Scalable block methods and preconditioning for hardware efficient sparse eigensolutions

Abstract

Block variants of the Jacobi-Davidson method for computing a few extreme eigenpairs of a large sparse matrix are known to improve the robustness of the standard algorithm, but are generally shunned because the total number of floating-point operations increases. We present a scalable implementation of a block Jacobi-Davidson solver with blocked GMRES preconditioning. By detailed performance engineering and numerical experiments we demonstrate that the increase in operations is typically more than compensated by the performance gains of the block operations on modern parallel hardware architectures. Thus the block solver is both more efficient and robust than its single vector counterpart.

Moreover, we discuss a row projection method for the solution of ill-conditioned linear systems encountered in the FEAST algorithm for computing inner eigenpairs. We introduce a novel hybridly parallel and fully iterative implementation of the eigenvalue solver which exploits parallelism on several levels and present numerical and performance results.