Inexact matrix exponential preconditioner for implicitly restarted Arnoldi method in fluid dynamics stability problems for parallel heterogeneous architecture

Abstract

We wish to solve the problem of linear stability analysis of stationary solutions for a fluid dynamics problem governed by 3D Navier Stokes equations. The discrete problem is of order O(106) - O(107) so we use parallel computations to perform linear analysis. We aim at the heterogeneous computational architecture based on central (CPU) and graphical (GPU) processing units. In the paper we use the implicitly restarted Arnoldi method (IRAM) to recover the most dangerous eigenvalues. The IRAM has poor convergence without specific transformation of the matrix spectrum. Standard methods require either long computations or inverse of the transformed matrix. This can be troublesome in fluid dynamics problems where matrix is not available explicitly and is non-symmetric, ill-conditioned and can be fully filled. We propose the usage of the inexact matrix exponential with shift-inverse transformation using only matrix-vector product. The inverse is performed using Krylov methods using only matrix-vector product. We prove convergence bounds for the GMRES method and show that this type of matrix transformation accelerates GMRES convergence. We demonstrate application of the method in recovering eigenvalues close to the imaginary axis in some model problems on some matrices from the Matrix Market. We also demonstrate the application of the method for some fluid dynamics problems. We demonstrate convergence, acceleration and efficiency on the multiple GPU cluster.