RUI: Geometric Optimization Involving Partial Differential Equations

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Optimal geometric design provides a vast number of interesting and challenging mathematical problems. One of the famous problems goes back to 18th Century. J.-L. Lagrange formulated the problem to maximize the critical load of a rod of variable cross-sectional area with given length and volume. Another famous classic example is that L. Rayleigh conjectured that the disk should minimize the fundamental frequency of a membrane among all shapes of equal area, more than a century ago. Other recent applications include mechanical vibration, design of optical resonator, photonic crystal waveguides, determination of favorable and unfavorable regions in population dynamics, soap films and minimal surfaces, drug design, and image segmentation. Numerical approaches for these kinds of problems require both forward solvers and optimization solvers. The forward solvers are numerical approaches to solve problems on a given setting of geometric parameters or domain. The optimization solvers aim to find the optimal geometric design, which maximizes the design objective. In this proposal, the aim is to study geometric optimization of p-Laplacian Poisson’s equations, Laplace Beltrami operator, Steklov problems, and their applications in optimal radiotherapy design and free boundary minimal surfaces. The forward solvers are based on finite element methods and methods of particular solutions while the optimization solvers are based on rearrangement methods, shape derivatives, and sensitivity analysis of conformal factor, conformal classes, and metrics. The PI will study a wide class of problems arising from many applications including (1) optimization of total displacement, (2) convergence rate study of rearrangement methods for optimization problems, (3) optimal radiotherapy design, (4) maximizing conformal and topological Laplace-Beltrami eigenvalues on closed Manifolds, and (5) extremal Steklov eigenvalue problems and free boundary minimal surfaces. The project will advance the development of optimization solvers based on rearrangement methods, shape derivatives, and sensitivity analysis and provide tools to solve aforementioned applications. Also, the obtained results will be integrated to develop new curriculums on numerical analysis and partial differential equations at Claremont McKenna College. The PI will supervise both undergraduate and graduate students. In addition, the PI will organize applied math seminars, working group seminars, and a series of minisymposium at coming AIMS, ICIAM, SIAM, and other international conferences to engage interested scientists, including those from underrepresented groups. The PI and her students will also outreach to K-12 students via Gateway to Exploring Mathematical Sciences (GEMS) program at Claremont.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.