subject:(saddle point problems)

1. Nifa, Naoufal. Solveurs performants pour l'optimisation sous contraintes en identification de paramètres : Efficient solvers for constrained optimization in parameter identification problems.

Degree: Docteur es, Génie mécanique, 2017, Université Paris-Saclay (ComUE)

URL: http://www.theses.fr/2017SACLC066

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Cette thèse vise à concevoir des solveurs efficaces pour résoudre des systèmes linéaires, résultant des problèmes d'optimisation sous contraintes dans certaines applications de dynamique des… (more)

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Cette thèse vise à concevoir des solveurs efficaces pour résoudre des systèmes linéaires, résultant des problèmes d'optimisation sous contraintes dans certaines applications de dynamique des structures et vibration (la corrélation calcul-essai, la localisation d'erreur, le modèle hybride, l'évaluation des dommages, etc.). Ces applications reposent sur la résolution de problèmes inverses, exprimés sous la forme de la minimisation d'une fonctionnelle en énergie. Cette fonctionnelle implique à la fois, des données issues d'un modèle numérique éléments finis, et des essais expérimentaux. Ceci conduit à des modèles de haute qualité, mais les systèmes linéaires point-selle associés, sont coûteux à résoudre. Nous proposons deux classes différentes de méthodes pour traiter le système. La première classe repose sur une méthode de factorisation directe profitant de la topologie et des propriétés spéciales de la matrice point-selle. Après une première renumérotation pour regrouper les pivots en blocs d'ordre 2. L'élimination de Gauss est conduite à partir de ces pivots et en utilisant un ordre spécial d'élimination réduisant le remplissage. Les résultats numériques confirment des gains significatifs en terme de remplissage, jusqu'à deux fois meilleurs que la littérature pour la topologie étudiée. La seconde classe de solveurs propose une approche à double projection du système étudié sur le noyau des contraintes, en faisant une distinction entre les contraintes cinématiques et celles reliées aux capteurs sur la structure. La première projection est explicite en utilisant une base creuse du noyau. La deuxième est implicite. Elle est basée sur l'emploi d'un préconditionneur contraint avec des méthodes itératives de type Krylov. Différentes approximations des blocs du préconditionneur sont proposées. L'approche est implémentée dans un environnement distribué parallèle utilisant la bibliothèque PETSc. Des gains significatifs en terme de coût de calcul et de mémoire sont illustrés sur plusieurs applications industrielles.

This thesis aims at designing efficient numerical solution methods to solve linear systems, arising in constrained optimization problems in some structural dynamics and vibration applications (test-analysis correlation, model error localization,hybrid model, damage assessment, etc.). These applications rely on solving inverse problems, by means of minimization of an energy-based functional. This latter involves both data from a numerical finite element model and from experimental tests, which leads to high quality models, but the associated linear systems, that have a saddle-point coefficient matrices, are long and costly to solve. We propose two different classes of methods to deal with these problems. First, a direct factorization method that takes advantage of the special structures and properties of these saddle point matrices. The Gaussian elimination factorization is implemented in order to factorize the saddle point matrices block-wise with small blocks of orders 2 and using a fill-in reducing topological…

Advisors/Committee Members: Aubry, Denis (thesis director).

Subjects/Keywords: Optimisation sous contraintes; Systèmes point-Selle; Préconditionneurs par blocs; Problèmes inverses; Constrained optimization; Saddle point systems; Block preconditioners; Inverse problems

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

APA (6^th Edition):

Nifa, N. (2017). Solveurs performants pour l'optimisation sous contraintes en identification de paramètres : Efficient solvers for constrained optimization in parameter identification problems. (Doctoral Dissertation). Université Paris-Saclay (ComUE). Retrieved from http://www.theses.fr/2017SACLC066

Chicago Manual of Style (16^th Edition):

Nifa, Naoufal. “Solveurs performants pour l'optimisation sous contraintes en identification de paramètres : Efficient solvers for constrained optimization in parameter identification problems.” 2017. Doctoral Dissertation, Université Paris-Saclay (ComUE). Accessed February 27, 2021. http://www.theses.fr/2017SACLC066.

MLA Handbook (7^th Edition):

Nifa, Naoufal. “Solveurs performants pour l'optimisation sous contraintes en identification de paramètres : Efficient solvers for constrained optimization in parameter identification problems.” 2017. Web. 27 Feb 2021.

Vancouver:

Nifa N. Solveurs performants pour l'optimisation sous contraintes en identification de paramètres : Efficient solvers for constrained optimization in parameter identification problems. [Internet] [Doctoral dissertation]. Université Paris-Saclay (ComUE); 2017. [cited 2021 Feb 27]. Available from: http://www.theses.fr/2017SACLC066.

Council of Science Editors:

Nifa N. Solveurs performants pour l'optimisation sous contraintes en identification de paramètres : Efficient solvers for constrained optimization in parameter identification problems. [Doctoral Dissertation]. Université Paris-Saclay (ComUE); 2017. Available from: http://www.theses.fr/2017SACLC066

Texas A&M University

2. Ghesmati, Arezou. Residual and Goal-Oriented h- and hp-adaptive Finite Element; Application for Elliptic and Saddle Point Problems.

Degree: PhD, Mathematics, 2018, Texas A&M University

URL: http://hdl.handle.net/1969.1/173289

► We propose and implement an automatic hp-adaptive refinement algorithm for the Stokes model problem. In this work, the strategy is based on the earlier work… (more)

▼ We propose and implement an automatic hp-adaptive refinement algorithm for the Stokes model problem. In this work, the strategy is based on the earlier work done by Dörfler at al. in 2007 for the Poisson problem. Similar to any other adaptivity approach, an a posteriori estimator is needed to control the error in areas with high residuals. We define a family of residual-based estimators Ŋva a € [0; 1] for the hp-adaptive finite element approximation of the exact solution. Moreover, we show the reliability and efficiency of the estimators Ŋva. Finally, numerical examples illustrate the exponential convergence rate of the hp-AFEM in comparison with the h-AFEM. In many applications, such as analysis of fluid flows in our case, we are not interested in computing the solution itself, but instead the aim is finding a good approximation for some functional of interest. In these cases, the idea is to develop some a posteriori error estimates to generate a sequence of h- or hp-adaptive grids that minimize the error in our goal functional with respect to the problem size. In this work, we apply local averaging interpolation operators such as Scott-Zhang and Clément type operators to formulate the dual weight of our proposed goal-oriented error estimator. This idea was recently used in an application to the Poisson problem. We extend those results to saddle-point problems and provide a dual-weighted goal estimator for each cell. The reliability of the goal estimator is proved and numerical examples demonstrate the performance of the locally defined dual-weighted goal-estimator in terms of reliability, efficiency, and convergence. Another important aspect of this research is providing a goal-oriented adaptive finite element method for symmetric second-order linear elliptic problems. We prove that the product of primal and dual estimators, which is a reliable upper bound for the error in the goal functional, decays at the optimal rate. The results reported in the numerical experiments confirm the quasi-optimality behavior of our goal-oriented algorithm. Advisors/Committee Members: Bangerth, Wolfgang (advisor), Guermond, Jean Luc (committee member), Bonito, Andrea (committee member), Ragusa, Jean C (committee member).

