New Sampling Algorithms and Inverse Spectral Methods in Scattering

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Non-destructive testing has many applications in engineering and medical imaging. Innovations in non-destructive testing have given rise to new imaging methods such as electrical impedance tomography (EIT), which can be used in medical applications as an alternative to more expensive imaging modalities. In many other instances as well, one needs to determine the interior structure of an object with little a priori information using acoustic or electromagnetic waves. This project aims to develop new algorithms for fast and accurate reconstructions that will be able to determine interior features. The investigator and graduate students will study new methods and applications for imaging techniques such as EIT. The primary result of the project will be new mathematical techniques for shape recovery that are largely independent of unknown physical properties of an object. The research has three parts that are connected to the analytical and computational aspects of inverse scattering. The first project is to develop new theoretically rigorous and computationally simple regularization algorithms for the factorization method. The project will focus on electrical impedance tomography, where the theory will be extended for an operator mapping a Hilbert space into the dual space. The goal is to develop the analytical framework for a new algorithm for shape reconstruction. The second project is to extend the applicability of the direct sampling method (DSM) to near-field data as well as reduce the number of measurements needed. These are simple algorithms for shape reconstruction stable with respect to noise in the data but requiring measurements from many sources and receivers. The aim is to develop the resolution analysis for new DSMs. The new methods involve transforming near-field data into far-field data and applying factorization of the far-field operator. The third project is to study new transmission eigenvalue problems arising in inverse scattering. These eigenvalues can be used to estimate or determine if there are defects in a scatterer. The reconstruction methods studied in the first two parts of the project correspond to shape reconstruction, whereas the study of these spectral problems will lead to parameter identification. Results of the research are expected not only to allow recovery of a scattering object but also to provide information about its material parameters.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.