# MiniSymposium-10: Recent Developments in Inverse Scattering Theory

Broadly speaking, inverse scattering problems are the problems of determining information about an object (scatterer) from measurements of the field scattered from that object. These problems have potential applications to such diverse areas as nondestructive testing, geophysical exploration, medical imaging, and radar. During the past two decades, this area of research has been one of the fastest-growing areas in applied mathematics with significant developments in mathematical theory, computational algorithms, and applications. The goal of this mini-symposium is to bring together researchers working on different aspects of the field to discuss recent and new results in the field and to promote idea exchange as well as potential future collaborations.

## Organizer:

**Dinh-Liem Nguyen**, Kansas State University

## Saturday, October 2, 2021 at 2:40 – 4:00 pm (CST) (Session 1)

**Trung Truong**

Kansas State University

2:40 pm

*Inverse Born Series Method for a Periodic Inverse Scattering Problem*

Periodic inverse scattering problems have attracted an increasing amount of attention during the past few decades due to their applications in the study of photonic crystals. These problems are known to be non-linear and severely ill-posed. There have been several different approaches to tackle both shape reconstruction and material parameter reconstruction problems. The latter often requires advanced a priori information about the solution, for example, optimization-based methods need a good initial guess. Inverse Born series, on the other hand, is a direct method for reconstructing the material parameter with an analysis that can be rigorously justified under certain conditions. Numerical reconstructions also show high levels of accuracy and stability. This talk focuses on the analysis of the Forward and Inverse Born series as well as their modified versions to solve the periodic inverse scattering problem. Corresponding numerical results will also be presented. This is joint work with Dinh-Liem Nguyen and Colin Williams.

**Fatemeh Pourahmadian**

University of Colorado-Boulder

3:00 pm

*Imaging in Highly Scattering Solids*

This talk presents recent progress on in-situ differential tomography of progressive variations in heterogeneous materials of a priori unknown properties. The reported data analytic solution takes advantage of non-iterative sampling methods to inverse scattering. The analysis pertinent to energy-preserving and self-adjoint systems featuring random heterogeneities and discontinuities is presented and applied to both synthetic and experimental data. The challenges in extending such imaging solutions to multi-physics and lossy systems will also be discussed.

**Dinh-Liem Nguyen**

Kansas State University

3:20 pm

*Orthogonality Sampling Method for Electromagnetic Inverse Scattering Problems*

We consider the electromagnetic inverse scattering problem that aims to reconstruct the location and shape of anisotropic scattering objects from far-field measurements. We will discuss our recent results on the orthogonality sampling method for solving the inverse problem. Compared with classical sampling methods (e.g., the linear sampling method, the factorization method) the orthogonality sampling method is simpler to implement, can work with one-wave data, and its stability can be easily justified. This is joint work in part with Isaac Harris.

**Isaac Harris**

Purdue University

3:40 pm

*Regularization of the Factorization Method with Applications*

In this talk, we discuss a new regularized version of the Factorization Method. The Factorization Method uses Picard's Criteria to define an indicator function to image an unknown region. In most applications, the data operator is compact which gives that the singular values can tend to zero rapidly which can cause numerical instabilities. The regularization of the Factorization Method presented here seeks to avoid the numerical instabilities in applying Picard's Criteria. This method allows one to image the interior structure of an object with little a priori information in a computationally simple and analytically rigorous way. Here we will focus on an application of this method to diffuse optical tomography where will prove that this method can be used to recover an unknown subregion from the Dirichlet-to-Neumann mapping.

## Sunday, October 3, 2021 at 11:00 am - 12:20 pm (CST) (Session 2)

**Thu Le**

Kansas State University

11:00 am

*Imaging of 3D Objects with Experimental Data using Orthogonality Sampling Methods*

We consider the electromagnetic inverse scattering problem that aims to reconstruct the location and shape of an unknown object from the electromagnetic field scattered by that object. It has applications in radar and nondestructive testing. In this talk, we investigate a modified version of the Orthogonality Sampling Method (OSM) for Maxwell's equations. This modification allows the method to work with more types of polarization associated with the data. Numerical results testing against 3D experimental data from the Fresnel institute will be presented. The results show that the modified OSM performs better than its original version in real data verification. This is joint work with Dinh-Liem Nguyen, Hayden Schmidt, and Trung Truong.

**Alexander Mamonov**

University of Houston

11:20 am

*Reduced Order Model Approach to Quantitative Imaging with Waves*

We present an approach to the numerical solution of an inverse scattering problem for a generic hyperbolic system of equations with an unknown coefficient called the reflectivity. The system models waves (sound, electromagnetic or elastic), and the reflectivity models unknown scatterers embedded in a smooth and known medium. The inverse problem is to determine quantitatively the reflectivity from the time-domain measurements of the scattering data, performed at an array of sensors. Our approach is based on a reduced order model (ROM) of a wave propagator operator, which maps the wave from one time instant to the next, at an interval corresponding to the discrete time sampling of the data. While the propagator is unknown in the bulk, its ROM, a projection on the subspace of discretely sampled time-domain wavefield snapshots can be computed from the measured data only. Once computed, the ROM can be used to image the reflectors quantitatively. This is possible due to an almost affine dependency of the ROM on the reflectivity in a certain parametrization of the hyperbolic system. Such a dependency can be exploited to formulate a Gauss-Newton iterative solution scheme that converges in a few iterations. The proposed scheme compares favorably to the conventional approaches, as demonstrated by the numerical results. (Joint with L. Borcea, V. Druskin, M. Zaslavsky and J. Zimmerling.)

**Loc Nguyen**

University of North Carolina-Charlotte

11:40 am

*Carleman Estimates and the Contraction Principle for an Inverse Source Problem for Nonlinear Hyperbolic Equations*

Our main aim is to solve an inverse source problem for a general nonlinear hyperbolic equation. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the inverse problem. To find this fixed point, we define a recursive sequence with an arbitrary initial term by the same manner as in the classical proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution with the exponential rate. Therefore, our new method can be considered as an analog of the contraction principle. We rigorously study the stability of our method with respect to noise. Numerical examples are presented.

**Carlos Borges**

University of Central Florida

12:00 pm

*Recursive Linearization Algorithm to Recover High Resolution Features of the Shape and Impedance of Obstacles*

In this work, the recursive linearization algorithm (RLA) is used to recover high resolution features of both the shape and the impedance function of an obstacle from measurements of the scattered field at multiple frequencies. The RLA is applied framing at each frequency the inverse problem as a nonlinear optimization problem. The single frequency inverse problem is both nonlinear and ill-posed. To deal with the nonlinearity, we apply a Gauss-Newton iteration. We treat the ill-posedness by considering the approximation of the shape and impedance function to be bandlimited functions. Numerical examples are presented to demonstrate that the method can recover the shape and impedance function of the obstacle with high resolution.