Abstract:

The theory of A-analytic maps was developed in late 1990s to treat planar
tomography problems as boundary-value problems for elliptic equation with
operator coefficients. The classical Radon problem (1917) on the plane becomes
equivalent to the Cauchy problem for a Beltrami-type operator equation with
Cauchy data on a closed contour. The classical analytic theory is for complex
valued functions, while the A-analytic theory is for a sequence valued map
satisfying the Beltrami type operator equation. The solution of the Beltrami-
type operator equation have properties very similar with the classical analytic
maps. One of them is that if one know the values of the sequence on the closed
contour, then one can determine values of the sequence inside the domain via a
Cauchy-type integral formula. The A-analytic theory have been successfully
applied to investigate various planar inverse problems. In this talk, we will present
some tomographic inverse problems and show how the theory of A-analyticity is
employed to tackle these problems. In particular, we will present an inverse source
problem in the stationary radiating transport through an absorbing and scattering
medium. Of specific interest, the exiting radiation is measured on an arc. The
attenuation and scattering properties of the medium are assumed known. For
scattering kernels dependent on the angle of scattering, we show that a source can
be quantitatively recovered in the convex hull of the measuring arc.