$\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian
This provides a theoretical justification for using hierarchical matrix techniques to solve fractional Laplacian problems efficiently, benefiting computational scientists working on nonlocal operators.
The authors prove that the inverse of the stiffness matrix arising from a Galerkin discretization of the fractional Laplacian can be approximated by blockwise low-rank matrices with exponential convergence in the block rank.
The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasi-uniform mesh. We show that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.