Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients

Abstract

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domainD⊂ ℝd. The multilevel algorithmQL*which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuoet al.(Found. Comput. Math.15(2015) 411–449). The random coefficient is assumed to admit a representation withlocally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math.141(2019) 63–102). It also extends the MLQMC error analysis in Kuoet al.(Math. Comput.86(2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) inD, and to product weights in QMC integration. In particular, in polytopal domainsD⊂ ℝd,d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices ofD. This allows fornon-stationaryGRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators inDwith boundary conditions on∂D. In the weighted function spaces inD, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations ofDyield optimal asymptotic convergence rates. Comparison of theε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations,e.g.of Karhunen–Loève type. Our analysis yields general bounds for theε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.