Adaptive Hamiltonian Variational Integrators and Applications to Symplectic Accelerated Optimization

Abstract

It is well known that symplectic integrators lose their near energy preservation properties when variable time-steps are used. The most common approach to combining adaptive time-steps and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincaré transformation. Since the transformed Hamiltonian is typically degenerate, the use of Hamiltonian variational integrators based on Type II or Type III generating functions is required instead of the more traditional Lagrangian variational integrators based on Type I generating functions. Error analysis is provided, and numerical tests based on the Taylor variational integrator approach in [J. M. Schmitt, T. Shingel, and M. Leok, BIT, 58 (2018), pp. 457--488] to time-adaptive variational integration of Kepler's 2-body problem are presented. Finally, we use our adaptive framework together with the variational approach to accelerated optimization presented in [A. Wibisono, A. Wilson, and M. Jordan, Proc. Natl. Acad. Sci. USA, 113 (2016), pp. E7351--E7358] to design efficient variational and nonvariational explicit integrators for symplectic accelerated optimization.