Steven Mattis

Steven Mattis, Barbara Wohlmuth

Stochastic inverse problems are generally solved by some form of finite sampling of a space of uncertain parameters. For computationally expensive models, surrogate response surfaces are often employed to increase the number of samples used in approximating the solution. The result is generally a trade off in errors where the stochastic error is reduced at the cost of an increase in deterministic/discretization errors in the evaluation of the surrogate. Such stochastic errors pollute predictions based on the stochastic inverse. In this work, we formulate a method for adaptively creating a special class of surrogate response surfaces with this stochastic error in mind. Adjoint techniques are used to enhance the local approximation properties of the surrogate allowing the construction of a higher-level enhanced surrogate. Using these two levels of surrogates, appropriately derived local error indicators are computed and used to guide refinement of both levels of the surrogates. Three types of refinement strategies are presented and combined in an iterative adaptive surrogate construction algorithm. Numerical examples, including a complex vibroacoustics application, demonstrate how this adaptive strategy allows for accurate predictions under uncertainty for a much smaller computational cost than uniform refinement.

Mario Teixeira Parente, Steven Mattis, Shubhangi Gupta et al.

Methane gas hydrates have increasingly become a topic of interest because of their potential as a future energy resource. There are significant economical and environmental risks associated with extraction from hydrate reservoirs, so a variety of multiphysics models have been developed to analyze prospective risks and benefits. These models generally have a large number of empirical parameters which are not known a priori. Traditional optimization-based parameter estimation frameworks may be ill-posed or computationally prohibitive. Bayesian inference methods have increasingly been found effective for estimating parameters in complex geophysical systems. These methods often are not viable in cases of computationally expensive models and high-dimensional parameter spaces. Recently, methods have been developed to effectively reduce the dimension of Bayesian inverse problems by identifying low-dimensional structures that are most informed by data. Active subspaces is one of the most generally applicable methods of performing this dimension reduction. In this paper, Bayesian inference of the parameters of a state-of-the-art mathematical model for methane hydrates based on experimental data from a triaxial compression test with gas hydrate-bearing sand is performed in an efficient way by utilizing active subspaces. Active subspaces are used to identify low-dimensional structure in the parameter space which is exploited by generating a cheap regression-based surrogate model and implementing a modified Markov chain Monte Carlo algorithm. Posterior densities having means that match the experimental data are approximated in a computationally efficient way.

Steven Mattis, Kyle Robert Steffen, Troy Butler et al.

Dynamical systems arise in a wide variety of mathematical models from science and engineering. A common challenge is to quantify uncertainties on model inputs (parameters) that correspond to a quantitative characterization of uncertainties on observable Quantities of Interest (QoI). To this end, we consider a stochastic inverse problem (SIP) with a solution described by a pullback probability measure. We call this an observation-consistent solution, as its subsequent push-forward through the QoI map matches the observed probability distribution on model outputs. A distinction is made between QoI useful for solving the SIP and arbitrary model output data. In dynamical systems, model output data are often given as a series of state variable responses recorded over a particular time window. Consequently, the dimension of output data can easily exceed $\mathcal{O}(1E4)$ or more due to the frequency of observations, and the correct choice or construction of a QoI from this data is not self-evident. We present a new framework, Learning Uncertain Quantities (LUQ), that facilitates the tractable solution of SIPs for dynamical systems. Given ensembles of predicted (simulated) time series and (noisy) observed data, LUQ provides routines for filtering data, unsupervised learning of the underlying dynamics, classifying observations, and feature extraction to learn the QoI map. Subsequently, time series data are transformed into samples of the underlying predicted and observed distributions associated with the QoI so that solutions to the SIP are computable. Following the introduction and demonstration of LUQ, numerical results from several SIPs are presented for a variety of dynamical systems arising in the life and physical sciences. For scientific reproducibility, we provide links to our Python implementation of LUQ and to all data and scripts required to reproduce the results in this manuscript.