An Adaptive Finite Element Framework for Inverse Imaging
Wolfgang Bangerth
Inst. for Comp.Eng.and Sciences
and Institute for Geophysics
University of Texas, Austin


In many applications one wants to infer interior properties of a body without invasive procedures. Examples include medical imaging, electrical impedance tomography, and seismic inversion. In particular, optical imaging of tumors (tomography) will be discussed as one of the goals of the techniques to be presented and we will show the first adaptive results reported for this kind of application.

If there is a relationship between the sought parameters and observable quantities in the form of a partial differential equation, such parameter estimation (or inverse) problems can be cast as an optimization problem involving PDEs as constraints. Unfortunately, inverse problems are
numerically very challenging, involving the solution of a large number of PDEs as subproblems. Adaptive finite element methods are therefore an important tool to reduce the amount of work, or to achieve otherwise unattainable accuracy. We will present error estimation techniques to generate discretizations for such problems that independently adapt both the meshes for the (observable) state variables as well as those for the sought parameters. The error estimates are based on residuals of the optimality  conditions and are weighted with the solution of a dual problem. We will show results demonstrating that this yields significant savings in computational  work, as well as much increased resolution and accuracy compared to traditional approaches.