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You searched for subject:( Symplectic integrator). Showing records 1 – 2 of 2 total matches.

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1. Shen, Xuefeng. Geometric Integrators for Stiff Systems, Lie Groups and Control Systems.

Degree: Mathematics, 2019, University of California – San Diego

The main idea of a geometric integrator is to adopt a geometric viewpoint of the problem and to construct integrators that preserve the geometric properties of the continuous dynamical system. For classical mechanics, both the Lagrangian and the Hamiltonian formulations can be described using the language of geometry. Due to the rich conservation properties of mechanics, it is natural to study the construction of numerical integrators that preserve some geometric properties, such as the symplectic structure, energy, and momentum maps. Such geometric structure-preserving numerical integrators exhibit nice properties compared to traditional numerical methods. This is especially true in galaxy simulations and molecular dynamics, where long time simulations are required to answer the corresponding scientific questions. Variational integrators have attracted interest in the geometric integration community as it discretizes Hamilton's principle, as opposed to the corresponding differential equation, to obtain a numerical integrator that is automatically symplectic, and which exhibits a discrete Noether's theorem. Besides classical mechanics, such an approach has also been applied to other fields, such as optimal control~, partial differential equations~, stochastic differential equations~, and so on. In this thesis, we consider generalizations of geometric integrators that are adapted to three special settings. One is the case of stiff systems of the form, \dot{q} = Aq + f(q), where the coefficient matrix A has a large spectral radius that is responsible for the stiffness of the system, while the nonlinear term f(q) is relatively smooth. Traditionally, exponential integrators have been used to address the issue of stiffness. In Chapter~\ref{exp}, we consider a special semilinear problem with A=JD, f(q)=J∇ V(q), where JT = -J, DT=D, and JD=DJ. Then, the system is described by \dot{q} = J(Dq+∇ V(q)), which naturally arises from the discretization of Hamiltonian partial differential equations. It is a constant Poisson system with Poisson structure Jij\frac{\partial}{\partial xi}\otimes \frac{\partial}{\partial xj}, and Hamiltonian H(q) = \frac{1}{2}qTDq + V(q). Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs are presented. These geometric exponential integrators exhibit better long time stability properties as compared to non-geometric integrators, and are computationally more efficient than traditional symplectic integrators and energy-preserving methods based on the discrete gradient method. The other generalization is to Lie groups. When configuration manifold is a Lie group, we would like to utilize the group structure rather than simply regard it as embedded submanifold. This is particularly useful when…

Subjects/Keywords: Mathematics; geometric reduction; kalman filter; lie group; stiff system; symplectic integrator; variational integrator

…3.3 Lie group variational integrator… …3.3.2 Variational integrator on the Lagrangian side… …3.3.3 Variational integrator on the Hamiltonian side… …61 61 64 64 66 68 68 71 74 75 77 79 High-Order Symplectic Lie Group Methods on SO(n… …93 Hamiltonian variational integrator on the rotation group SO(n)… 

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

APA (6th Edition):

Shen, X. (2019). Geometric Integrators for Stiff Systems, Lie Groups and Control Systems. (Thesis). University of California – San Diego. Retrieved from http://www.escholarship.org/uc/item/9g2730gd

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

Chicago Manual of Style (16th Edition):

Shen, Xuefeng. “Geometric Integrators for Stiff Systems, Lie Groups and Control Systems.” 2019. Thesis, University of California – San Diego. Accessed December 05, 2020. http://www.escholarship.org/uc/item/9g2730gd.

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

MLA Handbook (7th Edition):

Shen, Xuefeng. “Geometric Integrators for Stiff Systems, Lie Groups and Control Systems.” 2019. Web. 05 Dec 2020.

Vancouver:

Shen X. Geometric Integrators for Stiff Systems, Lie Groups and Control Systems. [Internet] [Thesis]. University of California – San Diego; 2019. [cited 2020 Dec 05]. Available from: http://www.escholarship.org/uc/item/9g2730gd.

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

Council of Science Editors:

Shen X. Geometric Integrators for Stiff Systems, Lie Groups and Control Systems. [Thesis]. University of California – San Diego; 2019. Available from: http://www.escholarship.org/uc/item/9g2730gd

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


Universitat Politècnica de València

2. 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 December 05, 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. 05 Dec 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 Dec 05]. 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

.