New method for efficient Monte Carlo–Neumann solution of linear stochastic systems

•Monte Carlo–Neumann solution of linear stochastic systems.•λ Convergence parameter is introduced.•λ Convergence parameter is found as solution to error minimization problem.•λ Parameter yields almost exact solutions with first order Neumann expansions.

The Neumann series is a well-known technique to aid the solution of uncertainty propagation problems. However, convergence of the Neumann series can be very slow, often turning its use highly inefficient. In this article, a λ convergence parameter is introduced, which yields accurate and efficient Monte Carlo–Neumann solutions of linear stochastic systems using first order Neumann expansions. The λ convergence parameter is found as solution to a distance minimization problem, for an approximation of the inverse of the system matrix using the Neumann series. The method presented herein is called Monte Carlo–Neumann with λ convergence, or simply MC–N λ method. The accuracy and efficiency of the MC–N λ method is demonstrated in application to stochastic beam bending problems.