymptotic methods

…splitting methods we split INa into three subsystems. The first contains the fast transition rates… …42 44 44 46 49 51 . . . . 55 55 55 59 61 3 Asymptotic and Numerical Methods 3.1… …Asymptotic methods . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Toy Example of… …for Markov Chains . . . . . . . 74 3.2 Numerical Integration Methods… …Explicit Methods – Forward Euler . . . . . . . . . . . . . . . 80 3.2.3 Rush-Larsen Technique…

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Stary, T. (2016). Mathematical and computational study of Markovian models of ion channels in cardiac excitation. (Doctoral Dissertation). University of Exeter. Retrieved from http://hdl.handle.net/10871/24166

Chicago Manual of Style (16^th Edition):

Stary, Tomas. “Mathematical and computational study of Markovian models of ion channels in cardiac excitation.” 2016. Doctoral Dissertation, University of Exeter. Accessed January 16, 2021. http://hdl.handle.net/10871/24166.

MLA Handbook (7^th Edition):

Stary, Tomas. “Mathematical and computational study of Markovian models of ion channels in cardiac excitation.” 2016. Web. 16 Jan 2021.

Vancouver:

Stary T. Mathematical and computational study of Markovian models of ion channels in cardiac excitation. [Internet] [Doctoral dissertation]. University of Exeter; 2016. [cited 2021 Jan 16]. Available from: http://hdl.handle.net/10871/24166.

Council of Science Editors:

Stary T. Mathematical and computational study of Markovian models of ion channels in cardiac excitation. [Doctoral Dissertation]. University of Exeter; 2016. Available from: http://hdl.handle.net/10871/24166

University of Notre Dame

30. Oyekola Oyekole. Development and Numerical Analysis of Partitioned Algorithms for Fluid-Elastic / Poroelastic Structure Interaction Problems</h1>.

Degree: Applied and Computational Mathematics and Statistics, 2020, University of Notre Dame

URL: https://curate.nd.edu/show/1j92g73588j

► This work introduces novel, loosely-coupled and easy-to-implement partitioned algorithms for Fluid-Structure Interaction (FSI) and Fluid-Poroelastic Structure Interaction, based on spatial discretization using the standard… (more)

Subjects/Keywords: Fluid-Structure Interaction (FSI); Fluid-Poroelastic Structure Interaction; Poroelasticity; Deformable porous media; Structural deformation; Partitioned methods; Loosely-coupled methods; Staggered solution procedure; De-coupling algorithms; Operator splitting; Finite Element Method; Finite Difference Method; Stability analysis; Error Estimates; Biot model; Stokes equations; Navier-Stokes equations; Incompressible flow; Linear elasticity; Beavers-Joseph-Saffman slip condition; High performance computing; Continuum Mechanics.

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Oyekole, O. (2020). Development and Numerical Analysis of Partitioned Algorithms for Fluid-Elastic / Poroelastic Structure Interaction Problems</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/1j92g73588j

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Oyekole, Oyekola. “Development and Numerical Analysis of Partitioned Algorithms for Fluid-Elastic / Poroelastic Structure Interaction Problems</h1>.” 2020. Thesis, University of Notre Dame. Accessed January 16, 2021. https://curate.nd.edu/show/1j92g73588j.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Oyekole, Oyekola. “Development and Numerical Analysis of Partitioned Algorithms for Fluid-Elastic / Poroelastic Structure Interaction Problems</h1>.” 2020. Web. 16 Jan 2021.

Vancouver:

Oyekole O. Development and Numerical Analysis of Partitioned Algorithms for Fluid-Elastic / Poroelastic Structure Interaction Problems</h1>. [Internet] [Thesis]. University of Notre Dame; 2020. [cited 2021 Jan 16]. Available from: https://curate.nd.edu/show/1j92g73588j.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Oyekole O. Development and Numerical Analysis of Partitioned Algorithms for Fluid-Elastic / Poroelastic Structure Interaction Problems</h1>. [Thesis]. University of Notre Dame; 2020. Available from: https://curate.nd.edu/show/1j92g73588j

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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