# Poster Session and Reception

**Diffuse Optical Tomography in the Bayesian Framework**Kit Newton (University of Wisconsin, Madison)

Many naturally-occuring models in the sciences are well-approximated

by simplified models, using multiscale techniques. In such settings

it is natural to ask about the relationship between inverse problems

defined by the original problem and by the multiscale approximation.

We develop an approach to this problem and exemplify it in the context

of optical tomographic imaging.

Optical tomographic imaging is a technique for infering the properties of

biological tissue via measurements of the incoming and outgoing light intensity; it may be used as a

medical imaging methodology. Mathematically, light propagation is modeled

by the radiative transfer equation (RTE), and optical tomography amounts to

reconstructing the scattering and the absorption coefficients in the RTE

from boundary measurements. We study this problem in the Bayesian framework,

focussing on the strong scattering regime. In this regime the forward

RTE is close to the diffusion equation (DE).

We study the RTE in the asymptotic regime where the forward problem

approaches the DE, and prove convergence of the inverse RTE to the inverse DE in both nonlinear

and linear settings. Convergence is proved by studying the distance between

the two posterior distributions using the Hellinger metric, and using

Kullback-Leibler divergence.

**A Modified Inverse Born Series for the Calderon Problem**

Anuj Abhishek (Drexel University)The problem of electrical impedance tomography wherein one tries to recover the conductivity of a medium based on measurement of electric field on the boundary gives rise to the Calderon problem. In the article, "Inverse Born Series for Calderon Problem" (Inverse Problems, 2012), Arridge, Moskow and Schotland have proposed a direct reconstruction method for such problems using inversion of the Born series. They have also studied the convergence and stability of the method and have shown in particular, that for sufficiently low contrast the forward series is conditionally convergent. In this work we propose a modified Born series that is unconditionally convergent and have provide a reconstruction method based on inversion of such a modified series. Our preliminary numerical experiments suggest that the reconstruction based on the modified series is better for the high contrast case.

**Refraction Problems with Phase Discontinuities on nonflat Metasurfaces**

****Eric Stachura (Kennesaw State University)

For classical lens design, a usual problem is to find two surfaces so that the region bounded between them, filled with a homogeneous material, refracts light in a prescribed way. For the design of metalenses, usually a surface is given and the question is to find a phase discontinuity (a function defined on the surface) such that the surface and phase discontinuity refract light in a desired way.

We provide a mathematical approach to study metasurfaces in nonflat geometries. Analytical conditions between the curvature of the surface and the set of refracted directions are introduced to guarantee the existence of phase discontinuities. Both the far field and near field cases are considered. The starting point is a vector form of Snell's law in the presence of discontinuities on interfaces.

This is joint work with C. E. Gutierrez and L. Pallucchini (Temple University).

**The recovery of a parabolic equation from measurements at a single point**

Amin Boumenir (State University of West Georgia)

By measuring the temperature at an arbitrary single point located inside an unknown object or on its boundary, we show how we can uniquely reconstruct all the coefficients appearing in a general parabolic equation which models its cooling. We also can reconstruct the shape of the object. The proof hinges on the fact that we can detect infinitely many eigenfunctions whose Wronskian does not vanish. This allows us to evaluate these coefficients by solving a simple linear algebraic system. The geometry of the domain and its boundary are found by reconstructing the first eigenfunction.

Boumenir, Amin; Tuan, Vu Kim; Hoang, Nguyen The recovery of a parabolic equation from measurements at a single point. Evol. Equ. Control Theory 7 (2018), no. 2, 197–216.**Copula directional dependence for inference and statistical analysis of whole‐brain connectivity from fMRI data**

Jong-Min Kim (University of Minnesota, Morris)

Co-author: Namgil Lee

Introduction:

Inferring connectivity between brain regions has been raising a lot of attention in recent decades. Copula directional dependence (CDD) is a statistical measure of directed connectivity, which does not require strict assumptions on probability distributions and linearity.

Methods:

In this work, CDDs between pairs of local brain areas were estimated based on the fMRI responses of human participants watching a Pixar animation movie. A directed connectivity map of fourteen predefined local areas was obtained for each participant, where the network structure was determined by the strengths of the CDDs. A resampling technique was further applied to determine the statistical significance of the connectivity directions in the networks. In order to demonstrate

the effectiveness of the suggested method using CDDs, statistical group analysis was conducted based on graph theoretic measures of the inferred directed networks and CDD intensities. When the 129 fMRI participants were grouped by their age (3–5 year‐old, 7–12 year‐old, adult) and gender (F, M), nonparametric two‐way analysis of variance (ANOVA) results could identify which cortical regions and connectivity structures correlated with the two physiological factors.

Results:

Especially, we could identify that (a) graph centrality measures of the frontal eye fields (FEF), the inferior temporal gyrus (ITG), and the temporopolar area (TP) were significantly affected by aging, (b) CDD intensities between FEF and the primary motor cortex (M1) and between ITG and TP were highly significantly affected by aging, and (c) CDDs between M1 and the anterior prefrontal cortex (aPFC) were highly significantly affected by gender.**Using eigenvalues to detect anomalies in the exterior of a cavity**

Samuel Cogar (University of Delaware)

We use modified near field operators and a nonsymmetric version of the generalized linear sampling method to investigate an inverse scattering problem for anisotropic media with data measured inside a cavity. The aim is to determine information on possible changes in the material properties of the medium surrounding the cavity, and to this end we introduce a new class of eigenvalue problems for which the eigenvalues can be determined from the measured scattering data. We augment our analysis with numerical testing of both the computation of eigenvalues from near field data and the behavior of the eigenvalues following changes in the material properties of the medium.