A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations

TL;DR

This paper compares 16 sparse linear system solvers in the context of severely ill-conditioned matrices from geophysical simulations, introducing a new method for condition number estimation and providing guidance on solver reliability.

Contribution

It offers a comprehensive benchmark of solvers for ill-conditioned systems, introduces the Projected Adam method for condition number estimation, and identifies the most reliable solvers for such problems.

Findings

Intel oneAPI MKL PARDISO, UMFPACK, and MUMPS are most reliable.

The Projected Adam method efficiently estimates matrix condition numbers.

Benchmark results guide solver selection for ill-conditioned systems.

Abstract

Solving sparse linear systems is a critical challenge in many scientific and engineering fields, particularly when these systems are severely ill-conditioned. This work aims to provide a comprehensive comparison of various solvers designed for such problems, offering valuable insights and guidance for domain scientists and researchers. We develop the tools required to accurately evaluate the performance and correctness of 16 solvers from 11 state-of-the-art numerical libraries, focusing on their effectiveness in handling ill-conditioned matrices. The solvers were tested on linear systems arising from a coupled hydro-mechanical marker-in-cell geophysical simulation. To address the challenge of computing accurate error bounds on the solution, we introduce the Projected Adam method, a novel algorithm to efficiently compute the condition number of a matrix without relying on eigenvalues or singular values. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO, UMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This work serves as a resource for selecting appropriate solvers, understanding the impact of condition numbers, and improving the robustness of numerical solutions in practical applications.