Efficient Approximation of Parametric Parabolic Partial Differential Equations

Abstract

Parametric partial differential equations (PDEs) are often used to model physical problems with uncertain inputs that are represented as functions of random variables. We consider a parametric, time-dependent advection-diffusion problem. Methods such as stochastic collocation and stochastic Galerkin use global polynomials to approximate the solution as a function of the parameters. Naively chosen polynomial approximation spaces can result in a high computational cost, particularly for high dimensional parameter domains. In this thesis, we design a posteriori error estimators that are used to adaptively construct an appropriate polynomial approximation space. Adaptivity is particularly important for time-dependent parametric problems where the dependence of the solution on the input parameters will change over time. Our approximation is constructed as a global polynomial interpolant using sparse grid stochastic collocation (SC). For each parameter realisation in our sparse grid, a space--time approximation is constructed on a discrete-time grid using adaptive timestepping with local error control applied to the ODE system arising from a finite element discretisation. We define a novel a posteriori error estimation strategy for the spatially discrete problem. This combines a hierarchical interpolation error estimator with a scaling argument to estimate global timestepping errors. Our error estimator, and in particular a simplified version, is shown to be effective at estimating a norm of the approximation error pointwise in time. The estimators are used to drive a novel adaptive-in-time approximation algorithm that controls the interpolation error to a dynamic tolerance that is related to the estimated timestepping errors. In comparison to classical (non-adaptive) sparse grid approximation, we see a reduced computational cost for comparable approximation errors. Adaptivity is seen to be essential for efficiently approximating the solution to high-dimensional parametric, parabolic PDE problems. We also consider a residual-based error estimation strategy for the full parametric advection-diffusion problem. Our strategy develops existing results to include a parametric advection term and allows each space-time approximation to be computed using adaptive timestepping. This strategy includes a spatial error indicator that can be localised to mesh elements. These error indicators could be used to drive an adaptive-in-time algorithm for the spatial, temporal and parametric discretisations.

Date of Award	13 Nov 2023
Original language	English
Awarding Institution	The University of Manchester
Supervisor	David Silvester (Co Supervisor) & Catherine Powell (Main Supervisor)