Methods and Applications for PDE-Constrained Optimization

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This project involves a three-year program of research on severalfundamental computational issues involved in the parallel implicitsolution of optimization problems with partial differential equation(PDE) constraints. Such problems arise in many contexts inengineering and scientific computation, since physical reality isoften expressed through models involving PDEs. Accuratediscretizations of PDEs lead to very large sparse constrainedoptimization problems, where at least part of the structure reflectsthe discretization. The goals of the research on methods include, butare not limited to, advancing the state-of-the-art in four fundamentaltopics: (1) the formulation and analysis of algorithms for large-scalenonlinear optimization; (2) adaptive mesh generation for PDEs; (3)multilevel PDE solvers; and (4) parallel computation. Although eachof these topics can be investigated in isolation, the investigatorsbelieve that the exploitation of their interactions is crucial for thecreation of effective global algorithms. The research is motivatedand guided by three particularly challenging applications: (i)geophysical inverse problems; (ii) projection methods for evolutionPDEs with constraints; and (iii) constrained level-set methods. Thesetopics cover a number of important applications of computationalscience, including off-shore petroleum exploration, the numericalmodeling of black holes, the modeling of crystal growth andbiomembranes, capturing diffraction effects of waves and pathplanning. Two features common to all these applications are that thePDE constraints must be handled using modern adaptive multi-leveltechniques and that the underlying optimization problem is highlynonlinear and hence nonconvex.The Investigators are members of the Computational and AppliedMathematics (CAM) Group within the Department of Mathematics at UC SanDiego. They have a combined expertise in applied mathematics,numerical optimization, numerical partial differential equations andparallel computation. An important goal is the development ofsoftware embodying the above algorithms. Software developed as partof the project will provide an effective method of technology transferand will extend the scope and effectiveness of existing codes thathave been developed by the investigators at UC San Diego. Thesoftware component of the project will have a substantial impact onresearch involving the modeling of complex systems as it will providescientists and engineers with instant access to state-of-the-artmethods. Within the Computational and Applied Mathematics Group, theInvestigators offer a program of instruction and research thatemphasizes 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 advancedgraduate students into the area of computational science, which playsa vital role in the study of systems arising in manufacturing,engineering and the natural sciences.