High-Dimensional Nonlinear Systems: Circumventing the Curse of Dimensionality via a Reduced-Order Formulation

In Chap. 7, the rapid increase of the computational cost as a function of the number of dimensions of the problem, also referred to as curse-of-dimensionality in stochastic dynamics, is addressed by developing a Wiener path integral variational formulation with mixed fixed/free boundary conditions. In this regard, the response joint probability density function is marginalized in an a priori manner, and thus, the associated computational cost becomes independent of the total number of degrees of freedom or stochastic dimensions. The capabilities of the technique are demonstrated by considering selected numerical examples. They include a stochastically excited high-dimensional (100-degree-of-freedom) nonlinear dynamical system modeling the behavior of large arrays of coupled nano-mechanical oscillators.