Optimization with PDE Constraints

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The investigator and his colleagues focus on severalfundamental computational issues involved in the parallelimplicit solution of optimization problems with partialdifferential equation The goal of the research is to advance thestate of the art in four fundamental topics: (1) the formulationand analysis of algorithms for large-scale optimization; (2)adaptive mesh generation for partial differential equations; (3)multilevel partial differential equation solvers; and (4)parallel computation. Although each of these topics can beinvestigated in isolation, the exploitation of their interactionsis crucial for the creation of effective global algorithms. Theinvestigators are members of a Scientific Computation Group thatoffers a program of instruction and research emphasizing the roleof scientific computation in the formulation, modeling, andsolution of problems from diverse and changing areas. A majorpart of the project involves the development of software and itsdissemination within the manufacturing, engineering, andscientific community. Software developed as part of the projectprovides an effective method of technology transfer and extendsthe scope and effectiveness of the existing codes PLTMG, MC, andSNOPT developed by the investigators. Because partial differential equations convenientlycharacterize the physical laws of many complex systems occurringin science and engineering, they also lie at the heart of themathematical models used to simulate and predict the behavior ofthese systems. The need to optimize the performance of suchsystems is the common feature of practical applications thatrange over such diverse areas as the optimal design of the hullof an America's Cup yacht, the cleanup of toxic waste sites, theconstruction of bioartificial arteries in tissue engineering, andthe management of stock portfolios and hedge funds. Softwaredeveloped in this project provides engineers and scientists withinstant access to state-of-the-art methods for the modeling andoptimization of complex systems involving partial differentialequation constraints. The resulting improvements in theefficiency, accuracy and robustness of these models have asubstantial impact in areas of manufacturing and engineering thatare vital to US global competitiveness.