Iterative low-rank approximation solvers for the extension method for fractional diffusion

Abstract

We consider the numerical method for fractional diffusion problems which is based on an extension to a mixed boundary value problem for a local operator in a higher dimensional space. We observe that, when this problem is discretized using tensor product spaces as is commonly done, the solution can be very well approximated by low-rank tensors; we provide some analysis to support this observation. This fact motivates us to apply iterative low-rank approximation algorithms in order to efficiently solve this extended problem. In particular, we employ a recently proposed greedy Tucker approximation method as well as a more classical greedy rank one update method. Throughout, all objects of interest are kept in suitable low-rank approximations, which dramatically reduces the required amount of memory compared to the full formulation of the extended problem.Our approach can be used for general, non-structured space discretizations. If the space discretization itself has tensor product structure, we can further decompose the problem in order to deal with even lower-dimensional objects. We also note that the approach can be directly applied to higher-order discretizations both in space and the extended variable.A further contribution of our work is a rank one version of a diagonal preconditioner which can mitigate the severe ill-conditioning of the extended problem arising due to mesh grading and singularity of the coefficient. This preconditioner can be realized using a simple pre- and postprocessing step and does not affect the computational effort of our algorithm.In several numerical examples, we demonstrate the convergence behavior of the proposed methods. In particular, the Tucker approximation approach requires only a few iterations in order to reach the discretization error in all tested settings.