A directed Monte Carlo solution of linear stochastic algebraic system of equations

This paper proposes a modified Monte Carlo simulation method for the solution of a linear stochastic algebraic system of equations arising from the stochastic finite element modelling of linear elastic problems. The basic idea is to direct Monte Carlo samples along straight lines and then utilise their spatial proximity or order to provide high quality initial approximations in order to significantly accelerate the convergence of iterative solvers at each sample. The method, termed the directed Monte Carlo (DMC) simulation, is developed first for one random variable using the preconditioned conjugate gradient equipped with an initial approximation prediction scheme, and then extended to multiple Gaussian random variable cases by the adoption of a general hyper-spherical transformation. The eigenproperties of the linear system are also briefly discussed to reveal the suitability of several preconditioning schemes for iterative solvers. Two numerical examples with up to around 6000 DOFs are provided to assess the performance of the proposed solution strategy and associated numerical techniques in terms of computational costs and solution accuracy.