Two-level space-time domain decomposition methods for unsteady inverse problems

TL;DR

This paper introduces a highly parallel space-time domain decomposition method for unsteady inverse problems, significantly improving computational efficiency and scalability on supercomputers for 3D convection-diffusion equations.

Contribution

It proposes a novel space-time coupled algorithm with a mixed finite element/finite difference method and new preconditioners, enabling high parallelism and robustness in inverse problem solving.

Findings

Effective recovery of unsteady moving sources.

Achieved strong scalability on supercomputers with over 1,000 processors.

Validated robustness and efficiency of the proposed method.

Abstract

As the number of processor cores on supercomputers becomes larger and larger, algorithms with high degree of parallelism attract more attention. In this work, we propose a novel space-time coupled algorithm for solving an inverse problem associated with the time-dependent convection-diffusion equation in three dimensions. We introduce a mixed finite element/finite difference method and a one-level and a two-level space-time parallel domain decomposition preconditioner for the Karush-Kuhn-Tucker (KKT) system induced from reformulating the inverse problem as an output least-squares optimization problem in the space-time domain. The new full space approach eliminates the sequential steps of the optimization outer loop and the inner forward and backward time marching processes, thus achieves high degree of parallelism. Numerical experiments validate that this approach is effective and robust for recovering unsteady moving sources. We report strong scalability results obtained on a supercomputer with more than 1,000 processors.