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Quelques problèmes liés à l'erreur statistique en homogénéisation stochastique

Abstract : In this thesis, we design numerical techniques to address the homogenization of equations the coefficients of which exhibit small scale random heterogeneities. Solving such elliptic partial differential equations is prohibitively expensive. One may use stochastic homogenization theory to reduce the complexity of this task. We then substitute the random, fine scale oscillating coefficients of the equation with constant homogenized coefficients. These coefficients are defined through an ergodic average inaccessible to practical computation. Only random approximations thereof are available. The error committed in this approximation is significant. These issues are detailed in the introductory Chapter 1. In Chapter 2, we show how to reduce the error in this approximation, in a nonlinear case, by using an antithetic variable estimator that has a smaller variance than the standard Monte Carlo estimator. In Chapter 3, in a linear case, we show how to obtain an even better variance reduction with the control variate method. Such a method is based on a surrogate model. In Chapter 4, we use a selection method to reduce the global error. Chapter 5 is devoted to the analysis of an inverse problem, wherein we seek parameters at the fine scale whilst only being provided with a handful of macroscopic quantities, among which the homogenized coefficients
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William Minvielle. Quelques problèmes liés à l'erreur statistique en homogénéisation stochastique. Mathématiques générales [math.GM]. Université Paris-Est, 2015. Français. ⟨NNT : 2015PESC1128⟩. ⟨tel-01284893⟩



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