Efficient Wiener path integral most probable path determination based on extrapolation

Abstract

The Wiener path integral technique for determining the stochastic response of diverse nonlinear dynamical systems relies on a variational formulation that leads to a functional minimization problem. This takes the form of a deterministic boundary value problem (BVP) to be solved for the most probable path, which is used for determining approximately the system response joint transition probability density function (PDF). The BVP corresponds to a specific grid point of the response PDF effective domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations but, also, the boundary conditions differ only slightly. In this paper, the above unique aspect of the technique is explored further, and it is shown that solution of a BVP and estimation of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently the PDF values at neighboring points without the need for solving additional BVPs. Notably, the developed approach enhances significantly the computational efficiency of the WPI technique without affecting considerably the exhibited accuracy. An indicative numerical example relating to a Duffing nonlinear oscillator is considered for demonstrating the reliability of the technique. Comparisons with pertinent Monte Carlo simulation data are included as well.