Numerical recovery of source singularities via the radiative transfer equation with partial data

The inverse source problem for the radiative transfer equation is considered, with partial data. Here we demonstrate numerical computation of the normal operator X*_V XV, where XV is the partial data solution operator to the radiative transfer equation. The numerical scheme is based in part on a forward solver designed by Monard and Bal [J. Comput. Phys., 229 (2010), pp. 4952-4979]. We will see that one can detect quite well the visible singularities of an internal optical source f for generic anisotropic k and σ, with or without noise added to the accessible data XV f. In particular, we use a truncated Neumann series to estimate XV and X*_V, which provides a good approximation of X*_V XV with an error of higher Sobolev regularity. This paper provides a visual demonstration of the author's previous work [M. Hubenthal, Inverse Problems, 27 (2011), 125009] in recovering the microlocally visible singularities of an unknown source from partial data. We also give the theoretical analysis necessary to justify the smoothness of the remainder when approximating the normal operator.