On scalability behaviour of Monte Carlo sparse approximate inverse for matrix computations

This paper presents a Monte Carlo SPAI pre-conditioner. In contrast to the standard deterministic SPAI pre-conditioners that use the Frobenius norm, a Monte Carlo alternative that relies on the use of Markov Chain Monte Carlo (MCMC) methods to compute a rough matrix inverse (MI) is given. Monte Carlo methods enable a quick rough estimate of the non-zero elements of the inverse matrix with a given precision and certain probability. The advantage of this method is that the same approach is applied to sparse and dense matrices and that complexity of the Monte Carlo matrix inversion is linear of the size of the matrix. The behaviour of the proposed algorithm is studied, its performance is investigated and a comparison with the standard deterministic SPAI, as well as the optimized and parallel MSPAI version is made. Further Monte Carlo SPAI and MSPAI are used for solving systems of linear algebraic equations (SLAE) using BiCGSTAB and a comparison of the results is made.