Collaborative Research: Parabolic Monge-Ampère Equations, Computational Optimal Transport, and Geometric Optics

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This project is centered around the development of theoretical tools which will lead to efficient numerical methods for solving the optimal transport problem, which is a mathematical problem that seeks to minimize the total cost of transporting mass from one location to another. The theoretical study of this problem has advanced greatly in recent years, and the results obtained have been applied successfully to a number of disciplines outside of mathematics, such as the design of lenses with specific reflection properties in Physics, modeling the atmosphere near the earth's surface in Geology, and creating optimal assignments in Economics, among others. This litany of applications makes the development of effective computational tools an ever more urgent matter, and it is important to have tools that can be mathematically guaranteed to exhibit outstanding performance. The project will develop novel computational methods based on nonlinear partial differential equations and establish rigorous mathematical results about these equations in order to guarantee desirable performance of the corresponding numerical algorithms. The work of the project involves individual and collaborative research by the Principal Investigators (PIs), and research mentoring of graduate and undergraduate students, with appropriate problems having been identified for students. The PIs will also engage in outreach through co-supervising an undergraduate team in 2024 through the Lafayette College Summer REU program and in 2025 through the Summer Undergraduate Research Institute in Experimental Mathematics program at Michigan State University. Both programs aim to recruit students from schools with limited opportunities for undergraduate research, with an eye toward recruitment of students from traditionally underrepresented groups in the mathematical sciences. The project develops the theoretical foundations for establishing existence and characterizing long-time behavior of solutions to a class of degenerate-parabolic fully nonlinear partial differential equations (PDE) in singular settings. These PDE are time-dependent variants of the classical Monge-Ampere equation. Significant progress has been made over the last few decades in developing a theory of classical solutions for time-dependent Monge-Ampere equations with smooth data and Dirichlet boundary conditions. By comparison, the theory of generalized solutions for such evolutionary equations is severely underdeveloped, especially in the context of optimal transport and geometric optics, where the natural boundary condition is a non-standard one. The current project will create a theoretical foundation for viscosity and weak solutions of a class of degenerate parabolic, fully nonlinear equations of Monge-Ampere type with oblique boundary data. The work will also establish quantitative rates of convergence for such equations; this will provide a promising numerical method for the design and construction of reflector surfaces arising in engineering problems.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.