Quadrature methods for elliptic PDEs with random diffusion

Abstract

In this thesis, we consider elliptic boundary value problems with random diffusion coefficients. Such equations arise in many engineering applications, for example, in the modelling of subsurface flows in porous media, such as rocks. To describe the subsurface flow, it is convenient to use Darcy's law. The key ingredient in this approach is the hydraulic conductivity. In most cases, this hydraulic conductivity is approximated from a discrete number of measurements and, hence, it is common to endow it with uncertainty, i.e. model it as a random field. This random field is usually characterized by its mean field and its covariance function. Naturally, this randomness propagates through the model which yields that the solution is a random field as well. The thesis on hand is concerned with the effective computation of statistical quantities of this random solution, like the expectation, the variance, and higher order moments. In order to compute these quantities, a suitable representation of the random field which describes the hydraulic conductivity needs to be computed from the mean field and the covariance function. This is realized by the Karhunen-Loeve expansion which separates the spatial variable and the stochastic variable. In general, the number of random variables and spatial functions used in this expansion is infinite and needs to be truncated appropriately. The number of random variables which are required depends on the smoothness of the covariance function and grows with the desired accuracy. Since the solution also depends on these random variables, each moment of the solution appears as a high-dimensional Bochner integral over the image space of the collection of random variables. This integral has to be approximated by quadrature methods where each function evaluation corresponds to a PDE solve. In this thesis, the Monte Carlo, quasi-Monte Carlo, Gaussian tensor product, and Gaussian sparse grid quadrature is analyzed to deal with this high-dimensional integration problem. In the first part, the necessary regularity requirements of the integrand and its powers are provided in order to guarantee convergence of the different methods. It turns out that all the powers of the solution depend, like the solution itself, anisotropic on the different random variables which means in this case that there is a decaying dependence on the different random variables. This dependence can be used to overcome, at least up to a certain extent, the curse of dimensionality of the quadrature problem. This is reflected in the proofs of the convergence rates of the different quadrature methods which can be found in the second part of this thesis. The last part is concerned with multilevel quadrature approaches to keep the computational cost low. As mentioned earlier, we need to solve a partial differential equation for each quadrature point. The common approach is to apply a finite element approximation scheme on a refinement level which corresponds to the desired accuracy. Hence, the total computational cost is given by the product of the number of quadrature points times the cost to compute one finite element solution on a relatively high refinement level. The multilevel idea is to use a telescoping sum decomposition of the quantity of interest with respect to different spatial refinement levels and use quadrature methods with different accuracies for each summand. Roughly speaking, the multilevel approach spends a lot of quadrature points on a low spatial refinement and only a few on the higher refinement levels. This reduces the computational complexity but requires further regularity on the integrand which is proven for the considered problems in this thesis.