Subjects/Keywords: Adaptive Finite Elements; Elliptic Problems; Saddle Point Problems; Goal-Oriented Adaptivity; Aposteriori Error Estimate; Optimal Convergence Rate

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APA (6^th Edition):

Ghesmati, A. (2018). Residual and Goal-Oriented h- and hp-adaptive Finite Element; Application for Elliptic and Saddle Point Problems. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/173289

Chicago Manual of Style (16^th Edition):

Ghesmati, Arezou. “Residual and Goal-Oriented h- and hp-adaptive Finite Element; Application for Elliptic and Saddle Point Problems.” 2018. Doctoral Dissertation, Texas A&M University. Accessed February 27, 2021. http://hdl.handle.net/1969.1/173289.

MLA Handbook (7^th Edition):

Ghesmati, Arezou. “Residual and Goal-Oriented h- and hp-adaptive Finite Element; Application for Elliptic and Saddle Point Problems.” 2018. Web. 27 Feb 2021.

Vancouver:

Ghesmati A. Residual and Goal-Oriented h- and hp-adaptive Finite Element; Application for Elliptic and Saddle Point Problems. [Internet] [Doctoral dissertation]. Texas A&M University; 2018. [cited 2021 Feb 27]. Available from: http://hdl.handle.net/1969.1/173289.

Council of Science Editors:

Ghesmati A. Residual and Goal-Oriented h- and hp-adaptive Finite Element; Application for Elliptic and Saddle Point Problems. [Doctoral Dissertation]. Texas A&M University; 2018. Available from: http://hdl.handle.net/1969.1/173289

University of Manchester

3. Newsum, Craig. Efficient reduced basis methods for parameter-dependent saddle point problems.

Degree: PhD, 2019, University of Manchester

URL: https://www.research.manchester.ac.uk/portal/en/theses/efficient-reduced-basis-methods-for-parameterdependent-saddle-point-problems(c9fe675e-42b2-4306-a05c-b6b7ad96de8f).html ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.771491

► We are interested in the efficient numerical solution of parameter-dependent saddle point problems arising from discretisations of systems of PDEs with uncertain inputs. Two popular… (more)

▼ We are interested in the efficient numerical solution of parameter-dependent saddle point problems arising from discretisations of systems of PDEs with uncertain inputs. Two popular solution strategies for such problems are stochastic collocation mixed finite element methods and stochastic Galerkin mixed finite element methods. The former requires the solution of a long sequence of deterministic linear systems corresponding to different choices of the input parameters while the latter requires the solution of a single extremely large linear system. Typically, a fine spatial discretisation is required, and this can lead to large discrete problems that are extremely time and memory intensive to solve. These are significant bottlenecks in the forward propagation of uncertainty in PDE models. Reduced basis methods (RBMs) seek to overcome these bottlenecks by reducing the size of the so-called high fidelity problem through a projection onto a lower-dimensional space. The dimension of this space should be large enough so that the corresponding reduced solution is sufficiently accurate but small enough that the reduced problem is cheaper to solve. There are two novel contributions in this thesis. First, we develop an efficient RBM algorithm to reduce the computational costs associated with stochastic collocation mixed finite element methods. For saddle point problems, we need to construct two reduced spaces which must be compatible to provide a well-posed reduced problem. This is done in a potentially expensive offline stage. We also use a discrete empirical interpolation method (DEIM) to approximate the uncertain inputs as these may depend non-affinely on the input parameters. We are then able to solve the sequence of linear systems at a dramatically cheaper cost. In addition, our algorithm is faster than the standard greedy algorithm for constructing the reduced spaces. Second, we develop an efficient solution algorithm for solving the huge saddle point systems arising from stochastic Galerkin mixed finite element methods in cases where there is not enough memory to use standard Krylov subspace methods. Our approach uses ideas from rational Krylov subspace approximation to iteratively generate a reduced approximation space. This strategy provides a low-rank approximation to the solution of the stochastic Galerkin equations which can be obtained at a much lower cost and using less memory when compared to standard preconditioned iterative methods. Both of the new reduced basis algorithms proposed in this thesis allow for the efficient forward propagation of uncertainty in PDE models. We demonstrate the extent of savings achieved through comprehensive numerical experiments.

Subjects/Keywords: 510; PDEs with random inputs; Saddle point problems; Stochastic collocation; Stochastic Galerkin; Uncertainty quantification; Reduced basis methods

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

APA (6^th Edition):

Newsum, C. (2019). Efficient reduced basis methods for parameter-dependent saddle point problems. (Doctoral Dissertation). University of Manchester. Retrieved from https://www.research.manchester.ac.uk/portal/en/theses/efficient-reduced-basis-methods-for-parameterdependent-saddle-point-problems(c9fe675e-42b2-4306-a05c-b6b7ad96de8f).html ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.771491

Chicago Manual of Style (16^th Edition):

Newsum, Craig. “Efficient reduced basis methods for parameter-dependent saddle point problems.” 2019. Doctoral Dissertation, University of Manchester. Accessed February 27, 2021. https://www.research.manchester.ac.uk/portal/en/theses/efficient-reduced-basis-methods-for-parameterdependent-saddle-point-problems(c9fe675e-42b2-4306-a05c-b6b7ad96de8f).html ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.771491.

MLA Handbook (7^th Edition):

Newsum, Craig. “Efficient reduced basis methods for parameter-dependent saddle point problems.” 2019. Web. 27 Feb 2021.

Vancouver:

Newsum C. Efficient reduced basis methods for parameter-dependent saddle point problems. [Internet] [Doctoral dissertation]. University of Manchester; 2019. [cited 2021 Feb 27]. Available from: https://www.research.manchester.ac.uk/portal/en/theses/efficient-reduced-basis-methods-for-parameterdependent-saddle-point-problems(c9fe675e-42b2-4306-a05c-b6b7ad96de8f).html ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.771491.

