Collaborative Research: Multilevel Methods for Optimal Control of Partial Differential Equations and Optimization-Based Domain Decomposition

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Optimal control of differential equations (PDECO) plays an important role in an ever increasing number of real-life applications ranging from petroleum reservoir modeling to weather prediction and the optimal shape design of airplane wings. While traditional PDECO uses deterministic models, this project targets PDECO where the differential equations also include uncertainties, such as irregular fluctuations in the ground composition, or turbulent wind speeds. The ultimate aim is to dramatically improve the solution quality and the computing time of such optimal control problems. The novel algorithms resulted from this project will impact optimization problems arising in geophysics, weather modeling etc. These problems are generic and advances in solution techniques will also benefit other sciences. Open source software will be created and shared with the community. Four graduate students will benefit from the project. Special attention will be given to recruit students from underrepresented groups. The project is focused on developing robust, scalable multilevel solvers for mainly two classes of potentially large-scale PDECO problems: PDECOs constrained by stochastic partial differential equations (PDEs) and by nonlocal PDEs. An additional thrust is to develop multilevel solvers in support of optimization-based domain decomposition - another kind of PDECO - for the forward PDE-models themselves. Multilevel/multigrid solvers are known to be optimal for many classes of forward models. However, their application to solve PDECO problems is still in its infancy. A naive application of multilevel methods to solve such optimization problems can lead to dependence on resolution (mesh-dependence) and on other parameters of the problem such as the stochastic dimension or the number of subdomains. In addition, since each iterate involves at least one PDE solve, the cost of solving such optimization problems can be prohibitive for large-scale, high-resolution problems, especially for problems that are significantly more expensive than the traditional, deterministic ones. The algorithms developed in this project aim to set new standards of efficiency and robustness. Novel mathematical tools will further advance the knowledge in numerical analysis and optimization. New special topics courses will be developed based on the research generated in the project and the notes will be shared with the community. The results of the research will be actively disseminated via technical research papers and talks at national and international conferences.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.