Optimization with PDE Constraints

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The investigator and his colleagues focus on several fundamental computational issues involved in the parallel implicit solution of optimization problems with partial differential equation The goal of the research is to advance the state of the art in four fundamental topics: (1) the formulation and 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 be investigated in isolation, the exploitation of their interactions is crucial for the creation of effective global algorithms. The investigators are members of a Scientific Computation Group that offers a program of instruction and research emphasizing the role of scientific computation in the formulation, modeling, and solution of problems from diverse and changing areas. A major part of the project involves the development of software and its dissemination within the manufacturing, engineering, and scientific community. Software developed as part of the project provides an effective method of technology transfer and extends the scope and effectiveness of the existing codes PLTMG, MC, and SNOPT developed by the investigators. Because partial differential equations conveniently characterize the physical laws of many complex systems occurring in science and engineering, they also lie at the heart of the mathematical models used to simulate and predict the behavior of these systems. The need to optimize the performance of such systems is the common feature of practical applications that range over such diverse areas as the optimal design of the hull of an America's Cup yacht, the cleanup of toxic waste sites, the construction of bioartificial arteries in tissue engineering, and the management of stock portfolios and hedge funds. Software developed in this project provides engineers and scientists with instant access to state-of-the-art methods for the modeling and optimization of complex systems involving partial differential equation constraints. The resulting improvements in the efficiency, accuracy and robustness of these models have a substantial impact in areas of manufacturing and engineering that are vital to US global competitiveness.