Council of Science Editors:

Newsum C. Efficient reduced basis methods for parameter-dependent saddle point problems. [Doctoral Dissertation]. University of Manchester; 2019. Available from: https://www.research.manchester.ac.uk/portal/en/theses/efficient-reduced-basis-methods-for-parameterdependent-saddle-point-problems(c9fe675e-42b2-4306-a05c-b6b7ad96de8f).html ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.771491

4. Lemaire, Simon. Discrétisations non-conformes d'un modèle poromécanique sur maillages généraux : Nonconforming discretizations of a poromechanical model on general meshes.

Degree: Docteur es, Mathématiques, 2013, Université Paris-Est

URL: http://www.theses.fr/2013PEST1168

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Cette thèse s'intéresse à la conception de méthodes de discrétisation non-conforme pour un modèle de poromécanique. Le but de ce travail est de simplifier les… (more)

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Cette thèse s'intéresse à la conception de méthodes de discrétisation non-conforme pour un modèle de poromécanique. Le but de ce travail est de simplifier les couplages liant la géomécanique d'un milieu poreux à l'écoulement polyphasique compositionnel ayant cours en son sein tels qu'ils sont réalisés actuellement dans l'industrie pétrolière, en discrétisant sur un même maillage, typiquement non-conforme car à l'image de la lithologie, la mécanique et l'écoulement. La nouveauté consiste donc à traiter la mécanique par une méthode d'approximation non-conforme sur maillages généraux. Dans cette thèse, nous nous concentrons sur un modèle d'élasticité linéaire. Les difficultés inhérentes à son approximation non-conforme sont son manque de coercivité (se traduisant par la nécessité de satisfaire une inégalité de Korn sur un espace discret discontinu), ainsi que le phénomène de verrouillage numérique lorsque le matériau tend à devenir incompressible. Dans une première partie, nous construisons un espace d'approximation sur maillages généraux, s'apparentant à une extension de l'espace de Crouzeix-Raviart. Nous explicitons ses propriétés d'approximation et de conformité, et montrons que ce dernier est adapté à une discrétisation primale coercive et robuste au locking du modèle d'élasticité sur maillages généraux. La méthode proposée est moins coûteuse que son équivalent éléments finis (en termes de propriétés) P2. Nous nous intéressons dans une deuxième partie à l'approximation non-conforme d'un modèle couplé de poroélasticité. Nous étudions la convergence d'une famille de schémas numériques dont la discrétisation en espace utilise le formalisme des schémas Gradient, auquel appartient la méthode développée pour la mécanique. Nous prouvons la convergence de telles approximations vers la solution de régularité minimale du problème continu, indépendamment des paramètres physiques du système

This manuscript focuses on the conception of nonconforming discretization methods for a poromechanical model. The aim of this work is to ease the coupling between the geomechanics and the multiphase compositional Darcy flow in porous media by discretizing mechanics and flow on the same mesh, typically nonconforming as it represents the lithology. Hence, the novelty hinges on a nonconforming treatment of mechanics on general meshes. In this work, we focus on a linear elasticity model. The nonconforming approximation of such a model is not straightforward owing to its lack of coercivity (meaning that a discrete Korn's inequality must hold on a discontinuous discrete space) and to the numerical locking phenomenon occurring as the material becomes incompressible. In a first part, we design an approximation space on general meshes, which can be viewed as an extension of the so-called Crouzeix-Raviart space. We study its approximation and conformity properties, and prove that this latter is well-adapted to the design of a primal, coercive, and locking-free discretization of the elasticity model on general meshes. The proposed method is less…

Advisors/Committee Members: Eymard, Robert (thesis director).

Subjects/Keywords: Poroélasticité; Méthodes non-conformes; Maillages généraux; Volumes finis; Verrouillage numérique; Problèmes de point-selle; Poroelasticity; Nonconforming methods; General meshes; Finite volumes; Numerical locking; Saddle-point problems

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

APA (6^th Edition):

Lemaire, S. (2013). Discrétisations non-conformes d'un modèle poromécanique sur maillages généraux : Nonconforming discretizations of a poromechanical model on general meshes. (Doctoral Dissertation). Université Paris-Est. Retrieved from http://www.theses.fr/2013PEST1168

Chicago Manual of Style (16^th Edition):

Lemaire, Simon. “Discrétisations non-conformes d'un modèle poromécanique sur maillages généraux : Nonconforming discretizations of a poromechanical model on general meshes.” 2013. Doctoral Dissertation, Université Paris-Est. Accessed February 27, 2021. http://www.theses.fr/2013PEST1168.

MLA Handbook (7^th Edition):

Lemaire, Simon. “Discrétisations non-conformes d'un modèle poromécanique sur maillages généraux : Nonconforming discretizations of a poromechanical model on general meshes.” 2013. Web. 27 Feb 2021.

Vancouver:

Lemaire S. Discrétisations non-conformes d'un modèle poromécanique sur maillages généraux : Nonconforming discretizations of a poromechanical model on general meshes. [Internet] [Doctoral dissertation]. Université Paris-Est; 2013. [cited 2021 Feb 27]. Available from: http://www.theses.fr/2013PEST1168.

Council of Science Editors:

Lemaire S. Discrétisations non-conformes d'un modèle poromécanique sur maillages généraux : Nonconforming discretizations of a poromechanical model on general meshes. [Doctoral Dissertation]. Université Paris-Est; 2013. Available from: http://www.theses.fr/2013PEST1168

Queensland University of Technology

5. Kempthorne, Daryl Matthew. The development of virtual leaf surface models for interactive agrichemical spray applications.

Degree: 2015, Queensland University of Technology

URL: http://eprints.qut.edu.au/84525/

► This project constructed virtual plant leaf surfaces from digitised data sets for use in droplet spray models. Digitisation techniques for obtaining data sets for cotton,… (more)

Subjects/Keywords: digitisation; 3D scanning; virtual leaf reconstruction; surface fitting; discrete smoothing D2 splines; thin plate spline finite element method; preconditioning saddle point problems; parameterisation techniques

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APA (6^th Edition):

Kempthorne, D. M. (2015). The development of virtual leaf surface models for interactive agrichemical spray applications. (Thesis). Queensland University of Technology. Retrieved from http://eprints.qut.edu.au/84525/

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):

Kempthorne, Daryl Matthew. “The development of virtual leaf surface models for interactive agrichemical spray applications.” 2015. Thesis, Queensland University of Technology. Accessed February 27, 2021. http://eprints.qut.edu.au/84525/.

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

MLA Handbook (7^th Edition):

Kempthorne, Daryl Matthew. “The development of virtual leaf surface models for interactive agrichemical spray applications.” 2015. Web. 27 Feb 2021.

Vancouver:

Kempthorne DM. The development of virtual leaf surface models for interactive agrichemical spray applications. [Internet] [Thesis]. Queensland University of Technology; 2015. [cited 2021 Feb 27]. Available from: http://eprints.qut.edu.au/84525/.

