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.

翻译：求解稀疏线性系统是众多科学与工程领域中的关键挑战，尤其是在系统严重病态的情况下。本研究旨在对针对此类问题设计的多种求解器进行全面比较，为领域科学家和研究人员提供有价值的见解与指导。我们开发了必要的工具，以准确评估来自11个先进数值库的16种求解器的性能与正确性，重点关注其在处理病态矩阵方面的有效性。这些求解器在耦合流体力学标记点网格地球物理模拟所产生的线性系统上进行了测试。为应对计算解精确误差界的挑战，我们提出了投影Adam方法，这是一种无需依赖特征值或奇异值即可高效计算矩阵条件数的新算法。我们的基准测试结果表明，Intel oneAPI MKL PARDISO、UMFPACK和MUMPS在所测试场景中是最可靠的求解器。本工作为选择合适的求解器、理解条件数的影响以及提升实际应用中数值解的鲁棒性提供了参考资源。