Quasi-Monte Carlo Integration in Uncertainty Quantification for PDEs with log-Gaussian Random Field Inputs

Abstract

Partial differential equations (PDEs) with incomplete knowledge on differential operators arise in science and engineering. This lack of knowledge results in uncertainty in the respective solutions. Mathematically, unknown coefficient functions of differential operators are modeled by random fields. These random fields describe physical quantities that constitute the PDE under consideration. In this thesis, elliptic PDEs with random coefficient functions will be studied. These describe for example subsurface flow in unknown media, which is modeled by a random field. Statistical moments of random solutions are formally infinite-dimensional integrals over parameter vectors that represent the random field input. In the considered models, the logarithm of the random field input is expanded in a function system with random, independent and identically distributed parameters. Suitable decay in the function systems allows to control the error of truncating the expansion. The resulting high-dimensional integral may be approximated by quadrature methods, particularly by quasi-Monte Carlo (QMC) rules. Here, the approximation of the expectation of functionals of the random solution is investigated. A class of admissible parameter distributions is considered, which includes the normal distribution as a special case and accommodates distributions with densities that decay more weakly than the density of the normal distribution. In this setting, QMC by randomly shifted lattice rules with so-called product weights is applicable, which allows the fast component-by-component construction algorithm to have asymptotic computational cost that grows linearly with respect to the truncation dimension, i.e., the number of terms in the expansion of the random field input. The random solution is spatially discretized by the finite element method, which has asymptotically optimal convergence rates also in non-convex polygonal or polyhedral domains if suitable local mesh refinement is used. The computational cost that is required to achieve an error threshold is studied asymptotically. In this thesis it is shown that multilevel QMC is in certain cases able to achieve an error with computational cost that is asymptotically essential the computational cost of solving the respective elliptic PDE with known, deterministic coefficient function. In particular, also the computational cost of iterative methods to approximate the solution is incorporated.