Symplectic integrators and optimal control

When selecting a numerical method to integrate an ODE system, it is intuitively clear that preservation of geometric properties is desirable. The particular subclasses of ODE systems we will consider are Lagrangian and Hamiltonian systems. The dynamical equations for these derive from variational principles, and we obtain structure preserving integrators by discretizing the principles rather than the ODEs they generate. We demonstrate some advantages that these symplectic integrators have over methods that are more rudimentary by looking at some examples from optimal control theory. Our major motivation for considering symplectic integrators is solving an image registration problem, where, using the least effort, we associate a set of landmark points on one image to a corresponding set of points on another. A mathematical formulation of this problem is as a Hamiltonian system; this becomes apparent once we realize that we are computing the motion of particles (the landmark points) under some appropriate potential function. We investigate the performance of symplectic methods on this, more complex, problem. We show that by formulating the problem as a system of nonlinear equations rather than one of optimal control, the explicit Euler method performs better than the symplectic integrators, especially on a set of data points generated by a real experiment. We give some evidence that the higher-order methods in Matlab's ODE suite may be better still, but we do not pursue this line of investigation in any detail