Parallel Monte Carlo Preconditioning with Diagonal Dominant Approximate Inverse

摘要

A parallel precoiiditioner is presented tor the solution of general systems of linear equations(SLAE). The stochastic approximate in-verse(MI) of a diagonal dominant matrix is computed explicitly using parallel Markov chain Monte Carlo method(MCMC). This inverse is then used as a precoiiditioner where solving corresponding SLAE. Monte Carlo methods are used for the stochastic approximation, since it is known that they arc very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. In this paper we show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE with the precoiiditioner. Experimental results with dense and sparse matrices are presented.