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

Council of Science Editors:

Kempthorne DM. The development of virtual leaf surface models for interactive agrichemical spray applications. [Thesis]. Queensland University of Technology; 2015. Available from: http://eprints.qut.edu.au/84525/

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

6. Rossato, Rafael Antonio. Sistemas elípticos de tipo hamiltoniano perto da ressonância.

Degree: PhD, Matemática, 2014, University of São Paulo

URL: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13042015-164728/ ;

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Neste trabalho consideramos sistemas elípticos de tipo hamiltoniano, envolvendo o operador Laplaciano, com uma parte linear dependendo de dois parâmetros e uma perturbação sublinear. Obtemos… (more)

Subjects/Keywords: Aproximação finito dimensional; Finite dimensional approximation; Geometria de ponto de sela; Near resonance problems; Problemas de quase ressonância; Saddle point geometry; Semilinear elliptic systems; Sistemas elípticos semilineares

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APA (6^th Edition):

Rossato, R. A. (2014). Sistemas elípticos de tipo hamiltoniano perto da ressonância. (Doctoral Dissertation). University of São Paulo. Retrieved from http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13042015-164728/ ;

Chicago Manual of Style (16^th Edition):

Rossato, Rafael Antonio. “Sistemas elípticos de tipo hamiltoniano perto da ressonância.” 2014. Doctoral Dissertation, University of São Paulo. Accessed February 27, 2021. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13042015-164728/ ;.

MLA Handbook (7^th Edition):

Rossato, Rafael Antonio. “Sistemas elípticos de tipo hamiltoniano perto da ressonância.” 2014. Web. 27 Feb 2021.

Vancouver:

Rossato RA. Sistemas elípticos de tipo hamiltoniano perto da ressonância. [Internet] [Doctoral dissertation]. University of São Paulo; 2014. [cited 2021 Feb 27]. Available from: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13042015-164728/ ;.

Council of Science Editors:

Rossato RA. Sistemas elípticos de tipo hamiltoniano perto da ressonância. [Doctoral Dissertation]. University of São Paulo; 2014. Available from: http://www.teses.usp.br/teses/disponiveis/55/55135/tde-13042015-164728/ ;

7. A. Fiscella. VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS.

Degree: 2014, Università degli Studi di Milano

URL: http://hdl.handle.net/2434/245334

► My thesis deals with nonlinear elliptic problems involving a non-local integrodifferential operator of fractional type. Our main results concern the existence of weak solutions for… (more)

Subjects/Keywords: integrodifferential operators; fractional Laplacian; Saddle Point Theorem; Palais-Smale condition; Stationary Kirchhoff problems; critical nonlinearities; variational methods; Settore MAT/05 - Analisi Matematica

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APA (6^th Edition):

Fiscella, A. (2014). VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS. (Thesis). Università degli Studi di Milano. Retrieved from http://hdl.handle.net/2434/245334

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):

Fiscella, A.. “VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS.” 2014. Thesis, Università degli Studi di Milano. Accessed February 27, 2021. http://hdl.handle.net/2434/245334.

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

MLA Handbook (7^th Edition):

Fiscella, A.. “VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS.” 2014. Web. 27 Feb 2021.

Vancouver:

Fiscella A. VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS. [Internet] [Thesis]. Università degli Studi di Milano; 2014. [cited 2021 Feb 27]. Available from: http://hdl.handle.net/2434/245334.

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

Council of Science Editors:

Fiscella A. VARIATIONAL PROBLEMS INVOLVING NON-LOCAL ELLIPTIC OPERATORS. [Thesis]. Università degli Studi di Milano; 2014. Available from: http://hdl.handle.net/2434/245334

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

8. Ul Jabbar, Absaar. Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations.

Degree: 2018, Technische Universität Dortmund

URL: http://dx.doi.org/10.17877/DE290R-19859

► Multigrid methods belong to the best-known methods for solving linear systems arising from the discretization of elliptic partial differential equations. The main attraction of multigrid… (more)

▼ Multigrid methods belong to the best-known methods for solving linear systems arising from the discretization of elliptic partial differential equations. The main attraction of multigrid methods is that they have an asymptotically meshindependent convergence behavior. Multigrid with Vanka (or local multilevel pressure Schur complement method) as smoother have been frequently used for the construction of very effcient coupled monolithic solvers for the solution of the stationary incompressible Navier-Stokes equations in 2D and 3D. However, due to its innate Gauß-Seidel/Jacobi character, Vanka has a strong influence of the underlying mesh, and therefore, coupled multigrid solvers with Vanka smoothing very frequently face convergence issues on meshes with high aspect ratios. Moreover, even on very nice regular grids, these solvers may fail when the anisotropies are introduced from the differential operator. In this thesis, we develop a new class of robust and efficient monolithic finite element multilevel Krylov subspace methods (MLKM) for the solution of the stationary incompressible Navier-Stokes equations as an alternative to the coupled multigrid-based solvers. Different from multigrid, the MLKM utilizes a Krylov method as the basis in the error reduction process. The solver is based on the multilevel projection-based method of Erlangga and Nabben, which accelerates the convergence of the Krylov subspace methods by shifting the small eigenvalues of the system matrix, responsible for the slow convergence of the Krylov iteration, to the largest eigenvalue. Before embarking on the Navier-Stokes equations, we first test our implementation of the MLKM solver by solving scalar model problems, namely the convection-diffusion problem and the anisotropic diffusion problem. We validate the method by solving several standard benchmark problems. Next, we present the numerical results for the solution of the incompressible Navier-Stokes equations in two dimensions. The results show that the MLKM solvers produce asymptotically mesh-size independent, as well as Reynolds number independent convergence rates, for a moderate range of Reynolds numbers. Moreover, numerical simulations also show that the coupled MLKM solvers can handle (both mesh and operator based) anisotropies better than the coupled multigrid solvers. Advisors/Committee Members: Turek, Stefan (advisor), Blum, Heribert (referee).

Subjects/Keywords: Monolithic multilevel methods; Krylov subspaces; GMRES; FEM; Navier-Stokes equations; Saddle point problems; 510; Multi-level-Verfahren; Krylov-Verfahren; Finite-Elemente-Methode; Navier-Stokes-Gleichung

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APA (6^th Edition):

Ul Jabbar, A. (2018). Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations. (Doctoral Dissertation). Technische Universität Dortmund. Retrieved from http://dx.doi.org/10.17877/DE290R-19859

Chicago Manual of Style (16^th Edition):

Ul Jabbar, Absaar. “Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations.” 2018. Doctoral Dissertation, Technische Universität Dortmund. Accessed February 27, 2021. http://dx.doi.org/10.17877/DE290R-19859.

