Mathematical Sciences: Iterative Methods for Equations and Optimization

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9321938 Kelley The principal investigator will design and analyze algorithms for the large systems of linear and nonlinear equations and optimization problems that arise when infinite dimensional problems are discretized. Objectives include further development of a new multilevel method for compact fixed point problems, work on globally convergent methods for nonsmooth compact fixed point problems, particularly those that arise from parabolic control problems, and analysis of the sup-norm convergence of certain methods based on inner products, particularly GMRES, CG, and primal-dual interior point methods. The principal investigator will also continue work on special purpose methods for optimization and solution of nonlinear equations. These methods are directed at problems such as optimization of noisy functions, aomputer aided design of microwave devices, and optimal control. Large systems of equations arise in scientific computation in many ways. This project is to design and analyze efficient solution methods for classes of very large problems that arise in many applications. The particular objectives of and algorithms studied in this project are driven by applications and the principal investigator works closely with scientists and engineers in other disciplines. As an example, in computer aided design and manufacture of microwave semiconductor devices, such as the amplifiers used in radar and cellular communication, the principal investigator's work has been incorporated into software for analysis and optimization of process yield, design of more efficient devices, and bench testing of chips. In the course of the development of this software, a new algorithm was discovered for the optimization of noisy functions, that is, functions whose values are corrupted by noise or error. Process yield models often have that feature. One goal of the present project is to continue the work on noisy optimization.