Open Access Publications from the University of California

## Geometric Integrators for Stiff Systems, Lie Groups and Control Systems

• Author(s): Shen, Xuefeng
• Advisor(s): Leok, Melvin
• et al.
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 codimension of the embedding is large. For the rigid body problem, the configuration space is $\mathbb{R}^3\rtimes SO(3)$, which is a Lie group. \citet{LeMcLe2005} were the first to directly use the Lie group structure of the rotation group to construct a Lie group variational integrator. In contrast, most prior approaches used the unit quaternion representation of the rotation group and applied symplectic integrators for constrained systems with the unit length constraint. In Chapter~\ref{quater}, we adopt the approach used in constructing Lie group variational integrators for rigid body dynamics on the rotation group and applied it to the unit quaternion representation. A Lie group variational integrator in the unit quaternion representation is derived, and it can be shown that our method is related to the RATTLE method applied to the rotation representation by the projection from unit quaternions to rotation matrices. The numerical results for our Lie group quaternion variational integrator are presented. The integrators constructed in Chapter~\ref{quater} are only second-order, and in Chapter~\ref{polar}, variational integrators of arbitrarily high-order on special orthogonal group $SO(n)$ are constructed by using the polar decomposition. It avoids the second-order derivative of the exponential map that arises in the traditional Lie group variational integrator method. Also, a reduced Lie--Poisson integrator is constructed. The resulting algorithms can naturally be implemented using fixed-point iteration. Numerical results are given for the case of $SO(3)$.

The last generalization is to control systems. We studied the problem of uncertainty propagation and measurement update for systems that are partially unobservable. We construct a method that satisfies the chain property that the unobservable subspace remains perpendicular to the measurement $dh$ during propagation. We characterize the unobservable subspace in terms of the group-invariance of the control system, and obtain a reduced control system on the observable variables. By decomposing the system explicitly into unobservable and observable parts $(x_N, x_O)$, the chain property can be naturally satisfied. Also, we propose a reduced Bayesian framework, where the update from the measurement is only applied to the observable variables $x_O$. In Chapter~\ref{geometric_reduce}, we consider a planar robot model, which has one odometry sensor and one camera. Odometry is used for propagation and the camera is used for measurement. In this model, the two-dimensional position as well as the orientation are all unobservable. We applied our technique to this model and performed numerical simulations. We tested this on straight line, circle, and general trajectories and found that the reduced Kalman filter that we proposed outperforms the classical Kalman filter and modifications that were proposed in the literature. In particular, it estimates the angle quite well, and as a result, yields a better estimate of the position as well.