MLA Handbook (7^th Edition):

Ul Jabbar, Absaar. “Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations.” 2018. Web. 27 Feb 2021.

Vancouver:

Ul Jabbar A. Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations. [Internet] [Doctoral dissertation]. Technische Universität Dortmund; 2018. [cited 2021 Feb 27]. Available from: http://dx.doi.org/10.17877/DE290R-19859.

Council of Science Editors:

Ul Jabbar A. Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations. [Doctoral Dissertation]. Technische Universität Dortmund; 2018. Available from: http://dx.doi.org/10.17877/DE290R-19859

Freie Universität Berlin

9. Gräser, Carsten. Konvexe Minimierung und Phasenfeldmodelle.

Degree: 2011, Freie Universität Berlin

URL: http://dx.doi.org/10.17169/refubium-11584

► Phasenfeldmodelle sind ein weit verbreiteter Ansatz zur Beschreibung von Prozessen, die sich wesentlich durch dünne Interfaceregionen zwischen weitestgehend homogenen Bereichen auszeichnen. Ein wichtiges Anwendungsfeld von… (more)

▼ Phasenfeldmodelle sind ein weit verbreiteter Ansatz zur Beschreibung von Prozessen, die sich wesentlich durch dünne Interfaceregionen zwischen weitestgehend homogenen Bereichen auszeichnen. Ein wichtiges Anwendungsfeld von Phasenfeldmodellen ist die Modellierung von physikalischen Phasenübergangs- und Phasenseparationvorgängen. Eine wesentliche Eigenschaften ist dabei, dass die Trennung der Phasen durch ein Doppelmuldenpotential getrieben ist, welches voneinander getrennte Minima für jede Phase besitzt. Bereits 1958 haben Cahn und Hilliard ein logarithmisches Potential vorgeschlagen, das zwar differenzierbar ist, aber singuläre Ableitungen besitzt. Geht die Temperatur gegen Null, so degeneriert das temperaturabhängige logarithmische Potential gegen das nichtdifferenzierbare Hindernispotential. Ziel der vorliegenden Arbeit ist die Entwicklung von Methoden zur effizienten numerischen Lösung solcher Gleichungen, die auch im Fall nichtglatter Potentiale und anisotroper Oberflächenenergien robust sind. Diese Methoden werden für die Cahn-Hilliard-Gleichung entwickelt, die prototypisch für eine Vielzahl solcher Modelle ist. Das Hauptresultat der Arbeit ist die Entwicklung eines schnellen iterativen Verfahrens zur Lösung nichtlinearer Sattelpunktprobleme, wie sie bei der Diskretisierung anisotroper Cahn-Hilliard-Gleichungen mit Finite-Elemente-Methoden entstehen. Die Grundlage dieses Verfahrens ist eine Umformulierung des Sattelpunktproblems als äquivalentes duales Minimierungsproblem. Das Energiefunktional dieses Minimierungsproblems ist differenzierbar und seine Ableitung ist das nichtlineare Schur-Komplement des Sattelpunktproblems. Für dieses Schur- Komplement wird eine verallgemeinerte Linearisierung hergeleitet, die im Rahmen eines nichtglatten Newton-Verfahrens Verwendung findet. Für dieses so genannte "Schur Nonsmooth Newton" Verfahren wird die globale Konvergenz mittels der Äquivalenz zu Abstiegsverfahren für das duale Minimierungsproblem gezeigt. Ferner wird bewiesen, dass die globale Konvergenz auch bei inexaktem Lösen der linearen Newton Probleme erhalten bleibt. In jedem Schritt dieses Verfahrens ist ein nichtlineares konvexes Minimierungsproblem zu lösen. Für die effiziente Behandlung dieser Teilprobleme wird das so genannte "Truncated Nonsmooth Newton Multigrid" (TNNMG) Verfahren entwickelt. Dieses nichtlineare Mehrgitterverfahren zeichnet sich im Gegensatz zu verwandten Verfahren durch seine Einfachheit und die Anwendbarkeit auf anisotrope Probleme aus. Hinsichtlich der Konvergenzgeschwindigkeit ist es bereits bekannten Verfahren ebenbürtig. Numerische Beispiele zeigen, dass die entwickelten Verfahren gitterunabhängig konvergieren. Auch erweisen sich die Verfahren als robust für verschiedene Temperaturen, einschließlich dem Grenzfall Null. Der Grund für diese Robustheit ist, dass die Verfahren nicht auf Differenzierbarkeit sondern der konvexen Struktur der Probleme basieren. Advisors/Committee Members: [email protected] (contact), m (gender), Prof. Dr. Ralf Kornhuber (firstReferee), Prof. Dr. Charles M. Elliott (furtherReferee).

Subjects/Keywords: phase field models; Cahn-Hilliard equation; convex minimization; nonsmooth Newton method; multigrid; saddle point problems; 500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik

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

APA (6^th Edition):

Gräser, C. (2011). Konvexe Minimierung und Phasenfeldmodelle. (Thesis). Freie Universität Berlin. Retrieved from http://dx.doi.org/10.17169/refubium-11584

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):

Gräser, Carsten. “Konvexe Minimierung und Phasenfeldmodelle.” 2011. Thesis, Freie Universität Berlin. Accessed February 27, 2021. http://dx.doi.org/10.17169/refubium-11584.

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

MLA Handbook (7^th Edition):

Gräser, Carsten. “Konvexe Minimierung und Phasenfeldmodelle.” 2011. Web. 27 Feb 2021.

Vancouver:

Gräser C. Konvexe Minimierung und Phasenfeldmodelle. [Internet] [Thesis]. Freie Universität Berlin; 2011. [cited 2021 Feb 27]. Available from: http://dx.doi.org/10.17169/refubium-11584.

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

Council of Science Editors:

Gräser C. Konvexe Minimierung und Phasenfeldmodelle. [Thesis]. Freie Universität Berlin; 2011. Available from: http://dx.doi.org/10.17169/refubium-11584

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

University of Illinois – Urbana-Champaign

10. Huang, Yen-Wei. On the saddle-point solution and the large-coalition behavior of fingerprinting games.

Degree: MS, 1200, 2010, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/16836

► We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence… (more)

Subjects/Keywords: fingerprinting; traitor tracing; capacity; game theory; minimax theorem; Jeffreys' prior; saddle-point problems

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

APA (6^th Edition):

Huang, Y. (2010). On the saddle-point solution and the large-coalition behavior of fingerprinting games. (Thesis). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/16836

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):

Huang, Yen-Wei. “On the saddle-point solution and the large-coalition behavior of fingerprinting games.” 2010. Thesis, University of Illinois – Urbana-Champaign. Accessed February 27, 2021. http://hdl.handle.net/2142/16836.

