A Fast Convergence Parameter for Monte Carlo–Neumann Solution of Linear Stochastic Systems

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 making its use highly inefficient. In this article, a fast convergence parameter (λ) convergence parameter is introduced, which yields accurate and efficient Monte Carlo–Neumann (MC-N) solutions of linear stochastic systems using first-order Neumann expansions. The λ convergence parameter is found as a solution to the 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 the MC-N λ method. The accuracy and efficiency of the MC-N λ method are demonstrated in application to stochastic beam-bending problems.