Partial Differential Equation Constrained Optimization: Algorithms, Numerics, and Applications

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The state of the art in modeling of the physical processes in science and engineering confronts us with solving increasingly complex optimization problems in the presence of constraints. Think of designing the wing of an aircraft where it is essential that it can withstand a tremendous amount of pressure. This is just one of the example of an optimization problem with constrains. Such problems are inherently nonlinear due to constraints. This makes the development and analysis of algorithms and numerical techniques for the solution of these problems extremely challenging but rewarding. This project will create new optimization algorithms that allows us to incorporate constraints in a systematic and robust fashion. The targeted applications range from next generation micro- and nano-scale lab-on-chip devices to novel material and structural designs. Open source software will be created so that the research can be easily used by scientists working in other research areas. The applications under consideration can be modeled using partial differential equations (PDEs). These PDEs are nonlocal (fractional); nonsmooth (contact problems); geometric, nonlinear, multiscale with an unknown domain, i.e. free boundary problems (FBPs). This project focuses on optimization problems with such PDE constraints - the so-called PDE constrained optimization problems. It aims to create new optimization schemes that will enable the solution of currently intractable optimization problems with nonsmooth features, including: optimization problems with nonlinear inequality constraints, contact problems, and risk-averse PDE constrained optimization. The resulting optimization algorithms will provide new insights into nonconvex nonsmooth problems. Optimization problems with surface tension is a new research field with great potential to enhance our understanding at the micro and nano-scales where surface effects dominate bulk effects. This will impact the design of next generation lab-on-chip, forensics and liquid lenses in astronomical telescopes. Open source software will be developed, which will not only benefit scientists in optimization, FBPs, and nonlocal problems but also scientists in nonlinear PDEs and data driven optimization problems. The results will be disseminated via two special topics courses on (i) PDE constrained optimization under uncertainty; (ii) Deep learning and PDE constrained optimization. One female student will get her PhD.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.