WIAS Preprint No. 2060, (2014)

An adaptive multi level Monte--Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data


  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Merdon, Christian
    ORCID: 0000-0002-3390-2145
  • Neumann, Johannes

2010 Mathematics Subject Classification

  • 35R60 47B80 60H35 65C20 65N12 65N22 65J10


  • partial differential equations with random coefficients, uncertainty quantification, multilevel Monte Carlo, adaptivity




The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.

Appeared in

  • SIAM ASA J. Uncertain. Quantif., 4 (2016) pp. 1219--1245.

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