Abstract:
Fluorescence photoacoustic tomography (fPAT) is a molecular imaging modality that combines photoacoustic tomography (PAT) with fluorescence imaging to obtain high-resolution imaging of fluorescence distributions inside heterogeneous media. The objective of this work is to study inverse problems in the quantitative step of fPAT where we intend to reconstruct physical coefficients in a coupled system of radiative transport equations using internal data recovered from ultrasound measurements. We derive uniqueness and stability results on the inverse problems and develop some efficient algorithms for image reconstructions. Numerical simulations based on synthetic data are presented to validate the theoretical analysis. The results we present here complement these in [Ren-Zhao, SIAM J. Imag. Sci., 2013] on the same problem but in the diffusive regime.

Abstract:
We give a survey of author's results on the inverse hyperbolic problems with time-dependent and time-independent coefficients. We consider the case of hyperbolic equations with Yang-Mills potentials and the case of domains with obstacles. In particular, we gave a stability estimate for the broken X-ray transform in the case of one convex obstacle in $\mathbb{R}^2$.

Abstract:
In this paper, we establish a global Carleman estimate for stochastic parabolic equations. Based on this estimate, we solve two inverse problems for stochastic parabolic equations. One is concerned with a determination problem of the history of a stochastic heat process through the observation at the final time $T$, for which we obtain a conditional stability estimate. The other is an inverse source problem with observation on the lateral boundary. We derive the uniqueness of the source.

Abstract:
In this article we establish three stability results for some inverse problems. More precisely we consider the following boundary value problem: $Delta u + lambda u + mu = 0$ in $Omega$, $u = 0$ on $partialOmega$, where $lambda$ and $mu$ are real constants and $Omega subset mathbb R^2$ is a smooth bounded simply-connected open set. The inverse problem consists in the identification of $lambda$ and $mu$ from knowledge of the normal flux $partial u/partial u$ on $partialOmega$ corresponding to some nontrivial solution.

Abstract:
In this paper, we study the stability of two inverse boundary value problems in an infinite slab with partial data. These problems have been studied by Li and Uhlmann for the case of the Schrodinger equation and by Krupchyk, Lassas and Uhlmann for the case of the magnetic Schrodinger equation. Here we quantify the method of uniqueness proposed by Li and Uhlmann and prove a log-log stability estimate for the inverse problems associated to the Schrodinger equation. The boundary measurements considered in these problems are modelled by partial knowledge of the Dirichlet-to-Neumann map: in the first inverse problem, the corresponding Dirichlet and Neumann data are known on different boundary hyperplanes of the slab; in the second inverse problem, they are known on the same boundary hyperplane of the slab.

Abstract:
We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [M. Choulli-P. Stefanov, 1996 and 1999] and partial stability estimates were obtained in [J.-N. Wang, 1999] for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.

Abstract:
Hybrid inverse problems are mathematical descriptions of coupled-physics (also called multi-waves) imaging modalities that aim to combine high resolution with high contrast. The solution of a high-resolution inverse problem, a first step that is not considered in this paper, provides internal information combining unknown parameters and solutions of differential equations. In several settings, the internal information and the differential equations may be described as a redundant system of nonlinear partial differential equations. We propose a framework to analyze the uniqueness and stability properties of such systems. We consider the case when the linearization of the redundant system is elliptic and with boundary conditions satisfying the Lopatinskii conditions. General theories of elliptic systems then allow us to construct a parametrix for such systems and derive optimal stability estimates. The injectivity of the nonlinear problem or its linearization is not guaranteed by the ellipticity condition. We revisit unique continuation principles, such as the Holmgren theorem and the uniqueness theorem of Calder\'on, in the context of redundant elliptic systems of equations. The theory is applied to the case of power density measurements, which are internal functionals of the form $\gamma|\nabla u|^2$ where $\gamma$ is an unknown parameter and $u$ is the solution to the elliptic equation $\nabla\cdot\gamma\nabla u=0$ on a bounded domain with appropriate boundary conditions.

Abstract:
We consider a geometric inverse problems associated with interior measurements: Assume that on a closed Riemannian manifold $(M, h)$ we can make measurements of the point values of the heat kernel on some open subset $U \subset M$. Can these measurements be used to determine the whole manifold $M$ and metric $h$ on it? In this paper we analyze the stability of this reconstruction in a class of $n$-dimensional manifolds which may collapse to lower dimensions. In the Euclidean space, stability results for inverse problems for partial differential operators need considerations of operators with non-smooth coefficients. Indeed, operators with smooth coefficients can approximate those with non-smooth ones. For geometric inverse problems, we can encounter a similar phenomenon: to understand stability of the solution of inverse problems for smooth manifolds, we should study the question of uniqueness for the limiting non-smooth case. Moreover, it is well-known, that a sequence of smooth $n$-dimensional manifolds can collapse to a non-smooth space of lower dimension. To analyze the stability of inverse problem in a class of smooth manifolds with bounded sectional curvature and diameter, we study properties of the spaces which occur as limits of these collapsed manifolds and study uniqueness of inverse problems on collapsed manifolds. Combining these, we obtain stability results for inverse problems in the class of smooth manifolds with bounded sectional curvature and diameter.

Abstract:
In this paper, solution of inverse contaminant transport problems is studied, including nonlinear sorption inequilibrium and non-equilibrium mode. A precise numerical solver for the direct problem is discussed. The method isbased on time stepping and operator splitting with respect to the nonlinear transport, diffusion and adsorption. The nonlinear transport problem corresponds to a multiple Riemann problem and is solved by modified front tracking method. The diffusion problem is solved by a finite volume scheme and the sorption part is solved by an implicit numerical scheme. The solution of the inverse problem isbased on an iterative approach. The gradient of the cost functional with respect to the determined parameters is constructed by means of solution of the corresponding adjoint system. Numerical examples are presented for a 1D situation and for a dual-well setting with steady-stateflow between injection and extraction wells.

Abstract:
We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.