Time-dependent stochastic basis adaptation for uncertainty quantification

TL;DR

This paper introduces a novel method for adapting stochastic bases over time in SPDEs, improving uncertainty quantification for time-varying, spatially dependent quantities by coupling basis adaptation with domain decomposition.

Contribution

It extends stochastic basis adaptation to time-dependent SPDEs with spatial dependence, addressing error accumulation by adapting the basis at each time step.

Findings

Numerical results demonstrate improved accuracy in linear and nonlinear diffusion equations.

The method effectively reduces stochastic dimensionality in time-varying problems.

First application of dynamic basis adaptation to spatially dependent, time-varying quantities.

Abstract

We extend stochastic basis adaptation and spatial domain decomposition methods to solve time varying stochastic partial differential equations (SPDEs) with a large number of input random parameters. Stochastic basis adaptation allows the determination of a low dimensional stochastic basis representation of a quantity of interest (QoI). Extending basis adaptation to time-dependent problems is challenging because small errors introduced in the previous time steps of the low dimensional approximate solution accumulate over time and cause divergence from the true solution. To address this issue we have introduced an approach where the basis adaptation varies at every time step so that the low dimensional basis is adapted to the QoI at that time step. We have coupled the time-dependent basis adaptation with domain decomposition to further increase the accuracy in the representation of the QoI. To illustrate the construction, we present numerical results for one-dimensional time varying linear and nonlinear diffusion equations with random space-dependent diffusion coefficients. Stochastic dimension reduction techniques proposed in the literature have mainly focused on quantifying the uncertainty in time independent and scalar QoI. To the best of our knowledge, this is the first attempt to extend dimensional reduction techniques to time varying and spatially dependent quantities such as the solution of SPDEs.