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，用于解决弹性理论，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中的应用。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将是培养研究生的数值方法及其分析，同时也培养学生的理论。学生将进一步接受有效实施计算机代码的培训，以便他们为工业职业做好更好的准备。该项目是关于一类新的不连续Galerkin（DG）方法的设计，实施和严格分析，用于变分不等式和不等式约束的最优控制问题，这些问题是材料科学应用中产生的非线性问题建模的基础，机械工程、形状优化和金融科学。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大的正惩罚参数，这取决于网格的形状规则性和其他未知常数。该项目将为变分不等式，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性问题设计鲁棒、可靠和有效的后验误差估计器。该奖项反映了NSF的法定使命，并被认为值得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来支持。