This grant builds upon prior research on optimization algorithms and software to develop successful methods for medical imaging and other challenging applied problems. Some of the applications studied are image reconstruction via Positron Emission Tomography (PET), radiation therapy planning, x-ray crystallography, and the modeling of underwater aquifers. These problems are very large. The algorithms used are based upon well developed general-purpose methods for nonlinear optimization. They use as a foundation software for unconstrained optimization (both serial and parallel) based on a truncated-Newton method developed in prior research. The applied algorithms also use sophisticated ideas developed for barrier methods, techniques that ensure stable numerical computations capable of producing high-accuracy solutions. The algorithms are also based on primal-dual frameworks that are proving to be successful for wide classes of linear and nonlinear optimization problems. The work extends these powerful computational tools, by exploiting the special features of the applied problems. Parallelism (both in the underlying optimization method and in the calculations of the model functions) is also essential to achieve effective methods. The immediate goal of this research is to produce software that can be used by scientists and clinicians. This requires algorithms that are fast and reliable. The longer-term hope is that the techniques developed for these applied problems will lead to improved methods for general-purpose optimization. The overall intent is to produce reliable algorithms and software to solve these problems as rapidly as possible.