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

MLA Handbook (7^th Edition):

Huang, Yen-Wei. “On the saddle-point solution and the large-coalition behavior of fingerprinting games.” 2010. Web. 27 Feb 2021.

Vancouver:

Huang Y. On the saddle-point solution and the large-coalition behavior of fingerprinting games. [Internet] [Thesis]. University of Illinois – Urbana-Champaign; 2010. [cited 2021 Feb 27]. Available from: http://hdl.handle.net/2142/16836.

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

Council of Science Editors:

Huang Y. On the saddle-point solution and the large-coalition behavior of fingerprinting games. [Thesis]. University of Illinois – Urbana-Champaign; 2010. Available from: http://hdl.handle.net/2142/16836

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

Louisiana State University

11. Liu, Sijing. Multigrid Methods for Elliptic Optimal Control Problems.

Degree: PhD, Numerical Analysis and Computation, Louisiana State University

URL: https://digitalcommons.lsu.edu/gradschool_dissertations/5279

► In this dissertation we study multigrid methods for linear-quadratic elliptic distributed optimal control problems. For optimal control problems constrained by general second order elliptic… (more)

Subjects/Keywords: Multigrid methods; Elliptic optimal control problems; Saddle point problems; Primal-dual active set algorithms

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APA (6^th Edition):

Liu, S. (n.d.). Multigrid Methods for Elliptic Optimal Control Problems. (Doctoral Dissertation). Louisiana State University. Retrieved from https://digitalcommons.lsu.edu/gradschool_dissertations/5279

Note: this citation may be lacking information needed for this citation format:
No year of publication.

Chicago Manual of Style (16^th Edition):

Liu, Sijing. “Multigrid Methods for Elliptic Optimal Control Problems.” Doctoral Dissertation, Louisiana State University. Accessed February 27, 2021. https://digitalcommons.lsu.edu/gradschool_dissertations/5279.

Note: this citation may be lacking information needed for this citation format:
No year of publication.

MLA Handbook (7^th Edition):

Liu, Sijing. “Multigrid Methods for Elliptic Optimal Control Problems.” Web. 27 Feb 2021.

Note: this citation may be lacking information needed for this citation format:
No year of publication.

Vancouver:

Liu S. Multigrid Methods for Elliptic Optimal Control Problems. [Internet] [Doctoral dissertation]. Louisiana State University; [cited 2021 Feb 27]. Available from: https://digitalcommons.lsu.edu/gradschool_dissertations/5279.

Note: this citation may be lacking information needed for this citation format:
No year of publication.

Council of Science Editors:

Liu S. Multigrid Methods for Elliptic Optimal Control Problems. [Doctoral Dissertation]. Louisiana State University; Available from: https://digitalcommons.lsu.edu/gradschool_dissertations/5279

Note: this citation may be lacking information needed for this citation format:
No year of publication.

12. Newsum, Craig James. Efficient reduced basis methods for parameter-dependent saddle point problems.

Degree: 2019, University of Manchester

URL: http://www.manchester.ac.uk/escholar/uk-ac-man-scw:318439

► We are interested in the efficient numerical solution of parameter-dependent saddle point problems arising from discretisations of systems of PDEs with uncertain inputs. Two popular… (more)

▼ We are interested in the efficient numerical solution of parameter-dependent saddle point problems arising from discretisations of systems of PDEs with uncertain inputs. Two popular solution strategies for such problems are stochastic collocation mixed finite element methods and stochastic Galerkin mixed finite element methods. The former requires the solution of a long sequence of deterministic linear systems corresponding to different choices of the input parameters while the latter requires the solution of a single extremely large linear system. Typically, a fine spatial discretisation is required, and this can lead to large discrete problems that are extremely time and memory intensive to solve. These are significant bottlenecks in the forward propagation of uncertainty in PDE models. Reduced basis methods (RBMs) seek to overcome these bottlenecks by reducing the size of the so-called high fidelity problem through a projection onto a lower-dimensional space. The dimension of this space should be large enough so that the corresponding reduced solution is sufficiently accurate but small enough that the reduced problem is cheaper to solve. There are two novel contributions in this thesis. First, we develop an efficient RBM algorithm to reduce the computational costs associated with stochastic collocation mixed finite element methods. For saddle point problems, we need to construct two reduced spaces which must be compatible to provide a well-posed reduced problem. This is done in a potentially expensive offline stage. We also use a discrete empirical interpolation method (DEIM) to approximate the uncertain inputs as these may depend non-affinely on the input parameters. We are then able to solve the sequence of linear systems at a dramatically cheaper cost. In addition, our algorithm is faster than the standard greedy algorithm for constructing the reduced spaces. Second, we develop an efficient solution algorithm for solving the huge saddle point systems arising from stochastic Galerkin mixed finite element methods in cases where there is not enough memory to use standard Krylov subspace methods. Our approach uses ideas from rational Krylov subspace approximation to iteratively generate a reduced approximation space. This strategy provides a low-rank approximation to the solution of the stochastic Galerkin equations which can be obtained at a much lower cost and using less memory when compared to standard preconditioned iterative methods. Both of the new reduced basis algorithms proposed in this thesis allow for the efficient forward propagation of uncertainty in PDE models. We demonstrate the extent of savings achieved through comprehensive numerical experiments. Advisors/Committee Members: SILVESTER, DAVID DJ, Powell, Catherine, Silvester, David.

Subjects/Keywords: PDEs with random inputs; Reduced basis methods; Saddle point problems; Stochastic collocation; Stochastic Galerkin; Uncertainty quantification

…basis methods for parameter-dependent saddle point problems February 1, 2019 We are interested… …in the efficient numerical solution of parameter-dependent saddle point problems arising… …collocation mixed finite element methods. For saddle point problems, we need to construct two… …CHAPTER 1. INTRODUCTION 1.2 Stochastic finite element methods for saddle point problems In… …x29;), together with appropriate boundary conditions, lead to saddle point problems of…

3 more images…

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

APA (6^th Edition):

Newsum, C. J. (2019). Efficient reduced basis methods for parameter-dependent saddle point problems. (Doctoral Dissertation). University of Manchester. Retrieved from http://www.manchester.ac.uk/escholar/uk-ac-man-scw:318439

Chicago Manual of Style (16^th Edition):

Newsum, Craig James. “Efficient reduced basis methods for parameter-dependent saddle point problems.” 2019. Doctoral Dissertation, University of Manchester. Accessed February 27, 2021. http://www.manchester.ac.uk/escholar/uk-ac-man-scw:318439.

