Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the regularity of the solution's dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel collocation and multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution's mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

Elliptic partial differential equations with diffusion coefficients of lognormal form, that is $a=exp(b)$, where $b$ is a Gaussian random field, are considered. We study the $\ell^p$ summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of $b$. These summability results have direct consequences on the approximation rates of best $n$-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of $b$, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Lo\`eve representation of $b$ may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion

Sparse Quadrature for High-Dimensional Integration with Gaussian Measure

In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions of the exactness and the boundedness of univariate quadrature rules as well as the regularity of the parametric functions with respect to the parameters, we obtain the convergence rate $O(N^{-s})$, where $N$ is the number of indices, and $s$ is independent of the number of the parameter dimensions. Moreover, we propose both an a-priori and an a-posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates