Forward and inverse problems of a semilinear transport equation

Abstract

We study forward and inverse problems for a semilinear radiative transport model where the absorption coefficient depends on the angular average of the transport solution. Our first result is the well-posedness theory for the transport model with general boundary data, which significantly improves previous theories for small boundary data. For the inverse problem of reconstructing the nonlinear absorption coefficient from internal data, we develop stability results for the reconstructions and unify an $L^1$ stability theory for both the diffusion and transport regimes by introducing a weighted norm that penalizes the contribution from the boundary region. The problems studied here are motivated by applications such as photoacoustic imaging of multi-photon absorption of heterogeneous media.