Uncertainty quantification with Lévy-type random fields

Abstract

A countless number of models in the natural sciences, engineering and economics are based on partial differential equations (PDEs). Due to insufficient data or measurement errors, certain characteristics of the underlying PDE are subject to uncertainty, and are usually modeled by continuous and/or Gaussian random fields. Although analytically tractable, the applications of continuous or Gaussian random fields are limited: spatial and temporal discontinuities cannot be captured and Gaussian distributions notoriously underestimate the probability of rare events. To this end, the focus of this thesis is on uncertainty quantification with Lévy-type random fields, a certain class of discontinuous stochastic objects that provide a significant extension to the existing methodology. In a nutshell, this dissertation explains how to incorporate Lévy-type random fields into PDEs and how the corresponding solutions become accessible by the means of discretization and simulation. The first main contribution is the introduction of a novel type of random field, consisting of a Gaussian part and a spatially discontinuous jump field. This Lévy-type field serves as a coefficient in advection-diffusion equations and allows to model, for instance, sudden changes in the permeability of a porous medium far more realistically than state-of-the-art continuous models. In contrast to the few examples in the literature, the discontinuous random coefficient in this thesis provides a unique flexibility as it is able to generate virtually any stochastic geometry. Apart from random PDEs with discontinuous coefficients, hyperbolic transport equations with Lévy noise as source term are considered. The noise processes take values in a (infinite-dimensional) Hilbert space and involve temporal discontinuities. Therefore, heavy-tailed random perturbations are introduced to the transport problem, and the resulting stochastic equation may be utilized, e.g., as a model for the dynamics in commodity forward markets. Due to the lack of tractable discretization schemes for the underlying stochastic PDE, this models have been, up to now, only of theoretical interest. This thesis paves the way to finally apply the forward model with Lévy noise in practice, as it provides the first fully discrete scheme for the corresponding stochastic transport problem. In both cases, PDEs with random jump coefficients or with Lévy noise as source term, regularity is inherently low due to the discontinuities and state-of-the-art numerical algorithms are prohibitive. To remedy this issue, several advanced schemes for discontinuous random problems are introduced that outperform standard methods in terms of convergence rates and computational effort. A comprehensive numerical analysis is provided and the superior performance of the new approaches is validated by numerous numerical experiments. Moreover, an emphasis is put on the approximation of infinite-dimensional, discontinuous random fields, a crucial part in the discretization of stochastic PDEs. A part of this thesis covers the sampling of Hilbert space-valued Lévy processes, and introduces a sampling technique combining truncated Karhunen-Loève expansions with discrete Fourier inversion. This algorithm stands out as the arguably most flexible of a very limited number of methods for the approximation and simulation of Lévy fields.