CAREER: Solving Inverse Problems via Transport and Measure Theory

This project develops new mathematical and computational methods for solving inverse problems that arise when scientists and engineers infer hidden causes from indirect, incomplete, or noisy observations. Such problems are central to many areas of national importance, including geoscience, medical imaging, and cancer treatment planning. The project advances the national interest by creating tools that can support faster, more accessible radiation therapy planning, improve the reliability of scientific computing with limited data, and strengthen the foundations of data-driven discovery. By improving methods for extracting useful information from uncertain observations, the project promotes the progress of science and supports advances in national health, prosperity, and welfare. The work also supports Presidential priorities in AI by developing learning-based methods that are reliable, interpretable, and grounded in mathematics. Education and outreach activities broaden participation in science, technology, engineering, and mathematics by providing student mentoring, curriculum development, public engagement, and data science workshops for incarcerated learners and reentry participants. This project establishes a transport-based, measure-theoretic framework for inverse problems by formulating unknown quantities, data, and solution methods over spaces of probability measures. The investigator studies three connected research directions. The first develops dynamical system models and inverse methods for settings in which the governing dynamics are unstable, the inverse problem is ill-posed, data are sparse, and observations are noisy or indirect. The second develops approaches for stochastic inverse problems in which the unknown parameters are inherently random, with an emphasis on uncertainty quantification beyond standard Bayesian formulations. The third develops inverse operator learning methods that use modern machine learning architectures to construct fast solvers for repeated inverse problem instances. The project combines optimal transport, variational analysis, functional analysis on spaces of probability measures, statistical learning, measure-valued flows, sampling methods, and computational algorithms. Expected contributions include new mathematical theory for nonlinear operators on spaces of probability measures, robust algorithms for inverse problems with limited data, scalable methods for uncertainty quantification, and efficient computational tools for applications in geoscience and medicine. 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.