A Multigrid-like Algorithm for Probabilistic Domain Decomposition

TL;DR

This paper introduces a multigrid-inspired iterative algorithm for Probabilistic Domain Decomposition that accelerates convergence by using improved approximations as control variates, significantly reducing Monte Carlo error and computational time.

Contribution

The paper proposes a novel multigrid-like iterative scheme for PDD that predicts speedup efficiently and enhances stability, achieving substantial performance improvements.

Findings

Achieved 10-100x speedup over previous PDD methods.

Demonstrated effectiveness through a numerical example.

Provided theoretical insights into PDD stability.

Abstract

We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control variates in order to reduce the Monte Carlo error of the following iterates--resulting in an overall acceleration of PDD for a given error tolerance. The key ingredient of the proposed algorithm is the ability to approximately predict the speedup with little computational overhead and in parallel. Besides, the theoretical framework allows to explore other aspects of PDD, such as stability. One numerical example is worked out, yielding an improvement of between one and two orders of magnitude over the previous version of PDD.