Time Harmonic Inverse Scattering for Linear and Nonlinear Media

In numerous domains of national significance, such as renewable energy, non-invasive medical diagnosis, underground exploration, infrastructure integrity, and the manufacturing of novel materials including 3-D printing, the ability to conduct fast imaging using electromagnetic, acoustic, or elastic waves is crucial. Developing effective methods for testing complex materials to detect structural defects or identify unknown targets efficiently and with minimal a priori information is highly desirable.

This need is particularly pronounced for materials exhibiting directional properties, multi-periodicities with non-commensurable periods, nonlinear interactions with probing waves, or peculiar geometric structures, all prevalent in modern applications across the mentioned domains. This project involves the development of novel techniques in inverse scattering theory to address contemporary imaging challenges, aiming for reliable target signatures or useful information about examined objects in computationally efficient ways.

The objective is to reduce reliance on a priori information describing the physics and geometry of targets, as well as on mathematical and computational complexities arising from complex background environments. Graduate students will be trained and participate in this research.

This project will investigate a non-iterative approach for solving inverse scattering problems for both linear and nonlinear inhomogeneous media. The generalized linear sampling method and interior eigenvalues form the unified mathematical framework for four interconnected projects. 1) Imaging of Inhomogeneous Media with Interior Eigenvalues: Motivated by the theory of transmission eigenvalues, the investigator and collaborators have developed a framework for modifying the scattering data, leading to new eigenvalue problems related to injectivity of the relative scattering operator.

These eigenvalues are determined from scattering data and show versatile potential to image changes/faults in the media. The goal of this project is to address mathematical and computational questions in this framework to broaden the applicability of these techniques to anisotropic/absorbing/dispersive media, meta-surfaces, and clusters of defects. 2) Transmission Eigenvalues and Non-scattering Phenomena: A fundamental challenge in scattering theory is whether incoming time harmonic waves at a particular wave number are not scattered by a given inhomogeneity.

The investigator will address mathematical questions on this topic involving non-selfadjoint spectral theory and free boundary regularity. These questions are important in applications since at a non-scattering wave number, the scattering operator is not injective. 3) Qualitative and Spectral Imaging Approach for Nonlinear Media: This project aims to establish mathematical foundations of inverse scattering for second-order harmonic generation in nonlinear optics as well as to develop the generalized linear sampling method and related spectral imaging for this model.

This approach is promising for nonlinear models since it does not require solving the nonlinear partial differential equations. 4) Scattering by Almost-periodic Layers and Imaging of Local Defects: Periodicity plays an important role in the engineering of exotic materials in contemporary applications. Mathematically, almost periodic coefficients in the equations that govern wave propagation are viewed as traces along a hyperplane with an irrational normal of higher-dimensional periodic functions.

This leads to a degenerate augmented principal differential operator in the model. Having understood the direct scattering problem, the investigator plans to extend the generalized linear sampling method to reconstruct local perturbation within an almost periodic layer without knowing a priori the almost periodic coefficients.

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.