Two novel classes of arbitrary high-order structure-preserving algorithms for canonical Hamiltonian systems

In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes

The bracket geometry of statistics

In this thesis we build a geometric theory of Hamiltonian Monte Carlo, with an emphasis on symmetries and its bracket generalisations, construct the canonical geometry of smooth measures and Stein operators, and derive the complete recipe of measure-constraints preserving dynamics and diffusions on arbitrary manifolds. Specifically, we will explain the central role played by mechanics with symmetries to obtain efficient numerical integrators, and provide a general method to construct explicit integrators for HMC on geodesic orbit manifolds via symplectic reduction. Following ideas developed by Maxwell, Volterra, Poincaré, de Rham, Koszul, Dufour, Weinstein, and others, we will then show that any smooth distribution generates considerable geometric content, including ``musical" isomorphisms between multi-vector fields and twisted differential forms, and a boundary operator - the rotationnel, which, in particular, engenders the canonical Stein operator. We then introduce the ``bracket formalism" and its induced mechanics, a generalisation of Poisson mechanics and gradient flows that provides a general mechanism to associate unnormalised probability densities to flows depending on the score pointwise. Most importantly, we will characterise all measure-constraints preserving flows on arbitrary manifolds, showing the intimate relation between measure-preserving Nambu mechanics and closed twisted forms. Our results are canonical. As a special case we obtain the characterisation of measure-preserving bracket mechanical systems and measure-preserving diffusions, thus explaining and extending to manifolds the complete recipe of SGMCMC. We will discuss the geometry of Stein operators and extend the density approach by showing these are simply a reformulation of the exterior derivative on twisted forms satisfying Stokes' theorem. Combining the canonical Stein operator with brackets allows us to naturally recover the Riemannian and diffusion Stein operators as special cases. Finally, we shall introduce the minimum Stein discrepancy estimators, which provide a unifying perspective of parameter inference based on score matching, contrastive divergence, and minimum probability flow.Open Acces

On the Geometry of the Hamilton-Jacobi equation

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 27-11-2015The classical Hamilton-Jacobi theory is well-understood from the symplectic geometry viewpoint. By Hamilton-Jacobi theory we mean the relation between certain PDE, the Hamilton-Jacobi equation, and Hamilton's equations, an ODE. These relations provide means to integrate the Hamilton's equations (or approximate them through canonical transformations). The main goal of this work is to extend the Hamilton-Jacobi theory to di erent geometric frameworks (reduction, Poisson, almost-Poisson, presymplectic...) and obtain new ways, analytic and numeric, to integrate Hamilton's equations in the corresponding geometric settings. Furthermore, one of the main points of this work is to develop a geometric setting where new numerical methods can be built on. In Chapter 1 we sketch the historical development of the Hamilton-Jacobi theory. We brie y introduce some of the connections of the theory with optics and other analytical, geometrical and dynamical issues. In this chapter, we introduce several viewpoints and sketch their connections: analytic, geometric and dynamic. We emphasize the role of lagrangian submanifolds, because lagrangian submanifolds will be the keystone to achieve our goals. In Chapter 2 we develop a reduction theory for the Hamilton-Jacobi equation. Reduction is one of the oldest and most useful techniques in geometric mechanics, so it is natural to wonder if that theory can be extended to the Hamilton-Jacobi theory, and take advantage of it. Ge and Marsden attempted to solve that question in [33]. We propose a new approach, general enough to include the previous ones and to give new insights to develop more applications. In Chapter 3, based on symplectic groupoids, we construct a Hamilton-Jacobi theory for linear Poisson structures (duals of Lie algebroids). This framework is very interesting in order to integrate analytically and numerically Hamilton's equations, and it solves some previous questions of the area. We review and complete some previous works by Channell, Ge, Marsden, Scovel and Weinstein. In Chapter 4 we present a Hamilton-Jacobi theory for almost-Poisson manifolds. The main objective is to understand from a purely geometric way the Hamilton-Jacobi theory for non-holonomic systems in [18, 40, 48]. Chapter 5 includes some extensions of the Hamilton-Jacobi theory, in order to deal with singular lagrangians. Singular lagrangians are common in classical eld theory, and so understanding them in the classical mechanics context seems to be a natural step. Finally, some conclusions and future work are analyzed in the last part of this thesisLa teoría clásica de Hamilton-Jacobi es, hoy en día, bien conocida desde el punto de vista de la geometría simpléctica. A lo largo de esta memoria por teoría de Hamilton-Jacobi se entiende la relación entre cierta EDP, la ecuación de Hamilton-Jacobi, y las ecuaciones de Hamilton (EDO). Ello proporciona medios para integrar las ecuaciones de Hamilton, o aproximarlas a través de transformaciones canónicas. La meta principal de este trabajo es extender esa teoría a otros contextos (reducción, Poisson, almost-Poisson, presimpléctico...) y obtener nuevas formas, analíticas y numéricas, de integrar las ecuaciones de Hamilton en otros marcos geométricos. Más aún, uno de los puntos principales tratados aquí es el desarrollo de herramientas geométricas para la implementación de nuevos métodos numéricos. En el Capítulo 1 esbozamos el desarrollo histórico de la teoría de Hamilton-Jacobi. Introducimos brevemente algunas de las conexiones de la teoría con la óptica y otros temas analíticos y dinámicos. Enfatizamos el papel de las subvariedades lagrangianas, porque dichas subvariedades serán la piedra angular para alcanzar nuestros objetivos. En el Capítulo 2 presentamos una teoría de reducción de la ecuación de Hamilton- Jacobi. La teoría de reducción es una de las más antiguas y útiles técnicas de la mecánica geométrica, por lo que es muy natural preguntarse si dicha teoría puede ser combinada con la teoría de Hamilton-Jacobi y así obtener nuevos resultados basándose en ella. Ge y Marsden dieron los primeros pasos en esta dirección en [33]. Nosotros proponemos una nueva aproximación, suficientemente general como para incluir los resultados previos, pero al mismo tiempo damos una nueva visión y más aplicaciones. En el Capítulo 3, usando grupoides simplécticos construimos una teoría de Hamilton- Jacobi para estructuras de Poisson linales (en el dual de un algebroide de Lie). Esta visión es muy interesante de cara a la integración analítica y numérica de las ecuaciones de Hamilton y resuelve algunas de las cuestiones del área. Revisamos y completamos trabajos previos de Channell, Ge, Marsden, Scovel y Weinstein. En el Capítulo 4 presentamos una teoría de Hamilton-Jacobi para estructuras almost-Poisson La meta principal es entender desde un punto de vista enteramente geométrico la teoría de Hamilton-Jacobi para sistemas no-holónomos desarrollada en [18, 40, 48]. El Capítulo 5 incluye algunas extensiones de Hamilton-Jacobi para tratar con lagrangianos singulares. Los lagrangianos singulares son comunes en la teoría clásica de campos y por ello entenderlos en el ámbito de la mecánica clásica parece el primer paso a tomar. Finalmente, discutimos las conclusiones y una perspectiva de trabajo futur

