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–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 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.

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    We omit their proofs, as the first lemma essentially follows from the linearity of the Fréchet derivative, the second can be proved by an iterated use of the Leibniz formula for Fréchet derivatives and the third one is a simple modification of the proof shown in [27, proof of Proposition 1.4.2] for the composition of real analytic functions from \({\mathbb {R}}\) to \({\mathbb {R}}\).

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    Clearly, in practice the Karhunen–Loève expansion also has to be truncated after the first M summands for some \(M \in {\mathbb {N}}\). However, the Taylor expansion of u at the point \({\varvec{0}}\) and the bounds from Theorem 3 imply that the error incurred by such a truncation tends to 0 as M tends to infinity. Thus, a large enough M can always be choosen to give the desired accuracy, while, as the quadrature considered has constants that do not depend on M, increasing the M also does not deteriorate the accuracy of the quadrature error.


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The work of the authors was supported by the Swiss National Science Foundation (SNSF) through the project “Multilevel Methods and Uncertainty Quantification in Cardiac Electrophysiology” (Grant 205321_169599).

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Correspondence to Marc Schmidlin.

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Harbrecht, H., Schmidlin, M. Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion. Stoch PDE: Anal Comp 8, 54–81 (2020).

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  • Uncertainty quantification
  • Anisotropic diffusion
  • Regularity estimates
  • Multilevel methods

Mathematics Subject Classification

  • 35R60
  • 65N30
  • 60H35