Distinguished Seminar in Computational Science and Engineering
April 4, 2024, 12 PM
When Optimal Transportation Meets PDE-Based Inverse Problems
Yunan Yang
Goenka Family Assistant Professor in Mathematics
The College of Arts & Sciences
Cornell University
Abstract:
Optimal transportation (OT) has emerged as an important area in mathematical analysis and applied mathematics since Monge introduced it in 1781. Over the last century, its deep ties with differential geometry and kinetics theory have been uncovered, with Kantorovich’s groundbreaking work in 1942 highlighting its effectiveness in addressing real-world challenges. In recent years, we have applied the concept of quadratic Wasserstein distance, derived from OT theory, to a range of inverse problems and complex, high-dimensional kinetic PDE-constrained optimization problems, e.g., waveform inversion and dynamical system modeling. This approach overcomes traditional methods’ limitations, such as the least-squares method, by addressing issues like nonconvexity and sensitivity to noise, and it introduces a novel geometric framework for gradients. The benefits of incorporating OT in PDE-based inverse problems are twofold. First, it offers a radically different measure for data misfit in a wide range of data-matching scenarios, bringing forth unique properties not found in norms for linear spaces. Second, employing the Wasserstein gradient enhances both the computational and analytical aspects of gradient dynamics, leading to improved inversion results in practical algorithms.
Bio:
Yunan Yang is an applied mathematician working on inverse problems and optimal transportation. Currently, she is a tenure-track Goenka Family Assistant Professor in the Department of Mathematics at Cornell University. Yunan Yang earned a Ph.D. in mathematics from the University of Texas at Austin in 2018, supervised by Prof. Bjorn Engquist. From September 2018 to August 2021, Yunan was a Courant Instructor at the Courant Institute of Mathematical Sciences, NYU, followed by an Advanced Fellowship at the Institute for Theoretical Studies at ETH Zurich. Yunan won the 19th IMA Leslie Fox Prize in Numerical Analysis as a First Prize and was honored as one of the Rising Stars in Computational and Data Sciences by the Oden Institute.
April 4, 2024 CSE Distinguished Seminar
Yunan Yang
Goenka Family Assistant Professor in Mathematics
Cornell University