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International Journal for Uncertainty Quantification
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ISSN Imprimer: 2152-5080
ISSN En ligne: 2152-5099

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International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2017019428
pages 189-205

A FULLY ADAPTIVE INTERPOLATED STOCHASTIC SAMPLING METHOD FOR LINEAR RANDOM PDES

Felix Anker
Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany
Christian Bayer
Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany
Martin Eigel
Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany
Johannes Neumann
Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany
John Schoenmakers
Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Germany

RÉSUMÉ

A numerical method for the fully adaptive sampling and interpolation of linear PDEs with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The spatial domain is decomposed by a nonuniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.


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