Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion
Provides theoretical justification for using multilevel methods in uncertainty quantification for anisotropic random diffusion problems, which is incremental for the field of computational PDEs.
The paper analyzes the regularity of solutions to elliptic PDEs with random anisotropic diffusion, showing that the decay of the Karhunen-Loève expansion determines parameter regularity. This enables multilevel methods to reduce computational complexity for uncertainty quantification while maintaining convergence rates, validated in 3D.
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-Loève 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-Loève 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.