A parallel quasi-monte carlo method for solving systems of linear equations

Abstract

This paper presents a parallel quasi-Monte Carlo method for solving general sparse systems of linear algebraic equations. In our parallel implementation we use disjoint contiguous blocks of quasirandom numbers extracted from a given quasirandom sequence for each processor. In this case, the increased speed does not come at the cost of less thrust-worthy answers. Similar results have been reported in the quasi-Monte Carlo literature for parallel versions of computing extremal eigenvalues [8] and integrals [9]. But the problem considered here is more complicated - our algorithm not only uses an s-dimensional quasirandom sequence, but also its k-dimensional projections (k = 1,2, . . . , s-1) onto the coordinate axes. We also present numerical results. In these test examples of matrix equations, the martrices are sparse, randomly generated with condition numbers less than 100, so that each corresponding Neumann series is rapidly convergent. Thus we use quasirandom sequences with dimension less than 10. © Springer-Verlag Berlin Heidelberg 2002.