A hybrid probabilistic domain decomposition algorithm suited for very large-scale elliptic PDEs

TL;DR

This paper introduces PDDSparse, a novel probabilistic domain decomposition method for large-scale elliptic PDEs that enhances scalability, fault tolerance, and GPU suitability by using stochastic linear systems solved via Monte Carlo simulations.

Contribution

The paper presents PDDSparse, a new probabilistic domain decomposition algorithm that replaces traditional iterative methods with a stochastic approach for improved scalability and parallelism.

Findings

Achieved scalability with up to 1536 cores in a proof of concept.

Demonstrated fault tolerance and GPU compatibility.

Reduced interfacial problem size to O(√M) for efficient parallel computation.

Abstract

State of the art domain decomposition algorithms for large-scale boundary value problems (with $M\gg 1$ degrees of freedom) suffer from bounded strong scalability because they involve the synchronisation and communication of workers inherent to iterative linear algebra. Here, we introduce PDDSparse, a different approach to scientific supercomputing which relies on a "Feynman-Kac formula for domain decomposition". Concretely, the interfacial values (only) are determined by a stochastic, highly sparse linear system $G(\omega){\vec u}={\vec b}(\omega)$ of size ${\cal O}(\sqrt{M})$, whose coefficients are constructed with Monte Carlo simulations-hence embarrassingly in parallel. In addition to a wider scope for strong scalability in the deep supercomputing regime, PDDSparse has built-in fault tolerance and is ideally suited for GPUs. A proof of concept example with up to 1536 cores is discussed in detail.