WIAS Preprint No. 2200, (2015)

Adaptive SDE based interpolation for random PDEs


  • Anker, Felix
  • Bayer, Christian
    ORCID: 0000-0002-9116-0039
  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Neumann, Johannes
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266


  • random PDE, stochastic differential equation, Feynman-Kac, interpolation, finite element, a posteriori error estimator, adaptive method, Euler Maruyama




A numerical method for the fully adaptive sampling and interpolation of PDE 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 physical domain is decomposed subject to a non-uniform 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.

Appeared in

  • Int. J. Uncertain. Quantif., 7 (2017), pp. 189--205; changed title: A fully adaptive interpolated stochastic sampling method for linear random PDEs

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