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The formulation of Bayesian inverse problems involves choosing prior distributions; choices that seem equally reason-able may lead to significantly different conclusions. We develop a computational approach to understand the impact of the hyperparameters defining the prior on the posterior statistics of the quantities of interest. Our approach relies on global sensitivity analysis (GSA) of Bayesian inverse problems with respect to the prior hyperparameters. This, however, is a challenging problem-a naive double loop sampling approach would require running a prohibitive number of Markov chain Monte Carlo (MCMC) sampling procedures. The present work takes a foundational step in making such a sensitivity analysis practical by combining efficient surrogate models and a tailored importance sampling approach. In particular, we can perform accurate GSA of posterior statistics of quantities of interest with respect to prior hyperparameters without the need to repeat MCMC runs. We demonstrate the effectiveness of the approach on a simple Bayesian linear inverse problem and a nonlinear inverse problem governed by an epidemiological model. 

Variance-based global sensitivity analysis (GSA) can provide a wealth of information when applied to complex models.A well-known Achilles' heel of this approach is its computational cost, which often renders it unfeasible in practice. An appealing alternative is to instead analyze the sensitivity of a surrogate model with the goal of lowering computational costs while maintaining sufficient accuracy. Should a surrogate be simple enough to be amenable to the analytical calculations of its Sobol' indices, the cost of GSA is essentially reduced to the construction of the surrogate.We propose a new class of sparse-weight extreme learning machines (ELMs), which, when considered as surrogates in the context of GSA, admit analytical formulas for their Sobol' indices and, unlike the standard ELMs, yield accurate approximations of these indices. The effectiveness of this approach is illustrated through both traditional benchmarks in the field and on a chemical reaction network. 

We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by partialdifferential equations (PDEs) with infinite-dimensional parameters. In previous works, HDSA has been used to assessthe sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on an inverse problem governed by a PDE modeling heat conduction. 

We present a computational framework for dimension reduction and surrogate modeling to accelerate uncertainty quantification in computationally intensive models with high-dimensional inputs and function-valued outputs. Our driving application is multiphase flow in saturated-unsaturated porous media in the context of radioactive waste storage. For fast input dimension reduction, we utilize an approximate global sensitivity measure, for function-valued outputs, motivated by ideas from the active subspace methods. The proposed approach does not require expensive gradient computations. We generate an efficient surrogate model by combining a truncated Karhunen-Loeve (KL) expansion of the output with polynomial chaos expansions, for the output KL modes, constructed in the reduced parameter space. We demonstrate the effectiveness of the proposed surrogate modeling approach with a comprehensive set of numerical experiments, where we consider a number of function-valued (temporally or spatially distributed) QoIs.