Performance Study of Hierarchical Semi-separable Compression Solver for Parabolic Problems with Space-Fractional Diffusion

Abstract

AbstractEquations involving fractional diffusion operators are used to model anomalous processes in which the Brownian motion hypotheses are violated. In this work we utilize the Fractional Laplacian operator, defined through the Riesz potential and homogeneous Dirichlet boundary conditions. We explore a parabolic problem in a model square domain, using a backward Euler scheme for the discretization in time. The resulting series of systems of linear algebraic equations are dense and computationally expensive to solve. When utilizing the traditional Gaussian Elimination, the computational complexity is \(O(n^3)\) for the LU factorization and \(O(n^2)\) for each time step, where n is the number of unknowns. This can be improved by using Hierarchical Semi-Separable (HSS) compression. With a solver from STRUMPACK, the computational complexity is reduced to \(O(n^2r)\) for the factorization and O(nr) for each time step, where r is the maximum off-diagonal rank of the matrix. The presented numerical experiments show the advantages of the HSS method for the examined problem.KeywordsAnomalous diffusionFractional laplacianParabolicHSS compressionSTRUMPACK