A Bayesian numerical homogenization method for elliptic multiscale inverse problems

A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings

Ensemble Kalman filter for multiscale inverse problems

We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analysed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is provided, and convergence of the approximate conditional posterior distribution is proved with respect to the Wasserstein distance. A numerical experiment validates our methodology, with a particular emphasis on modelling error and computational cost

Drift Estimation of Multiscale Diffusions Based on Filtered Data

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure

Drift Estimation of Multiscale Diffusions Based on Filtered Data

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure

A Bayesian numerical homogenization method for elliptic multiscale inverse problems

A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings

Numerical Homogenization And Model Order Reduction For Multiscale Inverse Problems

A new numerical method based on numerical homogenization and model order reduction is introduced for the solution of multiscale inverse problems. We consider a class of elliptic problems with highly oscillatory tensors that varies on a microscopic scale. We assume that the micro structure is known and seek to recover a macroscopic scalar parameterization of the microscale tensor (e.g., volume fraction). Departing from the full fine-scale model, which would require mesh resolution for the forward problem down to the finest scale, we solve the inverse problem for a coarse model obtained by numerical homogenization. The input data, i.e., measurement from the Dirichlet-to-Neumann map, are solely based on the original fine-scale model. Furthermore, reduced basis techniques are used to avoid computing effective coefficients for the forward solver at each integration point of the macroscopic mesh. Uniqueness and stability of the effective inverse problem is established based on standard assumptions for the fine-scale model, and a link to this latter model is established by means of G-convergence. A priori error estimates are established for our method. Numerical experiments illustrate the efficiency of the proposed scheme and confirm our theoretical findings

Penalization and Bayesian numerical methods for multiscale inverse problems

In this thesis we consider inverse problems involving multiscale elliptic partial differential equations. The name multiscale indicates that these models are characterized by the presence of parameters which vary on different spatial scales (macroscopic, microscopic, mesoscopic, etc.). The variations at the smallest scales make these equations very difficult to approximate also when considering forward problems, since classical numerical methods require a mesh resolution at the finest scales, hence a computational cost that is often prohibitive. For this reason one prefers to apply homogenization or effective methods which, neglecting what happens at the smallest scales, are able to provide accurate macroscopic solutions to the problem. For what concerns the solution of inverse problems, we propose then a new numerical algorithm based on homogenization techniques, model order reduction and regularization methods. First, we consider elliptic operators whose tensor varies on a microscopic scale. Under the assumption that the nature of its micro structure is known, we aim at recovering a macroscopic parameterization of the tensor from measurements originating from the full multiscale model, using homogenization. Practical examples include multi-phase media whose constituents are known, but their respective volume fraction is unknown. We consider the Calderón's formulation of the inverse problem. We prove that, under some regularity assumptions on the fine scale tensor, the effective inverse problem, with observed data consisting of the homogenized Dirichlet to Neumann (DtN) map, is also well-posed. We then solve the problem by considering finite measurements of the multiscale DtN map and using Tikhonov regularization, and we establish a convergence result of the solution by means of G-convergence. In a second stage, we consider a Bayesian approach which allows for uncertainty quantification of the results. We prove existence and well-posedness of the effective posterior probability measure, obtained by homogenization of the observation operator. By means of G-convergence we characterize the discrepancy between the fine scale and the homogenized model, and we prove convergence of the effective posterior towards the fine scale posterior in terms of the Hellinger distance. We also propose a numerical procedure to estimate the homogenization error statistics, which, if included in the inversion process, allow to account for approximation errors. Finally, we deal with multiscale inverse problems for the linear elasticity equation. In this context we assume that the heterogeneity of the material is determined by its geometry rather than by the coefficients of the equation. In particular, we consider porous media with random perforations and, following the Bayesian approach, we solve the inverse problem of determining the elastic properties of an hypothetical isotropic material. We prove the existence and well-posedness of the effective posterior measure, as well as its convergence in the fine scale limit by means of G-convergence. We conclude by describing a new probabilistic numerical method which computes a new posterior measure that accounts for approximation errors and reveals the uncertainty intrinsic in the numerical method

A Bayesian approach for multiscale inverse problems

In this talk we discuss a Bayesian approach for inverse problems involving elliptic differential equations with multiple scales. Computing repeated forward problems in a multiscale context is computationnally too expensive and we propose a new strategy based on the use of "effective" forward models originating from homogenization theory. Convergence of the true posterior distribution for the parameters of interest towards the homogenized posterior is established via G-convergence for the Hellinger metric. A computational approach based on numerical homogenization and reduced basis methods is proposed for an efficient evaluation of the forward model in a Markov Chain Monte Carlo procedure. We also discuss a methodology to account for the modeling error introduced by the effective forward model and the combination of the Bayesian multiscale method with a probabilisitic approach to quantify the uncertainty in building the effective forward model for a multiscale elastic problem in random media. References: A. Abdulle, A. Di Blasio, Numerical homogenization and model order reduction for multiscale inverse problems, to appear in SIAM MMS. A. Abdulle, A. Di Blasio, A Bayesian numerical homogenization method for elliptic multiscale inverse problems, Preprint submitted for publication.Non UBCUnreviewedAuthor affiliation: Ecole Polytechnique Fédérale de LausanneFacult