In this talk, I will discuss the solution of a class of inverse problems that originate in applications from image reconstruction. These applications include microscopy, medical imaging and, our particular application, astronomical imaging. We pose our inverse problem as a optimization problem. Incorporated into our choice of objective function is prior statistical knowledge about the noise in our data, while a regularization functional is incorporated for stability. Finally, the optimization problem we wish to solve is constrained. The constraints arise from the fact that the objects we are imaging (stars) are photon densities, or light intensities, and are therefore nonnegative. Since high resolution images are desired, the problem is large-scale. Consequently, efficient numerical techniques are required. I will discuss the problem of minimizing both quadratic and convex cost functions with nonnegativity constraints. An existing algorithm will be presented for the quadratic minimization problem, and ideas from this algorithm are extended to the problem of minimizing the convex function. Implementation of an efficient sparse preconditioner will also be discussed, and a numerical study of these algorithms will be presented.

Spring 2003 Colloquium Schedule | Mathematical Sciences | The University of Montana