Bayesian inverse analysis can be computationally burdensome when dealing with large scale numerical models dependent on high-dimensional stochastic input, and especially when model derivatives are unavailable, as is the case for many high-fidelity legacy codes. To overcome these limitations, we introduce a novel approach called Bayesian multi-fidelity inverse analysis (BMFIA), which utilizes computationally inexpensive lower fidelity models to construct a multi-fidelity likelihood function. This function can be learned robustly, and potentially adaptively, from a few high-fidelity simulations (100-300). Our approach incorporates in the resulting, multi-fidelity posterior the epistemic uncertainty stemming from the limited high-fidelity data and the information loss caused by the multi-fidelity approximation. BMFIA can handle non-linear dependencies between low- and high-fidelity models. In particular, when the former are differentiable the solution of high-dimensional problems can be achieved while maintaining the high-fidelity accuracy by the multi-fidelity likelihood. Bayesian inference is performed using state-of the-art sampling-based or variational methods which require solely evaluations of the lower-fidelity model. We demonstrate the applicability of BMFIA to large-scale biomechanical and coupled multi-physics problems.

In this work, we describe a new approach that uses variational encoder-decoder (VED) networks for efficient goal-oriented uncertainty quantification for inverse problems. Contrary to standard inverse problems, these approaches are goal-oriented in that the goal is to estimate some quantities of interest (QoI) that are functions of the solution of an inverse problem, rather than the solution itself. Moreover, we are interested in computing uncertainty metrics associated with the QoI, thus utilizing a Bayesian approach for inverse problems that incorporates the prediction operator and techniques for exploring the posterior. By harnessing recent advancements in the field of machine learning for large-scale inverse problems, in particular, by exploiting VED networks, we describe a data-driven approach for real-time goal-oriented uncertainty quantification for inverse problems.

We introduce a new method to reduce the dimension of the parameter and data space of high-dimensional Bayesian inverse problems. Commonly, different dimension reduction methods are applied separately to the two spaces. However, choosing a low-dimensional informed parameter subspace influences which data subspace is informative and vice versa. We thus propose a coupled method that, in addition, naturally reveals optimal experimental designs. Our method is based on the gradient of the forward operator of a Gaussian likelihood. It computes two projectors with an efficient and simple alternating singular value decomposition. Moreover, we control the approximation error through a certified $L^2$-error bound on the forward operator. We demonstrate the method on a large-scale Bayesian inverse problem in ocean modelling and use it to derive optimal sensor placements.

We are presenting a method to solve large dimensional Bayesian inverse problems where the parameter vector is assumed to be sparse. To this end, we use a Laplace-prior and show how the posterior density can be approximated based on a suitable coordinate splitting. Using an upper bound on the Hellinger distance between the exact and approximated posterior density, we show how the coordinate splitting can be performed.

Sampling from the approximated posterior is then straightforward and very efficient. Our theoretical framework allows also for sampling the exact posterior using a pseudo-marginal MCMC. However, this algorithm is relying heavily on a good coordinate splitting which might not be feasible in practice. Therefore, we propose a modified random-scan MCMC algorithm to sample from the exact posterior which is more robust and flexible.

In the end, we first illustrate the methodology on a simple example and then present the practical applicability on a large-dimensional 2D deblurring problem.