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

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Date
2015
Volume
2060
Issue
Journal
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Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

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.

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Keywords
Partial differential equations with random coefficients, uncertainty quantification, multilevel Monte Carlo, adaptivity
Citation
Eigel, M., Merdon, C., & Neumann, J. (2015). An adaptive multi level Monte-Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data (Vol. 2060). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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