Geometrical algorithms for the inverse scattering of waves

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

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.