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, timedependent advectiondiffusion 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 timedependent 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 spacetime approximation is constructed on a discretetime 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 adaptiveintime approximation algorithm that controls the interpolation error to a dynamic tolerance that is related to the estimated timestepping errors. In comparison to classical (nonadaptive) 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 highdimensional parametric, parabolic PDE problems. We also consider a residualbased error estimation strategy for the full parametric advectiondiffusion problem. Our strategy develops existing results to include a parametric advection term and allows each spacetime 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 adaptiveintime algorithm for the spatial, temporal and parametric discretisations.
Date of Award  1 Aug 2024 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  David Silvester (Supervisor) & Catherine Powell (Supervisor) 

 Uncertainty quantification
 AdvectionDiffusion
 Error estimation
 Stochastic collocation
 Parametric PDEs
 Adaptivity
Efficient Approximation of Parametric Parabolic Partial Differential Equations
Kent, B. (Author). 1 Aug 2024
Student thesis: Phd