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

具有不等式约束的确定性和随机优化问题的新型间断伽辽金方法

• 批准号: 2111004
• 负责人: Yi Zhang
• 金额: $ 11.49万
• 依托单位: University of North Carolina Greensboro
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111004&HistoricalAwards=false
• 关键词: Novel; Discontinuous; Galerkin; Methods; Deterministic

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.

该项目旨在开发新的数值方法来解决优化问题，这些方法可应用于弹性理论、多孔介质中的流体过滤、约束加热、癌症治疗、形状优化和金融数学。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。该项目的另一个重点是对研究生进行数值方法及其分析的培训，同时对学生进行理论培训。学生将进一步接受计算机代码的有效实施方面的培训，以便他们为工业职业生涯做好更好的准备。该项目致力于设计、实施和严格分析一类新型不连续伽辽金 (DG) 方法，用于变分法不等式和带有不等式约束的最优控制问题是材料科学、机械工程、形状优化和金融科学应用中产生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，使得完整的数值分析更加微妙。经典 DG 方法的公式通常需要较大的正罚参数，这些参数取决于网格的形状规律性和其他未知常数。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异扰动和随机扰动问题设计新颖的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 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查进行评估，被认为值得支持标准。