Novel Discontinuous Galerkin Methods for Deterministic and Stochastic Optimization Problems with Inequality Constraints

该项目旨在开发新的数值方法来解决在弹性理论、多孔介质流体过滤、约束加热、癌症治疗、形状优化和金融数学中应用的优化问题。该项目的计算模拟将为理解具有随机扰动的复杂物理模型提供见解。这个项目的另一个重点是训练研究生的数值方法及其分析，同时也训练学生的理论。学生将进一步接受有效使用电脑程式码的训练，为日后从事工业工作做好准备。该项目旨在设计、实施和严格分析一类新的不连续伽辽金（DG）方法，用于解决变分不等式和具有不等式约束的最优控制问题，这些方法是材料科学、机械工程、形状优化和金融科学中应用的非线性问题建模的基础。此外，潜在的问题可能涉及小参数和随机扰动，从而使完整的数值分析更加微妙。经典DG方法的公式通常需要很大的正惩罚参数，这取决于网格的形状规则和其他未知常数。该项目将为变分不等式、具有偏微分方程约束的最优控制问题以及相关的奇摄动和随机摄动问题设计新的DG方法。该项目的另一个目标是为相应的确定性和随机问题设计稳健、可靠和高效的后验误差估计器。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。

The project aims to develop new numerical methods for solving optimization problems that have applications in elasticity theory, fluid filtration in porous media, constrained heating, cancer therapy, shape optimization, and financial mathematics. The computational simulations from this project will provide insights on the understanding of complicated physical models with random perturbations. Another emphasis of this project will be the training of graduate students in numerical methods and their analysis while also training the students in theory. The students will further be trained in the efficient implementation of the computer codes so that they are better prepared for careers in industry.The project is on the design, implementation, and rigorous analysis of a new class of discontinuous Galerkin (DG) methods for variational inequalities and optimal control problems with inequality constraints that are fundamental for the modeling of nonlinear problems arising from applications in materials science, mechanical engineering, shape optimization, and financial science. Furthermore, the underlying problems may involve small parameters and random perturbations such that the complete numerical analyses are more subtle. The formulations of classical DG methods usually require large positive penalty parameters that depend on the shape regularity of the mesh and other unknown constants. The project will design novel DG methods for variational inequalities, optimal control problems with partial differential equations constraints, and related singularly perturbed and stochastically perturbed problems. Another goal of the project is to design robust, reliable, and efficient a posteriori error estimators for the corresponding deterministic and stochastic 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.