We consider the Bayesian inverse homogenization problem of recovering the locally periodic two scale coefficient of a two scale elliptic equation, given limited noisy information on the solution. We consider both the uniform and the Gaussian prior probability measures. We use the two scale homogenized equation whose solution contains the solution of the homogenized equation which describes the macroscopic behaviour, and the corrector which encodes the microscopic behaviour. We approximate the posterior probability by a probability measure determined by the solution of the two scale homogenized equation. We show that the Hellinger distance of these measures converges to zero when the microscale converges to zero, and establish an explicit convergence rate when the solution of the two scale homogenized equation is sufficiently regular. Sampling the posterior measure by Markov Chain Monte Carlo (MCMC) method, instead of solving the two scale equation using fine mesh for each proposal with extremely high cost, we can solve the macroscopic two scale homogenized equation. Although this equation is posed in a high dimensional tensorized domain, it can be solved with essentially optimal complexity by the sparse tensor product finite element method, which reduces the computational complexity of the MCMC sampling method substantially. We show numerically that observations on the macrosopic behaviour alone are not sufficient to infer the microstructure. We need also observations on the corrector. Solving the two scale homogenized equation, we get both the solution to the homogenized equation and the corrector. Thus our method is particularly suitable for sampling the posterior measure of two scale coefficients.
In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale Bayesian inverse problems. The key idea is to exploit (model-based) and construct (data-based) the intrinsic approximate low-dimensional structure of the underlying problem which consists of two components - a training component that computes a set of data-driven basis to achieve significant dimension reduction in the solution space, and a fas…
We propose a multilevel Monte Carlo-FEM algorithm to solve elliptic Bayesian inverse problems with "Besov random tree prior". These priors are given by a wavelet series with stochastic coefficients, and certain terms in the expansion vanishing at random, according to the law of so-called Galton-Watson trees. This allows to incorporate random fractal structures and large deviations in the log-diffusion, which occur naturally in many applications from geophysics or medical imaging. This framework entails two main difficulties: First, the associated diffusion coefficient does not satisfy a unifor…
In this work, we present a novel error analysis for recovering a spatially dependent diffusion coefficient in an elliptic or parabolic problem. It is based on the standard regularized output least-squares formulation with an $H^1(Ω)$ seminorm penalty, and then discretized using the Galerkin finite element method with conforming piecewise linear finite elements for both state and coefficient, and backward Euler in time in the parabolic case. We derive \textit{a priori} weighted $L^2(Ω)$ estimates where the constants depend only on the given problem data for both elliptic and parabolic cases. Fu…
In this work, we investigate a class of elliptic inverse problems and aim to simultaneously recover multiple inhomogeneous inclusions arising from two different physical parameters, using very limited boundary Cauchy data collected only at one or two measurement events. We propose a new fast, stable and highly parallelable direct sampling method (DSM) for the simultaneous reconstruction process. Two groups of probing and index functions are constructed, and their desired properties are analyzed. In order to identify and decouple the multiple inhomogeneous inclusions of different physical natur…
Often, when solving forward, inverse or data assimilation problems, only a part of the solution is needed. As a model, we consider the stationary diffusion problem. We demonstrate an algorithm that can compute only a part or a functional of the solution, without calculating the full inversion operator and the complete solution. It is a well-known fact about partial differential equations that the solution at each discretisation point depends on the solutions at all other discretisation points. Therefore, it is impossible to compute the solution only at one point, without calculating the soluti…
We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a probability measure on standard piecewise linear FEM. We present a posteriori error estimators based uniquely on probabilistic information. A series of numerical experiments illustrates the potential of the RM-FEM for error estimation and validates our analysis. We furthermore demonstrate how employing the RM-FEM enhances the quality of the sol…
We explore the possibility for using boundary data to identify sources in elliptic PDEs. Even though the associated forward operator has a large null space, it turns out that box constraints, combined with weighted sparsity regularization, can enable rather accurate recovery of sources with constant magnitude/strength. In addition, for sources with varying strength, the support of the inverse solution will be a subset of the support of the true source. We present both an analysis of the problem and a series of numerical experiments. Our work only addresses discretized problems. The reason for …
