CAREER: Large-Scale Bayesian Inverse Problems Governed by Differential and Differential-Algebraic Equations

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Nontechnical explanation of the project's broader significance and importance: Model-based projections of real life applications will play a central role in prediction and decision-making, in environment and climate change applications, for instance, to anticipate ice sheet contribution to sea level rise, or in the context of energy applications, to predict faults and assess dynamic stability in a power grid. However, models are typically subject to considerable uncertainties stemming from uncertain inputs to the model (e.g., coefficient fields, constitutive laws, source terms, geometries, and initial and/or boundary conditions) as well as from noisy and limited observations. While many of these input quantities cannot be directly observed or measured, they can be inferred from observations, such as those of ice surface velocities in ice sheets. This typically leads to an extremely challenging mathematical problem. This project aims to enable the propagation of uncertainties from data/observations through inference to prediction and increase predictability of complex physical systems. The selected driving application (i.e., the ice sheet model) for research and education activities, capture important general and complex algorithmic challenges such as large-scale, nonlinearity, time-dependence, and ill-posedness. The research will be, therefore, applicable to a broader spectrum of problems. The algorithms, mathematical findings and open source codes will be shared through peer reviewed journal papers, and presentations at conferences and workshops. Technical description of the project: Bayesian inversion facilitates the integration of data with complex physics-based models to quantify and reduce uncertainties in model predictions. This opens the door to more advanced capabilities for prediction and decision-making under uncertainty. However, the algorithmic developments for Bayesian inversion are subject to several challenges. For instance, characterizing the posterior distributions of parameters or predictions inevitably requires repeated evaluations of (possibly) large-scale and complex forward models governed by differential equations. In addition, the posterior distribution has a complex structure stemming from the presence of possibly nonlinear forward models and heterogeneous sources of data. To overcome these computational challenges, it is essential to exploit problem structure (e.g., derivatives and local sensitivity of the data with respect to parameters). The objectives of this proposal is to conduct exploratory work in addressing the mathematical and computational barriers in solving large-scale Bayesian inverse problems governed by differential equations. Developing mathematically rigorous and computationally efficient and robust methods in the context of statistical inference has the potential of transformative research in the field of modern computational inverse problems. In particular, the PI and her student will work on the following vertically-integrated research areas: (i) scalable algorithms for large-scale inverse problems (here the focus will be on second derivative (i.e., Hessian) approximations for inverse problems and on developing efficient preconditioners for inexact Newton-Krylov systems to increase the computational efficiency of inverse solvers), and (ii) uncertainty quantification in high dimensions (here the focus will be on building Hessian- and reduced order model-based methods for efficient posterior exploration in high dimensions). The proposed research requires an interdisciplinary perspective, namely it brings together applied mathematics, scientific computing and statistics.