On Multilevel Quadrature for Elliptic Stochastic Partial Differential Equations

  • Helmut Harbrecht
  • Michael Peters
  • Markus Siebenmorgen
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 88)


In this article, we show that the multilevel Monte Carlo method for elliptic stochastic partial differential equations is a sparse grid approximation. By using this interpretation, the method can straightforwardly be generalized to any given quadrature rule for high dimensional integrals like the quasi Monte Carlo method or the polynomial chaos approach. Besides the multilevel quadrature for approximating the solution’s expectation, a simple and efficient modification of the approach is proposed to compute the stochastic solution’s variance. Numerical results are provided to demonstrate and quantify the approach.


Quadrature Rule Quadrature Point Sparse Grid Finite Element Space Polynomial Chaos 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Helmut Harbrecht
    • 1
  • Michael Peters
    • 1
  • Markus Siebenmorgen
    • 1
  1. 1.Mathematisches InstitutBaselSwitzerland

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