Wiener path integral most probable path determination: A computational algebraic geometry solution treatment

Abstract

Abstract The recently developed Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear systems relies on solving a functional minimization problem for the most probable path, which is then utilized for evaluating a specific point of the system joint response probability density function (PDF). However, although various numerical optimization algorithms can be employed for determining the WPI most probable path, there is generally no guarantee that the selected algorithm converges to a global extremum. In this paper, first, a Newton’s optimization scheme is proposed for determining the most probable path, and various convergence behavior aspects are elucidated. Second, the existence of a unique global minimum and the convexity of the objective function of the considered nonlinear system are demonstrated by resorting to computational algebraic geometry concepts and tools, such as Grobner bases. Several numerical examples pertaining to diverse nonlinear oscillators are considered, where it is proved that the associated objective functions are convex, and that the proposed Newton’s scheme converges to the globally optimum most probable path. Comparisons with pertinent Monte Carlo simulation data are included as well for demonstrating the reliability of the WPI technique.