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Adjoint Error Estimation for Stochastic Collocation Methods

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Sparse Grids and Applications - Munich 2012

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 97))

Abstract

This paper deals with partial differential equations with random input data. An efficient way of solving such problems is adaptive stochastic collocation on sparse grids. For higher efficiency and a better understanding of the method, we derive adjoint error estimates for nonlinear stochastic solution functionals. The resulting adjoint problem also involves random parameters and can be treated by stochastic collocation as well. Only a few adjoint evaluations are required in order to estimate the deterministic error, while the stochastic error requires much more effort to be detected. To overcome these substantial additional costs, we suggest to replace the adjoint problem by a reduced model and demonstrate the applicability of the approach for nonlinear solution functionals and up to nine random dimensions.

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Acknowledgements

This work was supported by the “Excellence Initiative” of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.

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Authors and Affiliations

  1. Numerical Analysis and Scientific Computing (Department of Mathematics), Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Bettina Schieche

  2. Numerical Analysis and Scientific Computing (Department of Mathematics) & Graduate School of Computational Engineering, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Jens Lang

Authors
  1. Bettina Schieche
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  2. Jens Lang
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Correspondence to Jens Lang .

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Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Jochen Garcke

  2. Institute for Parallel and Distributed Systems, Universität Stuttgart, Stuttgart, Germany

    Dirk Pflüger

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Schieche, B., Lang, J. (2014). Adjoint Error Estimation for Stochastic Collocation Methods. In: Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2012. Lecture Notes in Computational Science and Engineering, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-319-04537-5_12

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