The Carleman Contraction Principle for Three-Dimensional Phased and Phaseless Inverse Scattering

The identification from only external measurements of unknown targets that are fully occluded inside a region is a challenging interdisciplinary research effort among mathematicians, physicists, and engineers. Important applications of this field include nondestructive testing, for example, checking for cracks inside a product, seismic exploration, for example, searching for oil under the sea, security, for example, the identification of buried anti-personnel explosive devices, and others. Current, widely used optimization-based methods for such tasks require some a priori knowledge about the true solution, which is not always available, and their computational cost is expensive. This project aims to develop new, theoretically sound techniques that are designed to quickly deliver reliable solutions for estimating occluded targets that are independent of the initial estimates. Both undergraduate and graduate students will be trained through involvement in this research. In this project, data for identifying unknown targets inside a medium will be collected through a measured scattering wave. The methodology to be developed in this work will be able to handle data where the phase may or may not be able to be measured, the latter which occurs in many challenging situations, such as the case of nano imaging, where the wavelength of the incident wave is particularly small. The new methods combine Carleman estimates, the contraction mapping principle, and an algebraic formula to retrieve the lost phase when needed. The use of the contraction mapping principle will guarantee key strengths of the new approach: relaxed requirements on the initial guesses and inexpensive computational cost. These new methods will be applied to partial differential equations that govern wave propagation, including the three-dimensional Helmholtz equation and Maxwell’s system. The new methods will be tested on both simulated and experimental data.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.