Adaptive Hamiltonian Variational Integrators and Symplectic Accelerated Optimization

TL;DR

This paper develops a framework for adaptive symplectic integrators using Poincaré transformations, enabling energy-preserving variable step size integration and applications to accelerated optimization.

Contribution

It introduces a novel variational integrator construction for degenerate Hamiltonians via Poincaré transformation, with error analysis and applications to optimization.

Findings

Effective adaptive integrators for Kepler's problem demonstrated

Error bounds established for the proposed methods

Accelerated optimization algorithms show improved efficiency

Abstract

It is well known that symplectic integrators lose their near energy preservation properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincar\'e transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincar\'e 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 of Schmitt, Shingel, Leok (2018) 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 Wibisono, Wilson, Jordan (2016) to design efficient variational and non-variational explicit integrators for symplectic accelerated optimization.