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 Author Shen, Xuefeng Title Geometric Integrators for Stiff Systems, Lie Groups and Control Systems URL http://www.escholarship.org/uc/item/9g2730gd Publication Date 2019 Discipline/Department Mathematics University/Publisher University of California – San Diego Abstract 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~\cite{junge2005discrete,leyendecker2010discrete}, partial differential equations~\cite{marsden1998multisymplectic}, stochastic differential equations~\cite{bou2009stochastic}, 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 $A=JD$, $f(q)=J\nabla V(q)$, where $J^T = -J, D^T=D$, and $JD=DJ$. Then, the system is described by $\dot{q} = J(Dq+\nabla V(q))$, which naturally arises from the discretization of Hamiltonian partial differential equations. It is a constant Poisson system with Poisson structure $J_{ij}\frac{\partial}{\partial x_i}\otimes \frac{\partial}{\partial x_j}$, and Hamiltonian $H(q) = \frac{1}{2}q^TDq + V(q)$. 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 Language en Rights public Country of Publication us Format application/pdf Record ID california:qt9g2730gd Other Identifiers qt9g2730gd Repository california Date Retrieved 2019-04-03 Date Indexed 2019-04-03

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…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…

integrator. In contrast, most prior approaches used the unit quaternion representation xiv of the rotation group and applied symplectic integrators for constrained systems with the unit length constraint. In Chapter 3, we adopt the approach used in…

…condition that bi 6= 0 now implies that ai1 = bi . 1.4 Variational integrators Surprisingly, all the symplectic integrators we have introduced so far, including the constrained and unconstrained cases, can be derived using a variational integrator

…variational integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Continuous time equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Variational integrator on the Lagrangian…

…side . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Variational integrator on the Hamiltonian side . . . . . . . . . . . . . . . . . . . . . . 3.4 Implementation of the algorithm…

…64 64 66 68 68 71 74 75 77 79 High-Order Symplectic Lie Group Methods on SO(n) using the Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Introduction…

…variational integrator on the rotation group SO(n) . . . . . . . . . . . . . . . 101 Numerical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Chapter 4 4.1 4.2 4.3 4.4 4.5 Chapter 5…

…applied to the nonlinear Schrödinger equation, n = 161, h = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Error plots for the energy preserving exponential integrator applied to the nonlinear Schrödinger equation, n = 161, h = 0.1…
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