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

by L. Herrmann and Ch. Schwab

(Report number 2017-19)

Abstract
We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element (FE) Method for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain $D\subset \mathbb{R}^d$. The multilevel algorithm $Q^*_L$ which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in [Frances Y. Kuo, Christoph Schwab, and Ian H. Sloan: Multi-level quasi-Monte Carlo finite element methods for a class of elliptic PDEs with random coefficients, Found. Comput. Math. {\bf 15} (2015) pp. 411--449]. The random coefficient is assumed to admit a representation with \emph{locally 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 [Lukas Herrmann and Christoph Schwab: QMC integration for lognormal-parametric, elliptic PDEs: local supports and product weights, Numer. Math., published online 11 August 2018]. It also extends the MLQMC error analysis in [Frances Y. Kuo, Robert Scheichl, Christoph Schwab, Ian H. Sloan, and Elisabeth Ullmann: Multilevel quasi-Monte Carlo methods for lognormal diffusion problems, Math. Comp. {\bf 86} (2017) pp. 2827--2860], to locally supported basis functions in the representation of the Gaussian random field (GRF) in $D$, and to product weights in QMC integration. In particular, in polytopal domains $D\subset \mathbb{R}^d$, $d=2,3$, 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 of $D$. This allows for \emph{non-stationary} GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in $D$ with boundary conditions on $\partial D$. In the weighted function spaces in $D$, first order, Lagrangean Finite Elements on regular, locally refined, simplicial triangulations of $D$ yield optimal asymptotic convergence rates. Comparison of the $\varepsilon$-complexity for a class of Mat\'{e}rn-like GRF inputs indicates, for input GRFs with low path regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of \KL type. Our analysis yields general bounds for the $\varepsilon$-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

Keywords: Quasi-Monte Carlo methods, multilevel quasi-Monte Carlo, uncertainty quantification, error estimates, high-dimensional quadrature, elliptic partial differential equations with lognormal input

@Techreport{HS17_715,
  author = {L. Herrmann and Ch. Schwab},
  title = {Multilevel quasi-Monte Carlo integration with product weights for elliptic PDEs with lognormal coefficients},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-19.pdf },
  year = {2017}
}

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