Low-Rank SPIKE Framework for Solving Large Sparse Linear Systems with Applications

TL;DR

This work extends the SPIKE framework by introducing a low-rank SVD-based approximation of the SPIKE spikes, enabling scalable handling of wide sparse bands. The LR-SPIKE family (LR-SPIKE-I, LR-SPIKE-T, LR-SPIKE-OTF) provides flexible preconditioners and, in some cases, approximate direct solvers that outperform Block Jacobi and compete with ILU. Comprehensive experiments on SuiteSparse matrices, a Sherman5 case study, and FEAST-enabled eigenvalue problems within the NESSIE framework demonstrate improved convergence, parallel scalability, and effective integration with FEAST’s contour-based solves. The results indicate substantial practical impact for large-scale sparse systems in engineering simulations and electronic-structure calculations, with traceable gains in solve times and reduced iteration counts.

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 non-zeros 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. There 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.

Figures (7)

• Figure 1: Visualization of Sherman5 matrix before and after reordering. Darker areas indicate a greater concentration of relatively large magnitude elements. A blur has been applied and magnitude is plotted on a log scale, to better show overall structure.
• Figure 2: Characterizing the difference between using the SVD of the $\boldsymbol{T}_{i}$ and $\boldsymbol{W}_{i}$ versus $\boldsymbol{B}_{i}$ and $\boldsymbol{C}_{i}$. $\boldsymbol{T}_{i}$ and $\boldsymbol{W}_{i}$ result in a superior preconditioner due to faster singular value decay.
• Figure 3: Exploration of preconditioner quality for Sherman5
• Figure 4: Visualization of reordering for matrix sme3Da.
• Figure 5: Comparison of LR-SPIKE-T and Block Jacobi for preconditioning BiCGTtab iterations when solving NESSIE Poisson matrices. Markers indicate Conjugate Gradient or BiCGStab iterations.