# Poster session and reception

Tuesday, February 14, 2017 - 4:30pm - 6:00pm

Lind 400

**Joint Inversion of Gravity and Traveltime Data Using a Level-set-based Structural Parameterization**

Wenbin Li (Michigan State University)

We develope a level-set-based structural parameterization for joint inversion of gravity and traveltime data, so that density contrast and seismic slowness are simultaneously recovered in the inverse problem. Because density contrast and slowness are different model parameters of the same survey domain, we assume that they are similar in structure in terms of how each property changes and where the interface is located, so that we are able to use a level-set function to parameterize the common interface shared by these two model parameters. The inversion of gravity and traveltime data is carried out by minimizing a joint datafitting. An adjoint state method is used to compute the traveltime gradient efficiently. Numerical examples are provided to demonstrate the inversion algorithm.**The Inverse Spectral Problem for Transmission Eigenvalues**

Samuel Cogar (University of Delaware)

We consider the inverse medium problem of determining the spherically stratified index of refraction n(r) from given spectral data. We begin by introducing a modified transmission eigenvalue problem depending on a parameter eta and an associated modified far field operator, which is injective with dense range provided that the wave number is not a modified transmission eigenvalue. We show that the refractive index n(r) is uniquely determined by the modified transmission eigenvalues corresponding to eta whenever eta is an upper bound for the refractive index.**Optimization Approach for Tomographic Inversion from Multiple Data Modalities**

Zichao (Wendy) Di (Argonne National Laboratory)

Fluorescence tomographic reconstruction can be used to reveal the internal elemental composition of a sample. On the other hand, transmission tomography can be used to obtain the spatial distribution of the absorption coefficient inside the sample. In this work, we integrate both modalities and formulate an optimization approach to simultaneously reconstruct the composition and absorption effect in the sample.**Second Kind Integral Formulation for the Mode Calculation of Optical Waveguides**

Jun Lai (Zhejiang University)

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.**A Support Theorem for Integral Moments of a Symmetric m-Tensor Field**

Anuj Abhishek (Tufts University (Medford, MA, US))

We consider the q-th integral moment, I^(q)f, of any symmetric tensor field f of order m over a simple, real-analytic Riemannian manifold of dimension n. The zeroth integral moment of such tensor fields coincides with the usual notion of geodesic ray transform of the tensor fields. In this work, we use Analytic Microlocal techniques to prove that if we know I^(q)f = 0 for q = 0, 1,.., m over the open set of geodesics not intersecting a geodesically convex subset of the manifold, then the support of f lies within that convex subset.**Hyperbolic PDEs Modelling for Accurate 3D Scanning via Photometric Stereo**

Roberto Mecca (University of Cambridge (Cambridge, GB))

Photometric Stereo is a fundamental problems in Computer Vision aimed at reconstructing surface depth having multiple images taken under different illuminations. Photometric Stereo is an active research area, and a diverse range of approaches have been proposed in recent decades. However, devising a robust reconstruction technique still remains a challenging goal, as the image acquisition process is highly nonlinear. Recent Photometric Stereo variants rely on simplifying assumptions in order to make the problem solvable: light propagation is still commonly assumed to be uniform, and the Bidirectional Reflectance Distribution Function is assumed to be diffuse, with limited interest for specular materials. Here, a well-posed formulation based on Hyperbolic PDEs for a specular reflectance function is presented as well as numerous experiments to show strengths and weaknesses of the approach. This derivation is based on ratio of images, which makes the model independent from photometric invariants and yields a well-posed system of quasi-linear PDEs with discontinuous coefficients.**Shape Reconstruction of the Multi-scale Rough Surface**

Lei Zhang (Zhejiang University)

We consider the problem of reconstructing the shape of multi-scale sound-soft large rough surfaces from phases measurements of the scattered field generated by tapered waves with multiple frequencies impinging on a rough surface. To overcome both the ill-posedness and nonlinearity of this problem for a single frequency, the Landweber regularization method based on the adjoint of the nonlinear objective functional is used. When the multi-frequency data is available, an approximation method is introduced to estimate the large-scale structure of the rough surface using the data measurements at the lowest frequency. The obtained estimate serves as an initial guess for a recursive linearization algorithm in frequency, which is used to capture the small scale structure of the rough surface. Numerical experiments are presented to illustrate the effectiveness of the method.**Increasing Stability for a Solution of Helmholtz Equation by Using Cauchy Data**

Mozhgan (Nora) Entekhabi (Wichita State University (Wichita, KS, US))

Many problems related to steady-state oscillations (mechanical, acoustical, thermal, electromagnetic) lead to the Helmholtz equation. In this paper we will study increasing stability of the Cauchy problem for Helmholtz equation by using Cauchy data. Our analysis also answers the main question: How will stability of our solution increase if we are able to measure our data and what is the relationship between stable part and unstable part with wave number?**Perfectly-matched-layer Boundary Integral Equation Method for Wave Scattering in a Layered Medium**

Wangtao Lu (Michigan State University)

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green’s function of the background medium need to evaluate the expensive Sommerfeld integrals. Alternative BIE methods based on the free space Green’s function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the PML-transformed free-space Green’s function to define the boundary integral operators. The method is efficient, since the PML-transformed free-space Green’s function is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy.