Low‐Rank SPIKE Framework for Solving Large Sparse Linear Systems With Applications

ABSTRACT The SPIKE family of linear system solvers provides parallelism using a block tridiagonal partitioning. Typically SPIKE‐based solvers are applied to banded systems, resulting in structured off‐diagonal blocks with nonzeros elements restricted to relatively small submatrices comprising the band of the original matrix. In this work, a low‐rank SVD based approximation of the off‐diagonal blocks is investigated. This produces a representation which more effectively handles matrices with large, sparse bands. A set of flexible distributed solvers, the LR‐SPIKE variants, are implemented. These are applicable to a wide range of applications—from use as a “black‐box” preconditioner which straightforwardly improves upon the classic Block Jacobi preconditioner, to use as a specialized “approximate direct solver.” An investigation of the effectiveness of the new preconditioners for a selection of SuiteSparse matrices is performed, particularly focusing on matrices derived from 3D finite element simulations. In addition, the SPIKE approximate linear system solvers are also paired with the FEAST eigenvalue solver, where they are shown to be particularly effective due to the former's rapid convergence, and the latter's acceptance of loose linear system solver convergence, resulting in a combination which requires very few solver iterations.