Simulation of stochastic systems via polynomial chaos expansions and convex optimization

TL;DR

This paper presents a new convex optimization-based method for efficiently computing Polynomial Chaos Expansion coefficients in complex stochastic dynamical systems, reducing computational effort and avoiding extensive model manipulations.

Contribution

The authors introduce a novel regularization technique that leverages convex optimization to compute Polynomial Chaos coefficients for high-dimensional problems with fewer simulations.

Findings

Effective in three diverse applications: electric circuit, organizational behavior model, chemical oscillator.

Reduces computational cost compared to traditional methods.

Can incorporate additional stochastic information via convex constraints.

Abstract

Polynomial Chaos Expansions represent a powerful tool to simulate stochastic models of dynamical systems. Yet, deriving the expansion's coefficients for complex systems might require a significant and non-trivial manipulation of the model, or the computation of large numbers of simulation runs, rendering the approach too time consuming and impracticable for applications with more than a handful of random variables. We introduce a novel computationally tractable technique for computing the coefficients of polynomial chaos expansions. The approach exploits a regularization technique with a particular choice of weighting matrices, which allow to take into account the specific features of Polynomial Chaos expansions. The method, completely based on convex optimization, can be applied to problems with a large number of random variables and uses a modest number of Monte Carlo simulations, while avoiding model manipulations. Additional information on the stochastic process, when available, can be also incorporated in the approach by means of convex constraints. We show the effectiveness of the proposed technique in three applications in diverse fields, including the analysis of a nonlinear electric circuit, a chaotic model of organizational behavior, finally a chemical oscillator.