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Recent years have seen a massive explosion of datasets across all areas of science and engineering. The central questions are: How do we optimally learn from data through the lens of models? And how do we account for uncertainties in both data and models? These questions can be mathematically framed as Bayesian inverse problems. While powerful and sophisticated approaches have been developed to tackle these problems, such methods are often challenging to implement and typically require first and second order derivatives that are not always available in existing computational models. In this talk, we present an extensible software framework MUQ-hIPPYlib that overcomes these challenges by providing access to state-of-the-art algorithms that offer the capability to solve complex large-scale Bayesian inverse problems across a broad spectrum of scientific and engineering areas. 

We present an extensible software framework, hIPPYlib, for solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations (PDEs) with (possibly) infinite-dimensional parameter fields (which are high-dimensional after discretization). hIPPYlib overcomes the prohibitively expensive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The key property of the algorithms implemented in hIPPYlib is that the solution of the inverse problem is computed at a cost, measured in linearized forward PDE solves, that is independent of the parameter dimension. The mean of the posterior is approximated by the MAP point, which is found by minimizing the negative log-posterior with an inexact matrix-free Newton-CG method. The posterior covariance is approximated by the inverse of the Hessian of the negative log posterior evaluated at the MAP point. The construction of the posterior covariance is made tractable by invoking a low-rank approximation of the Hessian of the log-likelihood. Scalable tools for sample generation are also discussed. hIPPYlib makes all of these advanced algorithms easily accessible to domain scientists and provides an environment that expedites the development of new algorithms. 

Abstract. We consider the problem of inferring the basal sliding coefficientfield for an uncertain Stokes ice sheet forward model from syntheticsurface velocity measurements. The uncertainty in the forward modelstems from unknown (or uncertain) auxiliary parameters (e.g., rheologyparameters). This inverse problem is posed within the Bayesianframework, which provides a systematic means of quantifyinguncertainty in the solution. To account for the associated modeluncertainty (error), we employ the Bayesian approximation error (BAE)approach to approximately premarginalize simultaneously over both thenoise in measurements and uncertainty in the forward model. We alsocarry out approximative posterior uncertainty quantification based ona linearization of the parameter-to-observable map centered at themaximum a posteriori (MAP) basal sliding coefficient estimate, i.e.,by taking the Laplace approximation. The MAP estimate is found byminimizing the negative log posterior using an inexact Newtonconjugate gradient method. The gradient and Hessian actions to vectorsare efficiently computed using adjoints. Sampling from theapproximate covariance is made tractable by invoking a low-rankapproximation of the data misfit component of the Hessian. We studythe performance of the BAE approach in the context of three numericalexamples in two and three dimensions. For each example, the basalsliding coefficient field is the parameter of primary interest whichwe seek to infer, and the rheology parameters (e.g., the flow ratefactor or the Glen's flow law exponent coefficient field) representso-called nuisance (secondary uncertain) parameters. Our resultsindicate that accounting for model uncertainty stemming from thepresence of nuisance parameters is crucial. Namely our findingssuggest that using nominal values for these parameters, as is oftendone in practice, without taking into account the resulting modelingerror, can lead to overconfident and heavily biased results. We alsoshow that the BAE approach can be used to account for the additionalmodel uncertainty at no additional cost at the online stage. 

The deformations of several slender structures at nano-scale are conceivably sensitive to their non-homogenous elasticity. Owing to their small scale, it is not feasible to discern their elasticity parameter fields accurately using observations from physical experiments. Molecular dynamics simulations can provide an alternative or additional source of data. However, the challenges still lie in developing computationally efficient and robust methods to solve inverse problems to infer the elasticity parameter field from the deformations. In this paper, we formulate an inverse problem governed by a linear elastic model in a Bayesian inference framework. To make the problem tractable, we use a Gaussian approximation of the posterior probability distribution that results from the Bayesian solution of the inverse problem of inferring Young’s modulus parameter fields from available data. The performance of the computational framework is demonstrated using two representative loading scenarios, one involving cantilever bending and the other involving stretching of a helical rod (an intrinsically curved structure). The results show that smoothly varying parameter fields can be reconstructed satisfactorily from noisy data. We also quantify the uncertainty in the inferred parameters and discuss the effect of the quality of the data on the reconstructions.