Sparse-Grid Approximation of High-Dimensional Parametric PDEs

Abstract

In this thesis we analyse the approximation of countably-parametric functions $u$ and their expectation by sparse-grid polynomial interpolation and quadrature. It will be assumed that the target function $u$ allows holomorphic extensions in each of its arguments, which makes it naturally amenable to approximation by spectral methods. Functions of this type arise as solutions to parametric partial differential equations (PDEs) in Uncertainty Quantification. As such they play an increasingly important role in various areas of engineering and life sciences, which deal with PDE models that are subject to unknown or stochastic high-dimensional input data. The main contributions of this thesis concern several new theoretical results regarding the convergence rates that can be achieved by sparse-grid approximations. Exploiting symmetry properties, we show that sparse-grid quadrature can converge at more than twice of the best $N$-term rate. Sparse-grid interpolation on the other hand is known to be able to uniformly approximate the target function at the best $N$-term rate. In both cases, the efficiency of the approximation critically relies on suitable choices of the sparse-grid. We propose new apriori selected grids, which are proven to yield dimension independent convergence rates for both nested and nonnested interpolation/quadrature points. Additionally, an algorithm is provided to determine such sparse-grids in (essentially) linear complexity. In case of linear parametric PDEs we show improved summability of the generalized polynomial chaos coefficients, if the input functions used to model the uncertainty exhibit local supports. Contrary to previous results of this type, our new analysis may be applied to operators depending nonaffinely on the parameters, which occurs for example for the pullback of solutions to PDEs posed on parametric domains. To obtain a fully discrete analysis, we additionally address the error stemming from numerical PDE solvers and present a multilevel sparse-grid interpolation/quadrature framework. The scope of the corresponding convergence results comprises in particular multilevel interpolation/quadrature of finite element approximations to both linear and nonlinear parametric PDEs. We discuss in detail the error convergence for an exemplary diffusion problem in a polygonal domain, which compels us to deal with weighted Sobolev spaces to retain optimal convergence rates in the presence of corner singularities. As an example of a nonlinear PDE and a saddle point problem, solutions to the Navier-Stokes and the Stokes equations on parametrized domains are analysed. In the latter case our new results yield improved convergence rates if the expansion functions describing the parametric domain are locally supported. To substantiate the theoretical results, we report on various numerical experiments, with the main objective of investigating the observed convergence rates and comparing them with the proven ones. The numerical experiments show that our method is competitive and can be significantly more efficient than popular state-of-the-art alternatives such as higher-order quasi-Monte Carlo quadrature and adaptive sparse-grid quadrature.