Symplectic structure-preserving integrators for the two-dimensional Gross–Pitaevskii equation for BEC

AbstractSymplectic integrators have been developed for solving the two-dimensional Gross–Pitaevskii equation. The equation is transformed into a Hamiltonian form with symplectic structure. Then, symplectic integrators, including the midpoint rule, and a splitting symplectic scheme are developed for treating this equation. It is shown that the proposed codes fulfill the discrete charge conservation law. Furthermore, the global error of the numerical solution is theoretically estimated. The theoretical analysis is supported by some numerical simulations

Multi-Symplectic Integrators for Nonlinear Wave Equations

Symplectic (area-preserving) integrators for Hamiltonian ordinary differential equations have shown to be robust, efficient and accurate in long-term calculations. In this thesis, we show how symplectic integrators have a natural generalization to Hamiltonian PDEs by introducing the concept of multi-symplectic partial differential equations (PDEs). In particular, we show that multi-symplectic PDEs have an underlying spatio-temporal multi-symplectic structure characterized by a multi-symplectic conservation law MSCL). Then multi-symplectic integrators (MSIs) are numerical schemes that preserve exactly the MSCL. Remarkably, we demonstrate that, although not designed to do so, MSIs preserve very well other associated local conservation laws and global invariants, such as the energy and the momentum, for very long periods of time. We develop two types of MSIs, based on finite differences and Fourier spectral approximations, and illustrate their superior performance over traditional integrators by deriving new numerical schemes to the well known 1D nonlinear Schrödinger and sine-Gordon equations and the 2D Gross-Pitaevskii equation. In sensitive regimes, the spectral MSIs are not only more accurate but are better at capturing the spatial features of the solutions. In particular, for the sine-Gordon equation we show that its phase space, as measured by the nonlinear spectrum associated with it, is better preserved by spectral MSIs than by spectral non-symplectic Runge-Kutta integrators. Finally, to further understand the improved performance of MSIs, we develop a backward error analysis of the multi-symplectic centered-cell discretization for the nonlinear Schrödinger equation. We verify that the numerical solution satisfies to higher order a nearby modified multi-symplectic PDE and its modified multi-symplectic energy conservation law. This implies, that although the numerical solution is an approximation, it retains the key feature of the original PDE, namely its multi-symplectic structure