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We consider the discretization of time-space diffusion equations withfractional derivatives in space and either 1D or 2D spatial domains. The use ofimplicit Euler scheme in time and finite differences or finite elements inspace, leads to a sequence of dense large scale linear systems describing thebehavior of the solution over a time interval. We prove that the coefficientmatrices arising in the 1D context are rank structured and can be efficientlyrepresented using hierarchical formats ($\mathcal H$-matrices, HODLR).Quantitative estimates for the rank of the off-diagonal blocks of thesematrices are presented. We analyze the use of HODLR arithmetic for solving the1D case and we compare this strategy with existing methods that exploit theToeplitz-like structure to precondition the GMRES iteration. The numericaltests demonstrate the convenience of the HODLR format when at least areasonably low number of time steps is needed. Finally, we explain how theseproperties can be leveraged to design fast solvers for problems with 2D spatialdomains that can be reformulated as matrix equations. The experiments show thatthe approach based on the use of rank-structured arithmetic is particularlyeffective and outperforms current state of the art techniques.

Fast solvers for two-dimensional fractional diffusion equations using rank structured matrices