Nonlinear spectral problems in electromagnetics: asymptotics and inversion.

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Despite extraordinary advances in computing speed, the resolution of 3D imaging problems in electromagnetics continues to pose significant challenges in a wide range of applications, from biomedical imaging to geophysical exploration and non-destructive detection of material defects. By sending electromagnetic sources from probes in boreholes and reading back resulting potentials, for instance, one can in principle infer the location of oil deposits, which have different electromagnetic properties than the surrounding ground. Brute-force computational approaches, however, are often cumbersome or even intractable for large scale non-linear problems. In the proposed research, mathematical methods will be developed where we find explicit formulas for the dependence of the data on electromagnetic parameters, leading to fast algorithms fine-tuned to specific geometries and having clear physical interpretations. Several students will be trained in the course of the project, including a full time PhD student. The project will also provide undergraduate students the opportunity to spend six months delving deeply into a research problem, and may increase the likelihood they will pursue a higher degree in mathematics or a related field.The objective of this proposal is to solve several nonlinear spectral problems and related inverse problems in electromagnetics by developing novel asymptotic formulas based on material property perturbations. The project will involve five main components: (i) Asymptotic formulas for resonance values in the presence of linear and nonlinear small volume electromagnetic scatters. The asymptotics will be based on perturbations of simplified limiting problems whose solutions are easily computed. (ii) A dimensionally reduced formulation for the electric field and resonances in the presence of thin high contrast materials. This formulation will provide a high contrast Born approximation which we can invert. (iii) Explicit calculations of the perturbations of transmission eigenvalues for varying material properties. This will include small inhomogeneities, voids, or holes in the materials, and periodic microstructures. The formulas will reveal what material properties can be recovered from reading transmission eigenvalues in the far field. (iv) The development of a fast nonlinear inversion technique and a theory of uniqueness and stability in ultrasound modulated optical tomography, a novel hybrid imaging modality which provides nonlinear internal data for the recovery of optical parameters. (v) An inverse spectral method for nonlinear strings, a project for Drexel undergraduate students. The application of material perturbation asymptotics to nonlinear spectral problems is a unifying theme in all of these projects.