Efficient computation of matrix power-vector products: application for space-fractional diffusion problems (Q1726433)

The authors present a new approach for the computation of an action \(A^\alpha v\) of a symmetric positive definite sparse matrix \(A\) raised to power \(\alpha>0\) on a given vector \(v\). The proposed approach deals with a direct decomposition of \(\mathbb{R}^n\) into two mutually orthogonal subspaces. The first one is spanned by the eigenvectors of \(A\) corresponding to the smallest and the largest eigenvalues; the second subspace is spanned by the eigenvectors corresponding to all the remaining eigenvalues of \(A\). The matrix-vector product of the matrix-power \(A^\alpha\) with the vector \(v\) is done directly in the first subspace, by employing a spectral decomposition. In the other subspace, a power-series approximation of the matrix-power is employed. The authors provide several numerical experiments.