Geometric Variational Integrators for Multisymplectic PDEs and Adjoint Systems

Abstract

Variational integrators are a class of geometric structure-preserving numerical integrators that are based on a discretization of Hamilton’s variational principle. We construct, analyze and investigate the applications of variational integrators to multisymplectic partial differential equations and to adjoint systems.

The variational structure of multisymplectic PDEs encodes both the conservation laws admitted by these systems via Noether’s theorem and multisymplecticity, a covariant spacetime generalization of symplecticity. We develop variational integrators for these systems which preserve these properties at the discrete level, in both the Lagrangian and Hamiltonian settings. In the Lagrangian setting, we utilize compatible finite element spaces to develop these variational integrators and utilize their preservation of the de Rham complex to define discrete geometric structures associated to these integrators and naturally relate them to their continuous counterparts. In the Hamiltonian setting, we utilize a discrete Type II variational principle, based on the notion of a Type II generating functional for multisymplectic PDEs, to construct structure-preserving variational integrators for multisymplectic Hamiltonian PDEs.

Adjoint systems are ubiquitous in optimization and optimal control theory since they allow for efficient computation of sensitivities of cost functionals in optimization problems and arise as necessary conditions for optimality in optimal control problems via Pontryagin’s maximum principle. Adjoint systems admit a fundamental quadratic conservation law which is at the heart of the method of adjoint sensitivity analysis; this conservation law arises from the symplectic geometry of these systems. We develop a geometric theory for continuous and discrete adjoint systems associated to ordinary differential equations and differential-algebraic equations, by investigating their underlying symplectic and presymplectic structures, respectively. We develop a Type II variational principle for such systems at the continuous level. Subsequently, we discretize this variational principle to construct variational integrators for adjoint systems which preserve the quadratic conservation law at the discrete level and thus, allow for sensitivities of cost functions to be computed exactly. We further extend this framework to the Lie group setting and develop a variational integrator based on novel continuous and discrete Type II variational principles on cotangent bundles of Lie groups.