MLA Handbook (7^th Edition):

Newsum, Craig James. “Efficient reduced basis methods for parameter-dependent saddle point problems.” 2019. Web. 27 Feb 2021.

Vancouver:

Newsum CJ. Efficient reduced basis methods for parameter-dependent saddle point problems. [Internet] [Doctoral dissertation]. University of Manchester; 2019. [cited 2021 Feb 27]. Available from: http://www.manchester.ac.uk/escholar/uk-ac-man-scw:318439.

Council of Science Editors:

Newsum CJ. Efficient reduced basis methods for parameter-dependent saddle point problems. [Doctoral Dissertation]. University of Manchester; 2019. Available from: http://www.manchester.ac.uk/escholar/uk-ac-man-scw:318439

13. ZHAO RENBO. STOCHASTIC AND RANDOMIZED ALGORITHMS FOR LARGE-SCALE OPTIMIZATION IN MACHINE LEARNING.

Degree: 2018, National University of Singapore

URL: http://scholarbank.nus.edu.sg/handle/10635/148566

Subjects/Keywords: large-scale optimization; stochastic optimization; L-BFGS; primal-dual algorithms; saddle-point problems; first-order methods

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

APA (6^th Edition):

RENBO, Z. (2018). STOCHASTIC AND RANDOMIZED ALGORITHMS FOR LARGE-SCALE OPTIMIZATION IN MACHINE LEARNING. (Thesis). National University of Singapore. Retrieved from http://scholarbank.nus.edu.sg/handle/10635/148566

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):

RENBO, ZHAO. “STOCHASTIC AND RANDOMIZED ALGORITHMS FOR LARGE-SCALE OPTIMIZATION IN MACHINE LEARNING.” 2018. Thesis, National University of Singapore. Accessed February 27, 2021. http://scholarbank.nus.edu.sg/handle/10635/148566.

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

MLA Handbook (7^th Edition):

RENBO, ZHAO. “STOCHASTIC AND RANDOMIZED ALGORITHMS FOR LARGE-SCALE OPTIMIZATION IN MACHINE LEARNING.” 2018. Web. 27 Feb 2021.

Vancouver:

RENBO Z. STOCHASTIC AND RANDOMIZED ALGORITHMS FOR LARGE-SCALE OPTIMIZATION IN MACHINE LEARNING. [Internet] [Thesis]. National University of Singapore; 2018. [cited 2021 Feb 27]. Available from: http://scholarbank.nus.edu.sg/handle/10635/148566.

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

Council of Science Editors:

RENBO Z. STOCHASTIC AND RANDOMIZED ALGORITHMS FOR LARGE-SCALE OPTIMIZATION IN MACHINE LEARNING. [Thesis]. National University of Singapore; 2018. Available from: http://scholarbank.nus.edu.sg/handle/10635/148566

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

14. Wang, Bin. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.

Degree: PhD, Mathematics, 2017, University of Kansas

URL: http://hdl.handle.net/1808/27005

► Hybridizable Discontinuous Galerkin (HDG) is an important family of methods, which combine the advantages of both Discontinuous Galerkin in terms of flexibility and standard finite… (more)

Subjects/Keywords: Mathematics; BDDC; domain decomposition; hybridizable discontinuous Galerkin; saddle point problems; Stokes; weak Galerkin

…1.4 Saddle Point Problems Let V and Q be Hilbert spaces with inner products (… …of Saddle Point Problems We consider the approximation of (1.4) by finite… …x29;, hBv, qiQ∗ ,Q = b(v, q), u, v ∈ V, v ∈ V, q ∈ Q. Thus the saddle point… …saddle point problem (1.4) admits a solution (u, p) ∈ V × Q, where u ∈ V is… …solution of (1.8). Theorem 1.5.2. Existence and Uniqueness The saddle point problem…

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

APA (6^th Edition):

Wang, B. (2017). Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/27005

Chicago Manual of Style (16^th Edition):

Wang, Bin. “Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.” 2017. Doctoral Dissertation, University of Kansas. Accessed February 27, 2021. http://hdl.handle.net/1808/27005.

MLA Handbook (7^th Edition):

Wang, Bin. “Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations.” 2017. Web. 27 Feb 2021.

Vancouver:

Wang B. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. [Internet] [Doctoral dissertation]. University of Kansas; 2017. [cited 2021 Feb 27]. Available from: http://hdl.handle.net/1808/27005.

Council of Science Editors:

Wang B. Balancing domain decomposition by constraints algorithms for incompressible Stokes equations with nonconforming finite element discretizations. [Doctoral Dissertation]. University of Kansas; 2017. Available from: http://hdl.handle.net/1808/27005

15. Köster, Michael. A Hierarchical Flow Solver for Optimisation with PDE Constraints.

Degree: 2011, Technische Universität Dortmund

URL: http://hdl.handle.net/2003/29239

► Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt… (more)

▼ Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt has a significant impact on the quality of the crystal. Optimal control of the flow by electro-magnetic fields and/or boundary temperatures leads to optimisation problems with PDE constraints, which are frequently governed by the time-dependent Navier-Stokes equations. The mathematical formulation is a minimisation problem with PDE constraints. By exploiting the special structure of the first order necessary optimality conditions, the so called Karush-Kuhn-Tucker (KKT)-system, this thesis develops a special hierarchical solution approach for such equations, based on the distributed control of the Stokes – and Navier – Stokes. The numerical costs for solving the optimisation problem are only about 20-50 times higher than a pure forward simulation, independent of the refinement level. Utilising modern multigrid techniques in space, it is possible to solve a forward simulation with optimal complexity, i.e., an appropriate solver for a forward simulation needs O(N) operations, N denoting the total number of unknowns for a given computational mesh in space and time. Such solvers typically apply appropriate multigrid techniques for the linear subproblems in space. As a consequence, the developed solution approach for the optimal control problem has complexity O(N) as well. This is achieved by a combination of a space-time Newton approach for the nonlinearity and a monolithic space-time multigrid approach for 'global' linear subproblems. A second inner monolithic multigrid method is applied for subproblems in space, utilising local Pressure-Schur complement techniques to treat the saddle-point structure. The numerical complexity of this algorithm distinguishes this approach from adjoint-based steepest descent methods used to solve optimisation problems in many practical applications, which in general do not satisfy this complexity requirement. Advisors/Committee Members: Turek, Stefan.

