Distributed generic approximate sparse inverses

Abstract

The need for accuracy in the solution of linear systems derived from the discretization of partial differential equations leads to large sparse linear systems. The solution of sparse linear systems requires efficient scalable methods. Iterative solvers require efficient parallel preconditioning methods to solve effectively sparse linear systems. Herewith, a new parallel algorithm for the generic approximate sparse inverse matrix method for distributed memory systems is proposed. The computation of the distributed generic approximate sparse inverse matrix is based on a column-wise approach, which allows the separation to independent problems that can be handled in parallel without synchronization points or intermediate communications. This is achieved by reforming the generic approximate sparse inverse matrix algorithm and its process of computation with a new partial solution method for the computation of the nonzero elements of each column dictated by the approximate inverse sparsity pattern. Moreover, an algorithmic scheme is proposed for the efficient distribution of data amongst the available workstations, along with a load balancing scheme for problems with large standard deviation in the number of nonzero elements per column. Numerical results are presented for the proposed schemes for various model problems.