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You searched for +publisher:"Universitat Politècnica de València" +contributor:("Blanes Zamora, Sergio"). Showing records 1 – 3 of 3 total matches.

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Universitat Politècnica de València

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

Degree: 2014, Universitat Politècnica de València

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 (6th 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 (16th Edition):

Bader, Philipp Karl-Heinz. “Geometric Integrators for Schrödinger Equations .” 2014. Doctoral Dissertation, Universitat Politècnica de València. Accessed October 30, 2020. http://hdl.handle.net/10251/38716.

MLA Handbook (7th Edition):

Bader, Philipp Karl-Heinz. “Geometric Integrators for Schrödinger Equations .” 2014. Web. 30 Oct 2020.

Vancouver:

Bader PK. Geometric Integrators for Schrödinger Equations . [Internet] [Doctoral dissertation]. Universitat Politècnica de València; 2014. [cited 2020 Oct 30]. 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

2. Seydaoglu, Muaz. Splitting methods for autonomous and non-autonomous perturbed equations .

Degree: 2016, Universitat Politècnica de València

[EN] This thesis addresses the treatment of perturbed problems with splitting methods. After motivating these problems in Chapter 1, we give a thorough introduction in Chapter 2, which includes the objectives, several basic techniques and already existing methods. In Chapter 3, we consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods for several numerical examples and present some new improved schemes. In Chapter 4, we propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum A = D+epsilon*B of a sparse and efficiently exponentiable matrix D with sparse exponential exp(D) and a dense matrix epsilon*B which is of small norm in comparison with D. The predominant algorithm is based on scaling the large matrix A by a small number 2^(-s) , which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbation matrix B in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought. In Chapter 5, we consider the numerical integration of the perturbed Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this chapter we present new sixth-and eighth-order symplectic exponential integrators that are tailored to the Hill's equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. Several numerical examples illustrate the performance of the new methods. Conclusions and pointers to further research are detailed in Chapter 6.; [ES] Esta tesis aborda el tratamiento de problemas perturbados con métodos de escisión… Advisors/Committee Members: Bader, Philipp Karl-Heinz (advisor), Blanes Zamora, Sergio (advisor).

Subjects/Keywords: Geometric integration; Splitting method; Perturbed problems; Non-reversible systems; Matrix exponential; Hill's equation

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

APA (6th Edition):

Seydaoglu, M. (2016). Splitting methods for autonomous and non-autonomous perturbed equations . (Doctoral Dissertation). Universitat Politècnica de València. Retrieved from http://hdl.handle.net/10251/71358

Chicago Manual of Style (16th Edition):

Seydaoglu, Muaz. “Splitting methods for autonomous and non-autonomous perturbed equations .” 2016. Doctoral Dissertation, Universitat Politècnica de València. Accessed October 30, 2020. http://hdl.handle.net/10251/71358.

MLA Handbook (7th Edition):

Seydaoglu, Muaz. “Splitting methods for autonomous and non-autonomous perturbed equations .” 2016. Web. 30 Oct 2020.

Vancouver:

Seydaoglu M. Splitting methods for autonomous and non-autonomous perturbed equations . [Internet] [Doctoral dissertation]. Universitat Politècnica de València; 2016. [cited 2020 Oct 30]. Available from: http://hdl.handle.net/10251/71358.

Council of Science Editors:

Seydaoglu M. Splitting methods for autonomous and non-autonomous perturbed equations . [Doctoral Dissertation]. Universitat Politècnica de València; 2016. Available from: http://hdl.handle.net/10251/71358


Universitat Politècnica de València

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

Degree: 2019, Universitat Politècnica de València

[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 (6th 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 (16th Edition):

Kopylov, Nikita. “Magnus-based geometric integrators for dynamical systems with time-dependent potentials .” 2019. Doctoral Dissertation, Universitat Politècnica de València. Accessed October 30, 2020. http://hdl.handle.net/10251/118798.

MLA Handbook (7th Edition):

Kopylov, Nikita. “Magnus-based geometric integrators for dynamical systems with time-dependent potentials .” 2019. Web. 30 Oct 2020.

Vancouver:

Kopylov N. Magnus-based geometric integrators for dynamical systems with time-dependent potentials . [Internet] [Doctoral dissertation]. Universitat Politècnica de València; 2019. [cited 2020 Oct 30]. 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

.