A dynamic domain decomposition for a class of second order semi-linear equations

TL;DR

This paper introduces a parallel algorithm using dynamic domain decomposition for efficiently solving second order semi-linear equations from stochastic control, extending previous methods for deterministic cases and demonstrating improved speed under certain conditions.

Contribution

It extends the patchy domain decomposition method to second order equations with diffusion, incorporating a modified semi-Lagrangian scheme and demonstrating faster parallel computation.

Findings

Parallel dynamic decomposition outperforms static in suitable conditions.

The method effectively handles degenerate diffusion in stochastic control problems.

Numerical tests validate the efficiency and features of the proposed approach.

Abstract

We propose a parallel algorithm for the numerical solution of a class of second order semi-linear equations coming from stochastic optimal control problems, by means of a dynamic domain decomposition technique. The new method is an extension of the patchy domain decomposition method presented in a previous work for first order Hamilton-Jacobi-Bellman equations related to deterministic optimal control problems. The semi-Lagrangian scheme underlying the original method is modified in order to deal with (possibly degenerate) diffusion, by approximating the stochastic optimal control problem associated to the equation via discrete time Markov chains. We show that under suitable conditions on the discretization parameters and for sufficiently small values of the diffusion coefficient, the parallel computation on the proposed dynamic decomposition is faster than that on a static decomposition. To this end, we combine the parallelization with some well known techniques in the context of fast-marching-like methods for first order Hamilton-Jacobi equations. Several numerical tests in dimension two are presented, in order to show the features of the proposed method.