The anomalous (fractional) diffusion has a wide application in modeling many physical processes including superconductivity phenomena, protein diffusion within cells, diffusion through porous media, etc. The Fractional Laplacian is a non-local operator. Thus, after the discretization we get a dense system of linear equations. Solving it with traditional Gaussian elimination results in computational complexity of O(n3).
The hierarchical compression methods utilize the structure of the matrix A to compress it into a sparser approximation H. The Hierarchical Semi-Separable (HSS) compression is a derivative method that approximates the off-diagonal blocks as Ai j,i≠j Ui BijVj. If the approximation of Aij,i≠j is low-rank, the generator matrices Ui, Bij and Vj are much smaller than the non-compressed block Aij,i≠j. The diagonal blocks Aii can be compressed in a similar way and so on recursively. The computational complexity is O(r2n), where r is the maximum off-diagonal rank and, ideally, r approaches O(log n). However, the performance heavily depends on the ordering of the unknowns. In this work we analyze and compare the use of Recursive Bisection and Nested Dissection on the initial mesh to produce a more suitable ordering. Recursive bisection uses a divide and conquer approach to recursively divide the mesh in two and then each halve in two again, and so on recursively. Nested Dissection also divides the mesh in two recursively, but it does so along a set of separator vortices, which form a third part in each level of dissection.
We assess and compare the use of Recursive Bisection and Nested Dissection reorderings for improving the performance and accuracy of a parallel HSS solver from the STRUctured Matrix PACKage (STRUMPACK) for a fractional diffusion problem modeled with the Ritz potential. We also compare with results from the original and three other reorderings, as well as using a direct Gaussian elimination solver from Intel®’s Math Kernel Library (MKL).
