Γ -convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems

The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements

The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field. A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale. This paper contributes to the improvement of the computational efficiency, accuracy and robustness of such a method by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) of kinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used in a previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model in order to enhance both the robustness and the precision of the statistical inverse identification method. Finally, the proposed improved method is first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity (using multiscale simulated data obtained through numerical computations) and then exemplified on a real heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linear elasticity (using multiscale experimental data obtained through mechanical testing monitored by digital image correlation).

A scalable online algorithm for passive seismic tomography in underground mines

Understanding and monitoring the seismic responses of rock masses to massive mining are crucial for safety and economic viability of ever larger and deeper underground operations. Seismic monitoring can be used to detect stress variations and hazardous instabilities, but its effectiveness requires accurate estimations of the nonhomogeneous propagation velocity of microseismic waves. While predetermined velocity models are not accurate enough and might bias hypocenter localization, using active-source seismic tomography methods to estimate the velocity field provides limited spatial coverage. Thus, passive seismic tomography using first-arrival traveltimes of mining-induced microseisms (of unknown hypocenters) constitutes a promising tool. However, available methods solving this high-dimensional statistical inverse problem do not scale well with the data set size and cannot easily refine or update estimations with new data. We have developed a novel passive seismic tomography method able to dynamically learn the nonhomogeneous velocity field from a streaming of noisy first-arrival times, online (in real time) or from catalogs. We have developed a new Bayesian approach that avoids linearizing the forward problem and allows for general 3D velocity models. This is combined with the use of the stochastic gradient descent (SGD) method, which underlies much of the recent progress in machine learning and provides increasing accuracy at a cost scaling linearly with the data set size. Moreover, we introduce an adaptive variant of SGD based on raypath density, which significantly improves the speed of the algorithm, and we implement a parallel version of our method enabling its systematic use in real applications. These include the design of optimal sensor locations, the dynamic update of velocity estimates in production conditions, and the real-time determination of hypocenters and their uncertainty. Our method’s reach and effectiveness are illustrated with simulated seismic data on 3D checkerboards, using synthetic and real acquisition geometries, and on a dense 2D velocity grid.

Bayesian statistical approach to binary asteroid orbit determination

The problem of binary asteroids orbit determination is of particular interest, given knowledge of the orbit is the best way to derive the mass of the system. Orbit determination from observed points is a classic problem of celestial mechanics. However, in the case of binary asteroids, particularly with a small number of observations, the solution is not evident to derive. In the case of resolved binaries the problem consists in the determination of the relative orbit from observed relative positions of a secondary asteroid with respect to the primary. In this work, the problem is investigated as a statistical inverse problem. Within this context, we propose a method based on Bayesian modelling together with a global optimisation procedure that is based on the simulated annealing algorithm.