Stability for some inverse problems for transport equations

In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed condition on the principal part

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