On Monte Carlo and Quasi-Monte Carlo for Matrix Computations

This paper focuses on minimizing further the communications in Monte Carlo methods for Linear Algebra and thus improving the overall performance. The focus is on producing set of small number of covering Markov chains which are much longer that the usually produced ones. This approach allows a very efficient communication pattern that enables to transmit the sampled portion of the matrix in parallel case. The approach is further applied to quasi-Monte Carlo. A comparison of the efficiency of the new approach in case of Sparse Approximate Matrix Inversion and hybrid Monte Carlo and quasi-Monte Carlo methods for solving Systems of Linear Algebraic Equations is carried out. Experimental results showing the efficiency of our approach on a set of test matrices are presented. The numerical experiments have been executed on the MareNostrum III supercomputer at the Barcelona Supercomputing Center (BSC) and on the Avitohol supercomputer at the Institute of Information and Communication Technologies (IICT).