Nonlinear Equations and Optimization

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DMS 0070641ABSTRACT.The principal investigator will continue his research program in the numericalsolution of large systems of nonlinear equations and noisy optimization problems. In the area of nonlinear equations the PI will continue his research program on pseudo-transient continuation methods for nonlinear equations to the partial differential algebraic equation (PDAE) settingand design and analyze multilevel methods for fully iterative algorithms for parameter-dependent families of compact fixed point problems.In the area of optimization, the principal investigator will extend his work on sampling methods to problems with ``hidden constraints'', i.e. constraints whose violation can only be detected by failure of the objective function to return a value, continue his development of the implicitfiltering with the design of a Levenberg-Marquardt form of the algorithm, and investigate the performance of the DIRECT method both from the theoretical viewpoint and as a way to identify hidden constraints. This aspect of the PI's research involves undergraduates in a significant way.Many processes and models in engineering and science are expressed as nonlinearequations. Numerical simulation of these processes requires the rapid and accurate solution of these equations and a clear understanding of the methodsand their limitations. With a high-quality simulator in hand, optimizationcan be used in design. The optimization problems that arise in industrial design are often noisy because of measurement or simulation error. The principal investigator will continue his work on the computational solution of nonlinear equations and optimization problems. The work will consist of study of algorithms, needed to ensure robustness and reliability of simulations, implementation on distributed memory computers,which is important for for rapid turnaround, and applying the methodsin collaborations with scientists and engineers in industry, national laboratories, and academia. The work will be used in (a) environmental modeling, measurement, and remediation, (b) simulation, optimal design, and control in the aerospace industry, and in (c) optimization of gas pipeline networks.