We utilize a frequency-domain diffusion equation model for data generation. We parameterize the diffusion (related to the scattering) and absorption in terms of a small number of unknown parameters. The imaging problem we wish to solve becomes a weighted, non-linear least squares problem, with regularization provided through the parameterization. We use a damped Gauss-Newton method to solve the optimization problem. Unfortunately, each function and Jacobian evaluation requires the solution of several large-scale linear systems, where the size of each system depends on the number of voxels in the 3D image. Thus, the linear solves are a huge computational bottleneck for the imaging problem.
The goal of this talk is to analyze matrix characteristics and techniques for reducing the computational complexity of the forward solves. We consider Krylov-subspace methods for solving the systems at every outer (i.e. Gauss-Newton) iteration. In particular, we exploit relationships among the systems that exist at each Gauss-Newton and linesearch step to design a method that reuses information as appropriate from previous solves, thereby speeding convergence on subsequent systems. We give a numerical illustration of the promise of this approach.
This is joint work with Eric de Sturler, UIUC.