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Francis J. Chung

Publications and source records attributed to Francis J. Chung.

12 recordsLinked to original sources
arXiv2020

On diffusive scaling in acousto-optic imaging

Acousto-optic imaging (AOI) is a hybrid imaging process. By perturbing the to-be-reconstructed tissues with acoustic waves, one introduces the interaction between the acoustic and optical waves, leading to a more stable reconstruction of the optical properties. The mathematical model was described in [25], with the radiative transfer equation serving as the forward model for the optical transport. In this paper we investigate the stability of the reconstruction. In particular, we are interested in how the stability depends on the Knudsen number, Kn, a quantity that measures the intensity of the scattering effect of photon particles in a media. Our analysis shows that as Kn decreases to zero, photons scatter more frequently, and since information is lost, the reconstruction becomes harder. To counter this effect, devices need to be constructed so that laser beam is highly concentrated. We will give a quantitative error bound, and explicitly show that such concentration has an exponential dependence on Kn. Numerical evidence will be provided to verify the proof.

math.AP↗
arXiv2020

Inverse Radiative Transport with Local Data

We consider an inverse problem for a radiative transport equation (RTE) in which boundary sources and measurements are restricted to a single subset $E$ of the boundary of the domain $Ω$. We show that this problem can be solved globally if the restriction of the X-ray transform to lines through $E$ is invertible on $Ω$. In particular, if $Ω$ is strictly convex, we show that this local data problem can be solved globally whenever $E$ is an open subset of the boundary. The proof relies on isolation and analysis of the second term in the collision expansion for solutions to the RTE, essentially considering light which scatters exactly once inside the domain.

math.AP↗
arXiv2019

A Note on the Transport Method for Hybrid Inverse Problems

There are several hybrid inverse problems for equations of the form $\nabla \cdot D \nabla u - σu = 0$ in which we want to obtain the coefficients $D$ and $σ$ on a domain $Ω$ when the solutions $u$ are known. One approach is to use two solutions $u_1$ and $u_2$ to obtain a transport equation for the coefficient $D$, and then solve this equation inward from the boundary along the integral curves of a vector field $X$ defined by $u_1$ and $u_2$. It follows from an argument of Guillaume Bal and Kui Ren that for any nontrivial choices of $u_1$ and $u_2$, this method suffices to recover the coefficients on a dense set in $Ω$. This short note presents an alternate proof of the same result from a dynamical systems point of view.

math.AP↗
arXiv2019

A Transport Model for Multi-Frequency Acousto-Optic Tomography

In a medium where the dielectric permittivity is perturbed in the presence of an acoustic wave, optical scattering generates frequency-shifted light. In this paper we consider the inverse problem of recovering the optical properties of this medium from measurements of the frequency-shifted light, using a radiative transport equation (RTE) model for light propagation. Given some assumptions on the regularity and isotropicity of the coefficients of the RTE, we show that the absorption coefficient can be reconstructed from the boundary measurements of a single well chosen illumination, and that the scattering coefficients can be reconstructed from boundary measurements of a one-parameter family of illuminations.

math.AP↗
arXiv2016

The $L^p$ Carleman estimate and a partial data inverse problem

We construct an explicit Green's function for the conjugated Laplacian $e^{-ω\cdot x/h}Δe^{-ω\cdot x/h}$, which let us control our solutions on roughly half of the boundary. We apply the Green's function to solve a partial data inverse problem for the Schrödinger equation with potential $q \in L^{n/2}$. We also use this Green's function to derive $L^p$ Carleman estimates similar to the ones in Kenig-Ruiz-Sogge \cite{krs}, but for functions with support up to part of the boundary.

math.AP↗
arXiv2016

Optical tomography on graphs

We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.

math.CO↗
arXiv2016

Partial data inverse problems for the Hodge Laplacian

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. The method is based on Carleman estimates for the Hodge Laplacian with relative or absolute boundary conditions, and on the construction of complex geometric optics solutions which reduce the Calderon type problem to a tensor tomography problem for 2-tensors. The arguments in this paper allow to establish partial data results for elliptic systems that generalize the scalar results due to Kenig-Sjostrand-Uhlmann.

math.AP↗
arXiv2015

Partial Data Inverse Problems for Maxwell Equations via Carleman Estimates

In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim-Uhlmann and Kenig-Sjöstrand-Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.

math.AP↗
arXiv2014

Partial Data for the Neumann-Dirichlet Magnetic Schrödinger Inverse Problem

We show that an electric potential and magnetic field can be uniquely determined by partial boundary measurements of the Neumann-to-Dirichlet map of the associated magnetic Schrödinger operator. This improves upon previous results of the author by including the determination of a magnetic field. The main technical advance is an improvement on the Carleman estimate for the magnetic Schrödinger operator with the appropriate boundary conditions. This allows the construction of complex geometrical optics solutions with greater regularity, which are needed to deal with the first order term in the operator. This improved regularity of CGO solutions may have applications in the study of inverse problems in systems of equations with partial boundary data.

math.AP↗
arXiv2013

Partial Data for the Neumann-to-Dirichlet Map

We show that measurements of the Neumann-to-Dirichlet map, roughly speaking, on a certain part of the boundary of a smooth domain in dimension 3 or higher, for inputs with support restricted to the other part, determine an electric potential on that domain. Given a convexity condition on the domain, either the set on which measurements are taken, or the set on which input functions are supported, can be made to be arbitrarily small. The result is analogous to the result by Kenig, Sjöstrand, and Uhlmann for the Dirichlet-to-Neumann map. The main new ingredient in the proof is a Carleman estimate for the Schrödinger operator with appropriate boundary conditions.

math.AP↗
arXiv2011

A Partial Data Result for the Magnetic Schrodinger Inverse Problem

This article shows that knowledge of the Dirichlet-Neumann map on certain subsets of the boundary for input functions supported roughly on the rest of the boundary can be used to determine a magnetic Schrödinger operator. With some geometric conditions on the domain, either the subset on which the DN map is measured or the subset on which the input functions have support may be made arbitrarily small. This is an improvement on the partial data result in a paper by Dos Santos Ferreira, Kenig, Sjöstrand, and Uhlmann. The method involves modifying the Carleman estimate in that paper by conjugation with operators built from pseudodifferential pieces.

math.AP↗