Multiscale domain decomposition method for solving high-dimensional non-linear stochastic problems with localized uncertainties and non-linearities

Abstract : During the last few decades, functional approaches for uncertainty quantification and propagation (henceforth known as spectral stochastic methods) have been a subject of growing interest and are currently employed for solving complex multiscale stochastic problems. Such multiscale stochastic models may exhibit some variabilities in the material properties (affecting the operator), the boundary conditions (affecting the loading and the source term) or the geometry at different scales. Classical monoscale approaches based on adaptive remeshing (e.g. mesh refinement) or enrichement techniques may lead to high computational costs and memory storage. Conversely, multiscale approaches based on patches allow to take the high solution complexity into account by operating a separation of scales. A multiscale coupling statregy devoted to stochastic problems featuring localized uncertainties hase been recently proposed in [1]. It relies on an overlapping domain decomposition method and leads to a global-local (two-scale) formulation of the stochastic problem. The associated global-local iterative algorithm requires the successive solution of a series of simplified global problems (with deterministic operators) defined on a deterministic domain and complex local problems (with uncertain operators and/or geometry) defined on subdomains of interest called patches. Convergence and robustness properties of the algorithm have been analyzed in [1] for linear elliptic stochastic problems. In the present work, the method is extended to non-linear elliptic stochastic problems with locaized variabilities and non-linearities. Convergence and robustness results of the global-local iterative algorithm are shown for a class of non-linear elliptic stochastic problems. The multiscale coupling approach appears to be flexible and non-intrusive allowing to handle different models, approximation spaces and dedicated solvers at both local and global levels. The stochastic local problems are solved in parallel using a least-squares minimization method (non-intrusive sampling approach) which asks for the evaluation of samples of local solutions thanks to non-linear deterministic codes. Greedy algorithms proposed in [2] and dedicated to the solution of high-dimensional stochastic problems allow the adaptive construction of sparse polynomial approximations. The approximation error of local solutions is controlled by adapting the stochastic approximation basis and the number of samples. The performances of the multiscale domain decomposition method are illustrated through numerical examples carried out on a non-linear diffusion-reaction stochastic problem with localized random material heterogeneities. 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. Computer Methods in Applied Mechanics and Engineering, 255(0):255–274, 2013. [2] A. Chkifa, A. Cohen, R. DeVore, and C. Schwab. Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis, 47(1):253–280, 2013.
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Submitted on : Thursday, May 26, 2016 - 9:33:18 PM
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Florent Pled. Multiscale domain decomposition method for solving high-dimensional non-linear stochastic problems with localized uncertainties and non-linearities. Séminaire du Laboratoire de Mécanique des Sols, Structures et Matériaux (MSSMat), Nov 2015, Châtenay Malabry, France. ⟨hal-01308256⟩

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