Inverse Problems Related to Integral Geometry and a Parabolic Partial Differential Equation

Abstract:  We will discuss the following three inverse problems.

• Momentum ray transforms (MRT) in the Schwartz space: Momentum ray transform integrates symmetric tensor fields along straight lines with a power t k . Here we will discuss the injectivity question, kernel description, inversion algorithm and the range characterization of the momentum ray transforms. In this case we will show that given sufficient momentum ray transforms of a symmetric tensor field, it is possible to recover the entire tensor field.

• Restricted transverse ray transform (TRT): Next we study the transverse ray transform of symmetric m-tensor fields in R 3 restricted to all lines passing through a curve from microlocal viewpoint. More precisely, we are interested in an approximate inversion of a TRT acting on symmetric tensor fields restricted to all lines passing through a fixed curve in R 3 .

• Parabolic inverse problem: Finally we discuss the inverse problem of determining the convection coefficient and the time-dependent density coefficient appearing in a convection-diffusion equation from the boundary measurements.