Optimal solvers for linear systems with fractional powers of sparse SPD matrices

Summary In this paper, we consider efficient algorithms for solving the algebraic equation , 0< α <1, where is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in , d =1,2,3. This solution is then written as with with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function t β − α for 0< t ≤1 and on the assumption that one has at hand an efficient method (e.g., multigrid, multilevel, or other fast algorithms) for solving equations such as , c ≥0. The provided numerical experiments confirm the efficiency of the proposed algorithms.