Subjects/Keywords: Block-Glätter; Block smoother; CFD; Crank-Nicolson; Crystal growth; Czochralski; Distributed Control; Edge-oriented stabilisation; Elliptic; Elliptisch; EOJ stabilisation; EOJ Stabilisierung; FEAT; FEATFLOW; Finite Elemente; Finite Elements; First discretise then optimise; First discretize then optimize; First optimise then discretise; First optimize then discretize; Flow-Around-Cylinder; Full Newton-SAND; Heat equation; Hierarchical; Hierarchical solution concept; Hierarchisch; Hierarchisches Lösungskonzept; Inexact Newton; Inexaktes Newton-Verfahren; Instationär; Inverse Probleme; Inverse Problems; Kantenbasierte Stabilisierung; KKT system; Kristallwachstum; Krylov; Large-Scale; linear complexity; lineare Komplexität; Mehrgitter; Mehrgitter-Krylov; Monolithic; Monolithisch; Multigrid; Multigrid-Krylov; Multilevel; Navier-Stokes; Nichtparametrische Finite Elemente; Nonparametric finite elements; Nonstationary; OPTFLOW; Optimierung; Optimisation; Optimization; PDE Constraints; Raum-Zeit; saddle point; SAND; Sattelpunkt; Schur complement preconditioning; Schurkomplement-Vorkonditionierer; Space-time; SQP; Stokes; Theta schema; Theta scheme; Time-dependent; Transient; Unstructured Grids; Unstrukturierte Gitter; Vanka; Verteilte Kontrolle; Wärmeleitung; Wärmeleitungsgleichung; 510

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

APA (6^th Edition):

Köster, M. (2011). A Hierarchical Flow Solver for Optimisation with PDE Constraints. (Thesis). Technische Universität Dortmund. Retrieved from http://hdl.handle.net/2003/29239

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):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Thesis, Technische Universität Dortmund. Accessed February 27, 2021. http://hdl.handle.net/2003/29239.

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

MLA Handbook (7^th Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Web. 27 Feb 2021.

Vancouver:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Internet] [Thesis]. Technische Universität Dortmund; 2011. [cited 2021 Feb 27]. Available from: http://hdl.handle.net/2003/29239.

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

Council of Science Editors:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Thesis]. Technische Universität Dortmund; 2011. Available from: http://hdl.handle.net/2003/29239

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

16. Köster, Michael. A Hierarchical Flow Solver for Optimisation with PDE Constraints.

Degree: 2011, Technische Universität Dortmund

URL: http://dx.doi.org/10.17877/DE290R-6950

► Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt… (more)

▼ Active flow control plays a central role in many industrial applications such as e.g. control of crystal growth processes, where the flow in the melt has a significant impact on the quality of the crystal. Optimal control of the flow by electro-magnetic fields and/or boundary temperatures leads to optimisation problems with PDE constraints, which are frequently governed by the time-dependent Navier-Stokes equations. The mathematical formulation is a minimisation problem with PDE constraints. By exploiting the special structure of the first order necessary optimality conditions, the so called Karush-Kuhn-Tucker (KKT)-system, this thesis develops a special hierarchical solution approach for such equations, based on the distributed control of the Stokes – and Navier – Stokes. The numerical costs for solving the optimisation problem are only about 20-50 times higher than a pure forward simulation, independent of the refinement level. Utilising modern multigrid techniques in space, it is possible to solve a forward simulation with optimal complexity, i.e., an appropriate solver for a forward simulation needs O(N) operations, N denoting the total number of unknowns for a given computational mesh in space and time. Such solvers typically apply appropriate multigrid techniques for the linear subproblems in space. As a consequence, the developed solution approach for the optimal control problem has complexity O(N) as well. This is achieved by a combination of a space-time Newton approach for the nonlinearity and a monolithic space-time multigrid approach for 'global' linear subproblems. A second inner monolithic multigrid method is applied for subproblems in space, utilising local Pressure-Schur complement techniques to treat the saddle-point structure. The numerical complexity of this algorithm distinguishes this approach from adjoint-based steepest descent methods used to solve optimisation problems in many practical applications, which in general do not satisfy this complexity requirement. Advisors/Committee Members: Turek, Stefan (advisor), Meyer, Christian (referee).

Subjects/Keywords: Block-Glätter; Czochralski; Elliptisch; EOJ Stabilisierung; Finite Elemente; Hierarchisch; Hierarchisches Lösungskonzept; Inexaktes Newton-Verfahren; Instationär; Inverse Probleme; Kantenbasierte Stabilisierung; Kristallwachstum; Krylov; lineare Komplexität; Mehrgitter; Mehrgitter-Krylov; Monolithisch; Navier-Stokes; Nichtparametrische Finite Elemente; Optimierung; Raum-Zeit; Sattelpunkt; Schurkomplement-Vorkonditionierer; Stokes; Unstrukturierte Gitter; Vanka; Verteilte Kontrolle; Wärmeleitung; Wärmeleitungsgleichung; Block smoother; CFD; Crank-Nicolson; Crystal growth; Distributed Control; Edge-oriented stabilisation; Elliptic; EOJ stabilisation; FEAT; FEATFLOW; Finite Elements; First discretise then optimise; First discretize then optimize; First optimise then discretise; First optimize then discretize; Flow-Around-Cylinder; Full Newton-SAND; Heat equation; Hierarchical; Hierarchical solution concept; Inexact Newton; Inverse Problems; KKT system; Large-Scale; linear complexity; Monolithic; Multigrid; Multigrid-Krylov; Multilevel; Nonparametric finite elements; Nonstationary; OPTFLOW; Optimisation; Optimization; PDE Constraints; saddle point; SAND; Schur complement preconditioning; Space-time; SQP; Theta schema; Theta scheme; Time-dependent; Transient; Unstructured Grids; 510

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APA (6^th Edition):

Köster, M. (2011). A Hierarchical Flow Solver for Optimisation with PDE Constraints. (Doctoral Dissertation). Technische Universität Dortmund. Retrieved from http://dx.doi.org/10.17877/DE290R-6950

Chicago Manual of Style (16^th Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Doctoral Dissertation, Technische Universität Dortmund. Accessed February 27, 2021. http://dx.doi.org/10.17877/DE290R-6950.

MLA Handbook (7^th Edition):

Köster, Michael. “A Hierarchical Flow Solver for Optimisation with PDE Constraints.” 2011. Web. 27 Feb 2021.

Vancouver:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Internet] [Doctoral dissertation]. Technische Universität Dortmund; 2011. [cited 2021 Feb 27]. Available from: http://dx.doi.org/10.17877/DE290R-6950.

Council of Science Editors:

Köster M. A Hierarchical Flow Solver for Optimisation with PDE Constraints. [Doctoral Dissertation]. Technische Universität Dortmund; 2011. Available from: http://dx.doi.org/10.17877/DE290R-6950

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