CAREER: Scalable Algorithms for Nonlinear, Large-Scale Inverse Problems Governed by Dynamical Systems

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Large-scale inverse problems governed by dynamical systems are of paramount importance in numerous scientific disciplines. Examples include geoscience, medicine, climate science, manufacturing, national security, and economics. Inversion is an indispensable tool to infer knowledge from data in a consistent and predictable way, enabling scientific discovery, decision-making, and ultimately dependable model- and data-informed predictions. However, the practical use of inversion remains limited unless uncertainties can be quantified as they propagate through models and algorithms, Uncertainty quantification adds significant mathematical complications and massive computational costs to an already challenging problem. This project will develop a generic mathematical framework for large-scale, statistical inverse problems alongside software infrastructure with algorithms that scale on modern and future computing architectures. It blends mathematical methods and theory with data-intensive applications and good algorithmic practices to advance the frontiers of computational and data-enabled sciences, with the ultimate aspiration to promote data-driven scientific discovery and model-based prediction and by that, science in general. Alongside research activities, an educational and dissemination program is developed to communicate the results under this work to STEM students and researchers, and a broad audience of computational scientists and application specialists. The project will train students in areas that have seen exceptionally high industry demand in the US in recent years, such as optimization, statistical inference, data-enabled science, performance evaluation, and workload characterization. Educational activities include hands-on research experiences for graduate and undergraduate students, explicitly encouraging participation by minorities and underrepresented groups. Public domain software modules will be made available to a broad STEM audience and practitioners. Applications of this work include medicine, imaging, and geosciences.Fundamental mathematical and computational aspects of optimization under uncertainty, statistical inference, and the solution of large-scale inverse problems will be investigated in this project to promote the progress of scientific discovery and data exploration. The overarching aim is the design of fast computational kernels and scalable, black-box algorithms that rigorously follow mathematical and physical principles, have a sound theoretical basis, and provably converge to an optimal solution independent of the problem dimension. This includes the development of adaptive, hierarchical numerical schemes and mixed-precision algorithms, enabling high-accuracy computations if desired, and low-accuracy approximations when possible, targeting high data-throughput applications. The project explores (i) foundational mathematical aspects and the deployment of fast (scalable) algorithms for transport-based variational inference, (ii) the design of problem-informed regularization schemes for nonlinear inverse problems, and (iii) the integration of randomized algorithms and learning for the construction of low-order surrogate models for optimization, inference, sampling, and preconditioning. Effective numerical techniques and computational kernels for a fast evaluation of gradient and curvature information and their approximation are of paramount importance for the designed methodology. The performance will be assessed for parabolic (diffusion-dominated) and hyperbolic (advection-dominated) dynamical systems of varying complexity with applications in computational medicine, climate science, and geoscience.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.