Efficient Contour Integral-based Eigenvalue Computation Using an Iterative Linear Solver with Shift-Invert Preconditioning

TL;DR: In this article, a shift-invert preconditioning method was proposed to take advantage of the shift-invariance of the block Krylov subspace to reduce the number of parallel processes.

Abstract: Contour integral-based (CI) eigenvalue solvers are one of the efficient and robust approaches for sparse eigenvalue problems. They have attracted attention owing to their inherent parallelism. For implementing a CI eigensolver, the inner linear systems arising in the algorithm need to be solved using an efficient method. One widely-used method is to use a sparse direct linear solver provided by a well-established numerical library; it is numerically robust and presents good load balancing of parallel execution of the CI eigensolver. However, owing to high total computational and memory cost, the performance of the direct solver approach is suboptimal. In this study, we propose an alternative method that utilizes a block Krylov iterative linear solver and shift-invert preconditioning that can take advantage of the shift-invariance of the block Krylov subspace. Our approach adaptively sets a preconditioning parameter according to the number of parallel processes to reduce the iteration counts. Several numerical examples confirm that our method outperforms the direct solver approach.