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.

Probabilistic and Bayesian methods for uncertainty quantification of deterministic and stochastic differential equations

Giacomo Garegnani

In this thesis we explore uncertainty quantification of forward and inverse problems involving differential equations. Differential equations are widely employed for modeling natural and social phenomena, with applications in engineering, chemistry, meteorology, and economics. Mathematical models of complex systems in these fields require numerical methods, which introduce uncertainties in the outcome. Moreover, there has recently been a steep rise in the availability of data, which also come with an uncertainty. Therefore, blending mathematical models, data, and their respective uncertainties is nowadays of the utmost importance. The first part of this thesis is dedicated to two novel methods for multiscale inverse problems. We first consider an elliptic partial differential equation (PDE), with a diffusion tensor oscillating at a small scale. Given noisy observations of the solution, we consider the problem of inferring a slow-scale parametrization of the multiscale tensor. For this purpose, we combine numerical homogenization, which yields a single-scale surrogate of the full model, and the ensemble Kalman filter. The scheme we propose is accurate in the homogenized limit, and outperforms existing methods in terms of computational cost. We then study the error due to the mismatch between the full and the homogenized models, and show how to combine statistical techniques for model misspecification and our scheme. We then move to multiscale diffusion processes, and consider the problem of inferring effective dynamics from multiscale observations. A homogenized single-scale equation reproducing the full model exists also in this case. The effective model, though, is the subject of the inference procedure, and not only a computational tool. The resulting issue of model misspecification is usually bypassed by subsampling at an appropriate rate, which is non-trivial to choose, and which may give misleading results. We avoid subsampling by designing a novel technique based on filtered data, and show how to modify classical estimators and obtain an effective equation consistently with homogenization. Our technique is robust and can be employed as a black-box tool for inferring effective surrogates of complex stochastic models.In the second part we present two novel schemes belonging to the field of probabilistic numerics, whose purpose is to provide a statistical description of the uncertainty due to numerical discretization. We first consider ordinary differential equations (ODEs), and introduce a probabilistic integrator based on random time steps and Runge-Kutta methods (RTS-RK). Tuning the distribution of the time steps, we generate a probability measure on the solution which allows for a consistent uncertainty quantification of numerical errors. Unlike previous probabilistic methods in literature, our scheme inherits the geometric properties of the underlying deterministic integrators. In particular, we show long-time energy conservation when the RTS-RK is applied to Hamiltonian ODEs. We employ the idea of randomizing the discretization to propose a random mesh finite element method (RM-FEM) for elliptic PDEs. We prove that the measure induced by the RM-FEM on the solution can be employed to derive a posteriori error estimators. Hence, the RM-FEM provides a consistent statistical characterization of numerical errors. For both our novel schemes, we demonstrate the usefulness of the probabilistic approach in Bayesian inverse problems.

EPFL2021

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

Numerical methods for deterministic and stochastic differential equations with multiple scales and high contrasts

Giacomo Rosilho De Souza

Mathematical models involving multiple scales are essential for the description of physical systems. In particular, these models are important for the simulation of time-dependent phenomena, such as the heat flow, where the Laplacian contains mixed and indistinguishable fast and slow modes. Stationary problems can also exhibit a multiscale nature. For example, elliptic equations governed by a diffusion coefficient with strong discontinuities have solutions characterized by regions with a high gradient. Simulating such models is very demanding, as the computational cost of standard numerical methods is usually ruled by the fastest dynamics or the smallest scale. In the first part of this thesis, we develop multirate integration methods for deterministic and stochastic time-dependent problems with disparate time-scales. The cost of traditional schemes for such problems is prohibitive due to step size restrictions in the explicit case or solutions to large nonlinear systems in the implicit case. Existing multirate methods are either implicit or make use of interpolations, which trigger instabilities, or are based on a scale separation assumption, which is not satisfied by parabolic problems. Here we introduce a new framework based on modified equations which allows for the development of a whole new class of interpolation-free explicit multirate numerical methods, which do not need any scale separation, are stable and accurate. For deterministic problems, our methodology is based on the replacement of the original right-hand side by an averaged force, whose stiffness is reduced due to a fast but cheap auxiliary problem. Integrating the modified equation and the auxiliary problems by explicit schemes is generally cheaper than integrating the original problem. We thus introduce a multirate method based on stabilized explicit schemes and prove its efficiency, stability and accuracy. Numerical experiments show that standard schemes and our multirate approach provide essentially the same solutions; hence, the bottleneck caused by the stiffness of a few degrees of freedom is overcome without sacrificing accuracy. We also generalize the same framework to stochastic differential equations, where we need to introduce a damped diffusion term for which the resulting modified equation inherits the mean-square stability properties of the original problem. An interpolation-free stabilized explicit multirate method for stochastic equations is then derived. In the second part of this thesis, we consider elliptic problems with high gradients and develop a local adaptive discontinuous Galerkin scheme. Local methods for such problems already exist in literature; however, they are usually based on iterations and have several downsides. In particular, their a priori error analysis is based on rather strong and nonphysical assumptions and they lack a rigorous a posteriori error analysis. The scheme that we propose is based on a coarse solution on the full domain which is subsequently improved by solving local elliptic problems only once on subdomains with artificial boundary conditions. The a priori error analysis is performed under minimal regularity assumptions due to the gradient discretization framework. Furthermore, we derive a posteriori error estimators based on conforming fluxes and potential reconstructions which can be used to identify the local subdomains on the fly, are free of undetermined constants and robust in singularly perturbed regimes.

EPFL2020