Fast Solvers for Two-Dimensional Fractional Diffusion Equations Using Rank Structured Matrices

Abstract

We consider the discretization of time-space diffusion equations with fractional derivatives in space and either one-dimensional (1D) or 2D spatial domains. The use of an implicit Euler scheme in time and finite differences or finite elements in space leads to a sequence of dense large scale linear systems describing the behavior of the solution over a time interval. We prove that the coefficient matrices arising in the 1D context are rank structured and can be efficiently represented using hierarchical formats (H-matrices, HODLR). Quantitative estimates for the rank of the off-diagonal blocks of these matrices are presented. We analyze the use of HODLR arithmetic for solving the 1D case and we compare this strategy with existing methods that exploit the Toeplitzlike structure to precondition the GMRES iteration. The numerical tests demonstrate the convenience of the HODLR format when at least a reasonably low number of time steps is needed. Finally, we explain how these properties can be leveraged to design fast solvers for problems with 2D spatial domains that can be reformulated as matrix equations. The experiments show that the approach based on the use of rank-structured arithmetic is particularly effective and outperforms current state of the art techniques.

Rational Krylov for Stieltjes matrix functions: convergence and pole selection

“…In recent years, there has been an increasing interest in models involving fractional derivatives. For 2D problems on rectangular grids, discretizations by finite differences or finite elements lead to linear systems that can be recast as the solution of matrix equations with particularly structured coefficients [12,24]. However, a promising formulation which simplifies the design of boundary conditions relies on first discretizing the 2D Laplacian on the chosen domain, and then considers the action of the matrix function z −α (with the Laplacian as argument) on the right hand side.…”

“…This allows to rephrase the integrals of ( 24) in the more convenient form The above integral is well-defined even if we let → 0, we can can take the limit in (24) which yields exactly the value 1 2 for the first term, and we have reduced the problem to estimate f (t) = 1 2 + (−1) π ∞ 0 sin(st+θ(s)) s ds. To bound the integral, we split the integration domain in three parts:…”

Spectral properties of flipped Toeplitz matrices and related preconditioning

“…In the following lemmas we prove that {M n } n ∼ λ 0 and that { T n } n ∼ λ (g, [−π, π]) with g as in (1.1). The proof of Lemma 3.1 is based on the idea in Proposition 3.10 in [10].…”

Nonlocal pagerank

“…On the other hand, suitable choices of the smoothing functions may lead to structured transition matrices (as in the case of fractional derivatives e.g. [44]) and exploring this line of research may lead to efficient methods for using nonlocal PageRank on large scale problems, an issue that will be object of future investigations.…”