Variational integrators are geometric structure-preserving numerical
methods that preserve the symplectic structure, satisfy a discrete
Noether's theorem, and exhibit exhibit excellent long-time energy
stability properties. An exact discrete Lagrangian arises from
Jacobi's solution of the Hamilton-Jacobi equation, and it generates
the exact flow of a Lagrangian system. By approximating the exact
discrete Lagrangian using an appropriate choice of interpolation space
and quadrature rule, we obtain a systematic approach for constructing
variational integrators. The convergence rates of such variational
integrators are related to the best approximation properties of the
interpolation space.
Many gauge field theories can be formulated variationally using a
multisymplectic Lagrangian formulation, and we will present a
characterization of the exact generating functionals that generate the
multisymplectic relation. By discretizing these using
group-equivariant spacetime finite element spaces, we obtain methods
that exhibit a discrete multimomentum conservation law. We will then
briefly describe an approach for constructing group-equivariant
interpolation spaces that take values in the space of Lorentzian
metrics that can be efficiently computed using a generalized polar
decomposition. The goal is to eventually apply this to the
construction of variational discretizations of general relativity,
which is a second-order gauge field theory whose configuration
manifold is the space of Lorentzian metrics.
Melvin Leok is a professor in the Department of Mathematics at the
University of California, San Diego. His research interests are in
computational geometric mechanics, computational geometric control
theory, discrete geometry, and structure-preserving numerical schemes,
and particularly how these subjects relate to systems with symmetry.
He received his Ph.D. in 2004 from the California Institute of
Technology in Control and Dynamical Systems under the direction of
Jerrold Marsden. He is a three-time NAS Kavli Frontiers of Science
Fellow, and has received the NSF Faculty Early Career Development
(CAREER) award, the SciCADE New Talent Prize, the SIAM Student Paper
Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis.
He has given plenary talks at Foundations of Computational
Mathematics, NUMDIFF, and the IFAC Workshop on Lagrangian and
Hamiltonian Methods for Nonlinear Control. He serves on the editorial
boards of the Journal of Nonlinear Science, the SIAM Journal on
Control and Optimization, the LMS Journal of Computation and
Mathematics, the Journal of Geometric Mechanics, and the Journal of
Computational Dynamics.