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Multilevel Quasi-Monte Carlo Uncertainty Quantification for Advection-Diffusion-Reaction

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Book cover Monte Carlo and Quasi-Monte Carlo Methods (MCQMC 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 324))

Abstract

We survey the numerical analysis of a class of deterministic, higher-order QMC integration methods in forward and inverse uncertainty quantification algorithms for advection-diffusion-reaction (ADR) equations in polygonal domains \(D\subset {\mathbb {R}}^2\) with distributed uncertain inputs. We admit spatially heterogeneous material properties. For the parametrization of the uncertainty, we assume at hand systems of functions which are locally supported in D. Distributed uncertain inputs are written in countably parametric, deterministic form with locally supported representation systems. Parametric regularity and sparsity of solution families and of response functions in scales of weighted Kontrat’ev spaces in D are quantified using analytic continuation.

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Notes

  1. 1.

    For spaces V of real-valued functions, \(V^*\) denotes the Hilbertian dual; in the case that solutions \(q({\varvec{y}})\) are complex-valued, e.g. for Helmholtz problems, \(V^*\) denotes the antidual of V. Even for parametric models with real valued solutions, complexification is required for analytic continuation to complex parameters [5].

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Acknowledgements

This work was supported in part by the Swiss National Science Foundation (SNSF) under grant SNF 159940. This work was completed when LH was a member of the Seminar for Applied Mathematics at ETH Zürich. The authors thank the editors and one reviewer for his/her careful reading and the constructive remarks on our initial submission which improved the presentation.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science, Altenbergerstrasse 69, 4040, Linz, Austria

    Lukas Herrmann

  2. Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zurich, Switzerland

    Christoph Schwab

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  1. Lukas Herrmann
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  2. Christoph Schwab
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Correspondence to Christoph Schwab .

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  1. INRIA Rennes Bretagne-Atlantique, Campus Universtaire de Beaulieu, Rennes, France

    Bruno Tuffin

  2. Département d’informatique et de Recherche Opérationnelle (DIRO), Université de Montréal, Montreal, QC, Canada

    Pierre L'Ecuyer

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Herrmann, L., Schwab, C. (2020). Multilevel Quasi-Monte Carlo Uncertainty Quantification for Advection-Diffusion-Reaction. In: Tuffin, B., L'Ecuyer, P. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2018. Springer Proceedings in Mathematics & Statistics, vol 324. Springer, Cham. https://doi.org/10.1007/978-3-030-43465-6_2

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