The proposed research program is an interdisciplinary effort, aimed at implementing mathematically-informed computational solutions to inverse problems that involve volume scattering of elastic and electromagnetic waves in complex media. Seismology, for instance, is in great demand of new algorithms. The current preferred numerical methods--finite difference schemes that make no reference to the underlying geometry of waves--have a serious scalability issue, and give no hint on how to resolve the hard nonlinearity of the inverse problem. We argue that insights from microlocal and harmonic analysis suggest shifting much of the computational burden to a pre-processing predictive of wave kinematics: 1) by passing to a numerical representation in phase space for the operators related to the wave and Helmholtz equations, algorithms for the forward problem can be developed to restore near-linear complexity in the wave field data; and 2) fitting diffeomorphisms directly in phase-space by optimal transport ideas is predicted to resolve some of the nonconvexity issues otherwise arising when the inverse problem is solved using successive linearizations.
Moore's law of exponential increase in computing performance is not often matched by exponential progress in the computational sciences. The culprit is the lack of scalability of mainstream algorithms: the size of problems that can be solved grows more slowly than hardware capabilities. In increasingly many applications, the input of mathematicians is needed to help engineers and applied scientists rethink the design of numerical codes to avoid this curse of scalability. This proposal is an effort to take a step back and introduce new algorithmic ideas for seismic imaging, the discipline concerned with imaging the subsurface of the Earth. Seismic imaging is the energy sector's main predictive tool for hydrocarbon, water, and geothermal energy prospection. It is at the heart of monitoring techniques for reservoirs and carbon sequestration experiments. It has proved useful to geophysicists who debate the geological composition of the Earth's mantle. High-resolution seismic imaging is also starting to enable the Army and the Air Force to detect IEDs. All these remote imaging problems have by now become formidably complex computational questions that our generation will be responsible for solving.
The problem of imaging the Earth's interior is currently one of society's most formidable technical challenges. It took geophysicists decades to be able to find good formulas to leverage the knowledge of seismic waves -- noise from earthquakes -- to create plausible maps of the subsurface. Yet, the quest for a solution of the so-called "inverse scattering" problem is not over: there exist situations in which no known imaging method would succeed, even when sufficient information is in principle present in the recorded seismic data. The problem is of a combinatorial nature, i.e., it owes to our computers' inability to perform exhaustive searches in high-dimensional spaces. The group of Prof. Demanet is focused on approaching this challenge from a mathematical angle, using techniques of modern optimization such as convex relaxation. With the help of this NSF grant, two Ph.D. students in the group were given the opportunity to investigate new, important computational solutions to the inverse problems of imaging. Notable progress was made in the particular case when the iterative method starts from an initial guess which may be far away from the eventual image. The new imaging methods developed in Prof. Demanet's group have scientific and industrial applications in 1) determining the structure of the Earth from seismic waves, 2) visualizing the interior of biological tissues with ultrasound waves, and 3) locating reflecting objects from microwaves in radar systems. The interdisciplinary topic of imaging (at the intersection of math, statistics, physics, and EE), though in high demand in industry, is not normally taught in graduate school. An important component of the grant's activity was the organization of two graduate summer schools to help train groups of young computational engineers in the common mathematical language of wave-based imaging.
