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Bayesian inference provides a systematic framework for integration of datawith mathematical models to quantify the uncertainty in the solution of theinverse problem. However, the solution of Bayesian inverse problems governed bycomplex forward models described by partial differential equations (PDEs)remains prohibitive with black-box Markov chain Monte Carlo (MCMC) methods. Wepresent hIPPYlib-MUQ, an extensible and scalable software framework thatcontains implementations of state-of-the art algorithms aimed to overcome thechallenges of high-dimensional, PDE-constrained Bayesian inverse problems.These algorithms accelerate MCMC sampling by exploiting the geometry andintrinsic low-dimensionality of parameter space via derivative information andlow rank approximation. The software integrates two complementary open-sourcesoftware packages, hIPPYlib and MUQ. hIPPYlib solves PDE-constrained inverseproblems using automatically-generated adjoint-based derivatives, but it lacksfull Bayesian capabilities. MUQ provides a spectrum of powerful Bayesianinversion models and algorithms, but expects forward models to come equippedwith gradients and Hessians to permit large-scale solution. By combining thesetwo libraries, we created a robust, scalable, and efficient software frameworkthat realizes the benefits of each and allows us to tackle complex large-scaleBayesian inverse problems. To illustrate the capabilities of hIPPYlib-MUQ, wepresent a comparison of a number of MCMC methods on several inverse problems.These include problems with linear and nonlinear PDEs, various noise models,and different parameter dimensions. The results demonstrate that large ($\sim50\times$) speedups over conventional black box and gradient-based MCMCalgorithms can be obtained by exploiting Hessian information (from the logposterior), underscoring the power of the integrated hIPPYlib-MUQ framework.

hIPPYlib-MUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertainty