Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

TL;DR

This paper investigates how polynomial chaos methods propagate uncertainty in Hamiltonian, multi-time scale, and chaotic systems, revealing structural properties, limitations, and advantages in different dynamical regimes.

Contribution

It proves that polynomial chaos expansions preserve Hamiltonian structure, analyzes their limitations in long-term and chaotic systems, and demonstrates their effectiveness in multi-time scale problems.

Findings

Polynomial chaos preserves Hamiltonian structure in the expanded system.

Finite expansions fail at long times for Hamiltonian systems due to volume preservation.

Polynomial chaos effectively captures slow dynamics in multi-time scale systems.

Abstract

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, has multiple time scales, or displays chaotic dynamics. In particular, we prove that the differential equations for the expansion coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of the order of the expansion. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincar\'e section of the system and that, as the time scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.