We consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients μ(x), D(x), from a single measurement of the absorbed energy E(x) = μ(x)u(x), where u satisfies the elliptic partial differential equation
−∇ ⋅ (D(x)∇u(x)) + μ(x)u(x) =0 in Ω ⊂ ℝN .
This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. Using similar ideas as in , we show that when the coefficients μ, D are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of μ, D, we suggest a variational method based on an Ambrosio-Tortorelli approximation of a Mumford-Shah-like functional, which we implement numerically and test on simulated two-dimensional data.
In this paper we develop a convergence analysis in an infinite dimensional setting of the Levenberg–Marquardt iteration
for the solution of a hybrid conductivity imaging problem. The problem consists in determining the spatially varying conductivity σ
from a series of measurements of power densities for various voltage inductions. Although this problem has been very well studied in the literature, convergence and regularizing properties of iterative algorithms in an
infinite dimensional setting are still rudimentary. We provide a partial result under the assumptions that the
derivative of the operator, mapping conductivities to power densities, is injective and the data is noise-free. Moreover, we implemented the Levenberg–Marquardt algorithm and tested it on simulated data.