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

Abstract

Summary In this paper, we consider efficient algorithms for solving the algebraic equation Aαboldu=boldf, 0<α<1, where scriptA is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in Rd, d=1,2,3. This solution is then written as boldu=Aβ−αboldF with boldF=A−βboldf with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function tβ−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 false(scriptA+cscriptIfalse)boldu=boldF, c≥0. The provided numerical experiments confirm the efficiency of the proposed algorithms.

Solving time‐periodic fractional diffusion equations via diagonalization technique and multigrid

“…The action of on a vector can then be efficiently computed by the rational Krylov method. Alternatively, one can use a uniform best rational approximation, an exponentially convergent sinc quadrature, or a dimension extended PDE approach . Any of these techniques suffice for our multigrid method, particularly for the smoother Richardson iterations.…”

Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

“…We note here that similar test problems (only in 2D domains) with checkerboard functions on the right‐hand side are used to demonstrate the convergence of the selected numerical methods in many papers: M2, M3, and M4 . This test demonstrates the important feature of the fractional diffusion problems, how the convergence rate of numerical discretization methods depends on the smoothness of the solution.…”

“…Then, for each quadrature point, some specific 3D elliptic subproblem should be solved. M4 Alternative methods for solving algebraic systems (discrete problems). These methods are based on the best uniform rational approximations of the function t 1 − β for the BURA method and of the function t β for the R‐BURA method in the interval [0,1]. Computationally, a limited number of linear systems need to be solved, and the arising matrices are the positive diagonal shifts of the (original, non‐fractional) discretized elliptic operator.…”

“…This is the third method that we have selected for our comparison study. It is based on the best uniform rational approximations of the function t 1 − β for the BURA method and of the function t β for the R‐BURA method in the interval [0,1]. The approximate solution of the discrete problem is defined as where matrix A h and function on the right‐hand side are scaled as A h = h 2 /12 L h and , respectively.…”

“…This paper presents several important contributions to our previous research. First, for the pseudo‐parabolic method M2, we use the recently proposed geometrically graded time stepping scheme, which should resolve the singular behavior of the solution for pseudo‐time t close to 0 and give better convergence results compared to the previously used uniform time stepping scheme. Second, for the quadrature method M3, we employ a quadrature formula, which is different from the previously used formula with geometrically graded mesh and is expected to show superior convergence results. Third, for analysis of the BURA method M4, we include its recent modification R‐BURA, which has been recommended for the power coefficient β closer to 1. Finally, we compare the parallel solution times of the developed algorithms that are required to achieve the specified accuracy of the solution. We present and discuss the comparison plots, showing which parallel numerical algorithms are recommended for different levels of accuracy and different values of fractional power coefficient β . …”