On Monte Carlo method in distributed memory systems

Abstract

The work is devoted to solving systems of linear algebraic equations on distributed memory computers. It is supposed that there are M computing nodes, each of which has a limited fast memory, and communication between nodes takes considerable time. Provided that the matrix elements and vector of the right side can not be placed in its entirety in the memory of one node, the problem of effective use of equipment in between exchanges, i.e. whether each node to use the data available to him to be able to reduce the overall residual. In general assumptions about the matrix of the answer to this question is negative. An example is placed in the Appendix. We consider the case when the system has a large enough order and appropriate to apply the Monte Carlo method, the matrix is divided between computing nodes on non-overlapping blocks of rows with the same partition into blocks of indices of rows and columns. We also consider a modification of the method of simple iteration, based on this partition, consisting of two nested iterative process, so that only the outer iteration involve the exchange of messages between nodes. This iterative process leads naturally to a similar process using the Monte Carlo method that does not require a full copy of the storage matrix of the system on each compute node. The paper constructed and investigated for solving the linear estimates for the case and under certain additional conditions on the matrix system proved sufficient conditions for convergence. Refs 8.