Quantitative photoacoustic imaging in radiative transport regime

The objective of quantitative photoacoustic tomography (QPAT) is to reconstruct optical and thermodynamic properties of heterogeneous media from data of absorbed energy distribution inside the media. There have been extensive theoretical and computational studies on the inverse problem in QPAT, however, mostly in the diffusive regime. We present in this work some numerical reconstruction algorithms for multi-source QPAT in the radiative transport regime with energy data collected at either single or multiple wavelengths. We show that when the medium to be probed is non-scattering, explicit reconstruction schemes can be derived to reconstruct the absorption and the Gruneisen coefficients. When data at multiple wavelengths are utilized, we can reconstruct simultaneously the absorption, scattering and Gruneisen coefficients. We show by numerical simulations that the reconstructions are stable.Comment: 40 pages, 13 figure

Recovery of the absorption coefficient in radiative transport from a single measurement

In this paper, we investigate the recovery of the absorption coefficient from boundary data assuming that the region of interest is illuminated at an initial time. We consider a sufficiently strong and isotropic, but otherwise unknown initial state of radiation. This work is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium. We break the problem into two steps. First, in a linear framework, we seek the simultaneous recovery of a forcing term of the form $\sigma(t,x,\theta) f(x)$ (with $\sigma$ known) and an isotropic initial condition $u_{0}(x)$ using the single measurement induced by these data. Based on exact boundary controllability, we derive a system of equations for the unknown terms $f$ and $u_{0}$. The system is shown to be Fredholm if $\sigma$ satisfies a certain positivity condition. We show that for generic term $\sigma$ and weakly absorbing media, this linear inverse problem is uniquely solvable with a stability estimate. In the second step, we use the stability results from the linear problem to address the nonlinearity in the recovery of a weak absorbing coefficient. We obtain a locally Lipschitz stability estimate

Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations

When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model, for example, the orders of the fractional derivative or the source term, are often unknown, which requires one to discuss inverse problems to identify these physical quantities from some additional information that can be observed or measured practically. This chapter investigates several kinds of inverse coefficient problems for the fractional diffusion equation

Estimate of convection-diffusion coefficients from modulated perturbative experiments as an inverse problem

The estimate of coefficients of the Convection-Diffusion Equation (CDE) from experimental measurements belongs in the category of inverse problems, which are known to come with issues of ill-conditioning or singularity. Here we concentrate on a particular class that can be reduced to a linear algebraic problem, with explicit solution. Ill-conditioning of the problem corresponds to the vanishing of one eigenvalue of the matrix to be inverted. The comparison with algorithms based upon matching experimental data against numerical integration of the CDE sheds light on the accuracy of the parameter estimation procedures, and suggests a path for a more precise assessment of the profiles and of the related uncertainty. Several instances of the implementation of the algorithm to real data are presented.Comment: Extended version of an invited talk presented at the 2012 EPS Conference. To appear in Plasma Physics and Controlled Fusio