RUI: Geometric Optimization Involving Partial Differential Equations

最佳几何设计提供了大量有趣且具有挑战性的数学问题。其中一个著名的问题可以追溯到18世纪。J.-L。拉格朗日提出了给定长度和体积的变截面积杆的临界载荷最大化问题。另一个著名的经典例子是，一个多世纪以前，L.瑞利（L. Rayleigh）推测，在所有面积相等的形状中，圆盘应该使膜的基频最小。其他最近的应用包括机械振动、光学谐振器的设计、光子晶体波导、种群动力学中有利和不利区域的确定、肥皂膜和最小表面、药物设计和图像分割。这类问题的数值求解既需要前向求解，也需要优化求解。正演解是在给定几何参数或区域上求解问题的数值方法。优化解的目标是找到使设计目标最大化的最优几何设计。本文主要研究了p-Laplacian Poisson方程、Laplace Beltrami算子、Steklov问题的几何优化及其在放射治疗优化设计和自由边界极小曲面中的应用。正向求解基于有限元法和特解法，优化求解基于重排法、形状导数、保形因子、保形类和度量的灵敏度分析。PI将研究许多应用中产生的广泛问题，包括(1)总位移的优化，(2)优化问题重排方法的收敛率研究，(3)最佳放疗设计，(4)最大化闭流形上的共形和拓扑拉普拉斯-贝特拉米特征值，以及(5)极值Steklov特征值问题和自由边界最小曲面。该项目将推进基于重排方法、形状导数和灵敏度分析的优化求解器的开发，并为解决上述应用提供工具。同时，所得结果将被整合到克莱蒙特麦肯纳学院的数值分析和偏微分方程的新课程中。PI将监督本科生和研究生。此外，PI将在即将到来的AIMS， ICIAM， SIAM和其他国际会议上组织应用数学研讨会，工作组研讨会和一系列小型研讨会，以吸引感兴趣的科学家，包括那些来自代表性不足群体的科学家。PI和她的学生还将通过克莱蒙特大学探索数学科学之门（GEMS）项目向K-12学生推广。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.