We introduce a level set based approach to Bayesian geometric inverse problems. In these problems the interface between different domains is the key unknown, and is realized as the level set of a function. This function itself becomes the object of the inference. Whilst the level set methodology has been widely used for the solution of geometric inverse problems, the Bayesian formulation that we develop here contains two significant advances: firstly it leads to a well-posed inverse problem in which the posterior distribution is Lipschitz with respect to the observed data; and secondly it leads to computationally expedient algorithms in which the level set itself is updated implicitly via the MCMC methodology applied to the level set functionno explicit velocity field is required for the level set interface. Applications are numerous and include medical imaging, modelling of subsurface formations and the inverse source problem; our theory is illustrated with computational results involving the last two applications.

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This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schrödinger equation on the d-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension d, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.

This paper is concerned with the long time behavior of Langevin dynamics of Coulomb gases in R d with d ≥ 2, that is a second order system of Brownian particles driven by an external force and a pairwise repulsive Coulomb force. We prove that the system converges exponentially to the unique Boltzmann-Gibbs invariant measure under a weighted total variation distance. The proof relies on a novel construction of Lyapunov function for the Coulomb system.2010 Mathematics Subject Classification. Primary: 37A25, 60H10, 82C31,60K35; Secondary: 60B20.

Abstract. This paper concerns the approximation of probability measures on R d with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the best approximation, with respect to this divergence, from certain sets of Gaussian measures and Gaussian mixtures. The asymptotic behavior of such best approximations is then studied in the small parameter limit where the measure concentrates; this asympotic behavior is characterized using Γ-convergence. The theory developed is then applied to understand the frequentist consistency of Bayesian inverse problems in finite dimensions. For a fixed realization of additive observational noise, we show the asymptotic normality of the posterior measure in the small noise limit. Taking into account the randomness of the noise, we prove a Bernstein-Von Mises type result for the posterior measure. 1. Introduction. In this paper, we study the "best" approximation of a general finite dimensional probability measure, which could be non-Gaussian, from a set of simple probability measures, such as a single Gaussian measure or a Gaussian mixture family. We define "best" to mean the measure within the simple class which minimizes the Kullback-Leibler divergence between itself and the target measure. This type of approximation is central to many ideas, especially including the so-called variational inference [30], that are widely used in machine learning [3]. Yet such approximation has not been the subject of any substantial systematic underpinning theory. The purpose of this paper is to develop such a theory in the concrete finite dimensional setting in two ways: (i) by establishing the existence of best approximations and (ii) by studying their asymptotic properties in a measure concentration limit of interest. The abstract theory is then applied to study frequentist consistency [28] of Bayesian inverse problems.