On time-splitting methods for gradient flows with two dissipation mechanisms

TL;DR

This paper introduces a split-step method for solving gradient flows with two dissipation mechanisms, allowing solutions to be constructed by alternating between simpler gradient systems, with convergence guaranteed by energy-dissipation principles.

Contribution

It develops a novel split-step approach for gradient flows with dual dissipation potentials, enabling solution construction via semiflows and minimizing movements with proven convergence.

Findings

Solutions can be constructed by alternating simpler gradient flows.

Convergence is established using energy-dissipation principles.

Method applies to Banach space gradient systems with combined dissipation.

Abstract

We consider generalized gradient systems in Banach spaces whose evolutions are generated by the interplay between an energy functional and a dissipation potential. We focus on the case in which the dual dissipation potential is given by a sum of two functionals and show that solutions of the associated gradient-flow evolution equation with combined dissipation can be constructed by a split-step method, i.e. by solving alternately the gradient systems featuring only one of the dissipation potentials and concatenating the corresponding trajectories. Thereby the construction of solutions is provided either by semiflows, on the time-continuous level, or by using Alternating Minimizing Movements in the time-discrete setting. In both cases the convergence analysis relies on the energy-dissipation principle for gradient systems.