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.