A dynamical polynomial chaos approach for long-time evolution of SPDEs

Abstract

We propose a Dynamical generalized Polynomial Chaos (DgPC) method to solve time-dependent stochastic partial differential equations (SPDEs) with white noise forcing. The long-time simulation of SPDE solutions by Polynomial Chaos (PC) methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time. Exploiting the Markovian property of white noise, DgPC [1] implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise. The dimension of the representation is kept minimal by application of a Karhunen-Loeve (KL) expansion. Using frequent restarts and low degree polynomials on sparse multi-index sets, the method allows us to perform long time simulations, including the calculation of invariant measures for systems which possess one. We apply the method to the numerical simulation of stochastic Burgers and Navier Stokes equations with white noise forcing. Our method also allows us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations and show that the algorithm compares favorably with standard Monte Carlo methods.

Variance-based sensitivity analysis for time-dependent processes

“…viable alternatives to a pointwise-in-time CS-based strategy; see [43] and the references therein. While PC expansions are widely applicable, they present known shortcomings for certain classes of time-dependent problems; see e.g., [44,32,45,46,47] that address problems where straightforward implementation of a PC-based approach is not optimal. Depending on the application at hand, other types of surrogates might provide better alternatives for the purposes of Algorithm 1.…”

Dynamic stochastic finite element method using time‐dependent generalized polynomial chaos

“…This intrusive approach was first applied to coupled partial differential equations, where the response at regular times of updates is considered as a new random variable and is added to the existing set of random variables. The idea of regular updates of the set of basis functions was further extended to multidimensional stochastic dynamical systems by Ozen and Bal 7 who introduced dynamical PC. The idea behind this approach is to find an optimal set of basis polynomials that can accurately represent the entire solution in space and time between short‐time intervals.…”

“…The approach we present here is an intrusive approach based on the idea of dynamical PC from Ozen and Bal 7 . There are several motivations for why we choose an intrusive approach over a nonintrusive approach which are described below.…”

“…In this study, we first summarize the deterministic dynamic finite‐element method and extend it to the case of stochastic material models following the well‐known approach from the static case proposed by Ghanem and Spanos 1 . We next present the proposed algorithm to dynamically update the PC basis on the fly using an approach similar to dynamic PC from Ozen and Bal 7 to maintain the accuracy in the solution's representation during run time. Finally, the method is illustrated with the simple example of an impulse response on a one‐dimensional soil column with uncertain material properties.…”

Error Estimation of Polynomial Chaos Approximations in Transient Structural Dynamics

“…Using the same basis in each time step is called a time-frozen PC expansion and results in large errors in the evaluation of the long-term response [15]. Several strategies to add physical information to the variation of the spectral coefficients have been proposed in [16,17]. To control the error made by the PC expansion, which is the distance between the response and its projection in the space spanned by the chosen polynomials, error estimates are needed.…”