Liliana Borcea of the University of Michigan is the recipient of the 2017
AWM-SIAM Sonia Kovalevsky Lecture. Awarded yearly at the SIAM Annual Meeting, the lecture is intended to highlight significant contributions of women to applied or computational mathematics.
AWM and SIAM recognized Borcea for her distinguished scientific contributions to the mathematical and numerical analysis of wave propagation in random media, array imaging in complex environments, and inverse problems in high-contrast electrical impedance tomography, as well as model reduction techniques for parabolic and hyperbolic partial differential equations.
Borcea received the award at the 2017 SIAM Annual Meeting (AN17), held July 10-14, 2017 in Pittsburgh, Pennsylvania. She delivered the 2017 AWM-SIAM Sonia Kovalevsky Lecture, “Mitigating Uncertainty in Inverse Wave Scattering,” on July 10, 2017.
Borcea is the Peter Field Collegiate Professor of Mathematics at the University of Michigan, where she has been on the faculty since 2013. She received her MS and PhD in scientific computing and computational mathematics from Stanford University. She started her academic career as an NSF Postdoctoral Fellow at the California Institute of Technology. Before moving to the University of Michigan in 2013, she progressed from assistant professor to Noah Harding Professor in the Department of Computational and Applied Mathematics at Rice University. Currently, she is an elected member of the SIAM Council. She has also served on the editorial board of both SIAM/ASA Journal on Uncertainty Quantification and Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal.
Q: Why are you excited about winning the prize?
A: I am honored to be in the company of the outstanding women mathematicians who have won this award.
Q: What does your research mean to the public?
A: My research is in inverse problems, where the goal is to infer properties of a medium occupying some bounded domain, using measurements at the boundary. The mathematical model of the measurements is based on partial differential equations with unknown coefficients. These describe the unknown medium and are to be determined in inversion. For example, the wave equation models the propagation of sound waves and by probing the medium with pulses and measuring the scattered responses, we can determine variations of the acoustic impedance and wave speed, which model reflecting structures in the medium.
Inverse problems is an interdisciplinary area in applied mathematics with a broad spectrum of applications in medical imaging, nondestructive evaluation of large structures like bridges and airplane wings, inspection of welds, radar imaging and tracking of flying objects like airplanes, satellites or small debris in low-earth orbit, seismic exploration, ground water contamination monitoring, etc. The field is driven by advances in experimental science and in sensor technology, which have increased dramatically our ability to make new measurements and thus collect new types of data and lots of it. These advances create outstanding opportunities and new challenges for the field of inverse problems. My research is concerned with developing new inversion methodologies that address such challenges in the context of imaging with waves in heterogeneous media, and with using model reduction techniques for efficient numerical solutions of inverse problems for linear partial differential equations.
Q: Could you tell us a bit about the research that won you the prize?
A: Much of my research has been concerned with the following two themes:
(1) Sensor array imaging in media with random microstructure and (2) Development of inversion algorithms based on model reduction techniques.
Random microstructures are used as mathematical models of uncertainty of the media through which the waves propagate from the sensors (transducers, antennas) to the sought after reflecting structures. This uncertainty is due to small amplitude fluctuations of the wave speed and other material properties, on scales of at most of the order of the wavelength of the probing waves. These fluctuations model small inhomogeneities in the medium, like pebbles in concrete or air bubbles in water. They are weak scatterers in comparison with the sough after reflectors, and cannot be determined as part of the inversion, which is why they are modeled as random. As the waves propagate through such media, they interact with the microstructure and the wave field is randomized. My research is concerned with understanding the details of this randomization and then using this understanding to develop robust and efficient imaging algorithms for determining the sought after reflectors. With my colleagues Ricardo Alonso, Josselin Garnier, George Papanicolaou, Knut Solna, and Chrysoula Tsogka, I pursued this line of research in various settings for imaging in strongly scattering random media.
With my colleagues Vladimir Druskin, Fernando Guevara Vasquez, Alexander Mamonov, and Mikhail Zaslavsky, I have developed inversion algorithms based on reduced order models for elliptic, parabolic and hyperbolic linear partial differential equations. The reduced models are carefully designed to address the different challenges in inversion. For the elliptic and parabolic equations the challenge lies in the severe ill-posedness of the inverse problem, which stems from the low sensitivity of the data to changes of the coefficients in the equations at locations away from the boundary where the measurements are made. The reduced order models for these equations correspond to finite volume discretizations on special grids that are unknown and are computed as part of the inversion. These grids turn out to be refined near the boundary where the measurements are made, and coarsen inside the domain, thus capturing the lowered sensitivity of the data to changes of the coefficients away from the boundary. For hyperbolic problems, the challenge lies in dealing with multiple scattering effects of the waves in the medium. The reduced order models approximate the wave propagator operator and they can be used to filter the unwanted multiple scattering effects and improve the quality of the inversion.
Q: What does being a SIAM member mean to you?
A: My work and research interests are motivated by my view of applied mathematics as a science that reaches across disciplines to computer science, physics, statistics, and experimental science. SIAM is a unique forum for applied mathematicians to exchange ideas and to pursue interdisciplinary research. I joined SIAM when I was a graduate student, I regularly attend the SIAM meetings, and have been quite involved in the SIAM Activity Group on Imaging Science. I publish most of my research in SIAM journals and I have the honor to serve on a few of their editorial boards, as well as on the SIAM Council.