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Multiscale method with patches for the propagation of localized uncertainties in stochastic models

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HAL Id: hal-01057088

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Submitted on 29 Aug 2014

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Multiscale method with patches for the propagation of

localized uncertainties in stochastic models

Florent Pled, Mathilde Chevreuil, Anthony Nouy

To cite this version:

Florent Pled, Mathilde Chevreuil, Anthony Nouy. Multiscale method with patches for the propagation of localized uncertainties in stochastic models. International Conference on Extended Finite Element Methods (XFEM 2013), Sep 2013, Lyon, France. �hal-01057088�

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International Conference on Extended Finite Element Methods - XFEM 2013 September 11 – 13, 2013, Lyon, France A. Gravouil, Y. Renard and A. Combescure

MULTISCALE METHOD WITH PATCHES FOR THE PROPAGATION

OF LOCALIZED UNCERTAINTIES IN STOCHASTIC MODELS

F. Pled

GeM - LUNAM Universit´e, ´Ecole Centrale Nantes, Nantes, France, florent.pled@ec-nantes.fr M. Chevreuil

GeM - LUNAM Universit´e, Universit´e de Nantes, Nantes, France, mathilde.chevreuil@univ-nantes.fr A. Nouy

GeM - LUNAM Universit´e, ´Ecole Centrale Nantes, Nantes, France, anthony.nouy@ec-nantes.fr

Key Words: Uncertainty quantification, multiscale stochastic PDE, numerical zoom, fictitious domain methods, tensor approximation, high dimensional problems.

ABSTRACT

While methods based on functional approaches for uncertainty quantification in physical models have reached maturity, multiscale stochastic models have recently been the focus of new numerical develop-ments. Here we specifically take an interest in multiscale problems with numerous localized uncertain-ties at a micro level that can be associated with some variability in the operator or source terms, or even with some geometrical uncertainty. In order to handle the high dimensionality and the complexity that issue from such problems, a multiscale method based on patches has emerged as a relevant candidate for exploiting the localized side of uncertainties and has been extended to the stochastic framework in [1]. It proposes an efficient iterative global-local algorithm where the global problems at the macro level are made simple by introducing a fictitious patch that enables to define the (possibly coarse) global problem on a domain that contains no small scale geometrical details and that involves a deterministic operator. At the micro level, specific reformulations of local problems using fictitious domain methods [2] are introduced when the patch contains internal boundaries in order to formulate the local problem on a tensor product space. The global and local problems are solved using tensor based approximation methods [3] that allow the representation of high dimensional stochastic parametric solutions and at the same time make the stochastic methods non intrusive. In the present work, the approach is extended to problems with local non-linearities within the patches for which convergence properties are shown. We will also consider patches with variable positions which involve non conforming interfaces and for which rise questions of stability of approximation and optimal convergence with respect to the mesh.

REFERENCES

[1] M. Chevreuil, A. Nouy and E. Safatly. A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties. Comput. Methods Appl. Mech. Engrg., Vol. 255, 255–274, 2013.

[2] A. Nouy, M. Chevreuil, and E. Safatly. Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains. Computer Methods in Applied Mechanics and Engineering, Vol. 200, 3066–3082, 2011.

[3] W. Hackbusch. Tensor Spaces and Numerical Tensor Calculus, Vol. 42 of Series in Computa-tional Mathematics. Springer, 2012.

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