An efficient Monte Carlo solution for problems with random matrices

Abstract.

Random matrices, that is, matrices whose entries are measurable functions of a random vector Z, are encountered in finite element/difference formulations of a broad range of stochastic mechanics problems. Monte Carlo simulation, the only general method for solving this class of problems, is usual impractical when dealing with realistic problems. A new method is proposed for solving this class of problems. The method can be viewed as a smart Monte Carlo simulation. Like Monte Carlo, it calculates statistics for quantities of interest from deterministic matrices corresponding to samples of Z. In contract to Monte Carlo that uses a large number of samples of Z selected at random, the proposed method uses a small number of samples of this vector selected in an optimal manner. The method is based on stochastic reduced models (SROMs) for Z, i.e., random vectors with finite numbers of samples, and surrogate models expressing quantities of interest as known functions of Z. Theoretical arguments are followed by numerical examples providing statistics for inverses of random matrices, solutions of stochastic algebraic equations, and eigenvalues/eigenvectors of random matrices.