This project involves a three-year program of research on several fundamental computational issues involved in the parallel implicit solution of optimization problems with partial differential equation (PDE) constraints. Such problems arise in many contexts in engineering and scientific computation, since physical reality is often expressed through models involving PDEs. Accurate discretizations of PDEs lead to very large sparse constrained optimization problems, where at least part of the structure reflects the discretization. The goals of the research on methods include, but are not limited to, advancing the state-of-the-art in four fundamental topics: (1) the formulation and analysis of algorithms for large-scale nonlinear optimization; (2) adaptive mesh generation for PDEs; (3) multilevel PDE solvers; and (4) parallel computation. Although each of these topics can be investigated in isolation, the investigators believe that the exploitation of their interactions is crucial for the creation of effective global algorithms. The research is motivated and guided by three particularly challenging applications: (i) geophysical inverse problems; (ii) projection methods for evolution PDEs with constraints; and (iii) constrained level-set methods. These topics cover a number of important applications of computational science, including off-shore petroleum exploration, the numerical modeling of black holes, the modeling of crystal growth and biomembranes, capturing diffraction effects of waves and path planning. Two features common to all these applications are that the PDE constraints must be handled using modern adaptive multi-level techniques and that the underlying optimization problem is highly nonlinear and hence nonconvex.
The Investigators are members of the Computational and Applied Mathematics (CAM) Group within the Department of Mathematics at UC San Diego. They have a combined expertise in applied mathematics, numerical optimization, numerical partial differential equations and parallel computation. An important goal is the development of software embodying the above algorithms. Software developed as part of the project will provide an effective method of technology transfer and will extend the scope and effectiveness of existing codes that have been developed by the investigators at UC San Diego. The software component of the project will have a substantial impact on research involving the modeling of complex systems as it will provide scientists and engineers with instant access to state-of-the-art methods. Within the Computational and Applied Mathematics Group, the Investigators offer a program of instruction and research that emphasizes the role of computational science in the formulation, modeling, and solution of problems from diverse and changing areas. The activities associated with this project will help attract advanced graduate students into the area of computational science, which plays a vital role in the study of systems arising in manufacturing, engineering and the natural sciences.
