Algorithms and Numerical Methods for Optimization with Partial Differential Equation Constraints

偏微分方程约束优化的算法和数值方法

• 批准号: 2110263
• 负责人: Harbir Antil
• 金额: $ 34万
• 依托单位: George Mason University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2110263&HistoricalAwards=false
• 关键词: Algorithms; Numerical; Methods; Optimization; Partial

Optimization problems with constraints are ubiquitous in science and engineering. Some examples include designing a drug delivery mechanism to maximize the impact on cancerous cells, maximizing oil recovery from the wells, designing new materials that can be manufactured using additive manufacturing, and machine learning for solving inverse problems. Many of these problems are inherently non-linear and non-smooth, which makes the development of algorithms and their analysis extremely challenging. The project aims to create new optimization algorithms that will overcome these challenges and that are widely applicable. The precise target applications include, magnetic drug targeting, quantum spin chains, harmonic maps, structure design, and solving inverse problems using machine learning. Open-source software will be created and collaborations with practitioners will be carried out to maximize the impact of the work.The applications described above can be modeled using partial differential equations (PDEs). These PDEs are geometric (harmonic maps), nonlocal (fractional), multiphysics (magnetic drug delivery), multiscale with an unknown domain, i.e., free boundary problems (FBPs). The goal of this project is to study optimization problems with PDE constraints, i.e., PDE constrained optimization. Specifically, it aims to create new optimization methods based on Deep Learning and Augmented Lagrangian frameworks to solve several currently intractable optimization problems, for instance, problems constrained by advection dominated (also limiting transport equations) arising in magnetic targeted drug delivery. All these problems are nonlinear, nonconvex, and non-smooth in nature. Novel optimization algorithms will provide new insights into nonconvex non-smooth problems. In particular for optimization problems with state or gradient constraints, where concepts from set-valued analysis are typically needed. The deep learning work will help create new research directions. Cancerous cells absorb only a small amount of medicine, magnetic drug targeting has shown to increase this absorption rate without harming vital organs. This research will also improve our understanding of magnetic fluids and will create mathematical understanding of control of conservation laws. Nonlocal problems are increasingly important in science and engineering. They lead, for instance, to better models for quantum spin chains, cardiac electrical response, additive manufacturing (materials science), image denoising and phase separation. Open-source software will be created. This will not only benefit scientists in optimization, FBPs, and nonlocal problems but also scientists in nonlinear PDEs and data science. The results will be disseminated via a special topics course, research publications, and talks. Two PhD students will get PhDs. Reading seminars for students will be created.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.

带约束的优化问题在科学和工程中普遍存在。一些例子包括设计一种药物输送机制，以最大限度地提高对癌细胞的影响，最大限度地提高油井的采收率，设计可以使用增材制造制造的新材料，以及解决逆问题的机器学习。这些问题中的许多本质上是非线性和非光滑的，这使得算法的开发及其分析极具挑战性。该项目旨在创建新的优化算法，以克服这些挑战，并广泛适用。精确靶标应用包括：磁性药物靶向、量子自旋链、谐波图、结构设计以及使用机器学习解决逆问题。将创建开源软件，并与从业人员合作，以最大限度地发挥工作的影响。上面描述的应用程序可以使用偏微分方程（pde）建模。这些偏微分方程是几何（谐波映射）、非局部（分数）、多物理场（磁性药物输送）、具有未知域的多尺度，即自由边界问题（fbp）。本课题的目标是研究PDE约束下的优化问题，即PDE约束优化问题。具体而言，它旨在创建基于深度学习和增强拉格朗日框架的新优化方法，以解决当前一些棘手的优化问题，例如，在磁性靶向药物递送中出现的平流主导（也限制传输方程）约束的问题。所有这些问题本质上都是非线性、非凸和非光滑的。新的优化算法将为非凸非光滑问题提供新的见解。特别是对于具有状态或梯度约束的优化问题，通常需要集值分析的概念。深度学习工作将有助于创造新的研究方向。癌细胞只吸收少量药物，磁性药物靶向已被证明可以在不伤害重要器官的情况下增加这种吸收率。这项研究还将提高我们对磁流体的理解，并将建立对守恒定律控制的数学理解。非局部问题在科学和工程中越来越重要。例如，它们为量子自旋链、心脏电反应、增材制造（材料科学）、图像去噪和相位分离提供了更好的模型。开源软件将被创造出来。这不仅有利于优化、fbp和非局部问题的科学家，也有利于非线性偏微分方程和数据科学的科学家。研究结果将通过专题课程、研究出版物和讲座进行传播。两名博士生将获得博士学位。将为学生创建阅读研讨会。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。

项目成果

Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

• DOI: 10.1515/cmam-2021-0118
• 发表时间: 2022-02
• 期刊: Computational Methods in Applied Mathematics
• 影响因子: 1.3
• 作者: Tianyi Shi;Harbir Antil;D. Kouri

Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis

• DOI: 10.1137/20m1374122
• 发表时间: 2020-10
• 期刊: SIAM J. Sci. Comput.
• 作者: Harbir Antil;P. Dondl;Ludwig Striet

Parallel Deep ResNets for Chemically Reacting Flows

用于化学反应流的并行深度 ResNet

• DOI: 10.2514/6.2022-1076
• 发表时间: 2022
• 期刊: AIAA SciTech Forum
• 作者: Brown, Thomas S.;Antil, Harbir;Lohner, Rainald;Verma, Deepanshu;Togashi, Fumiya
• 通讯作者: Togashi, Fumiya

Sparse optimization problems in fractional order Sobolev spaces

分数阶 Sobolev 空间中的稀疏优化问题

• DOI: 10.1088/1361-6420/acbe5e
• 发表时间: 2023
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: Antil, Harbir;Wachsmuth, Daniel
• 通讯作者: Wachsmuth, Daniel

A deterministic pathogen transmission model based on high-fidelity physics

基于高保真物理的确定性病原体传播模型

• DOI: 10.1016/j.cma.2022.114929
• 发表时间: 2022
• 期刊: Computer Methods in Applied Mechanics and Engineering
• 影响因子: 7.2
• 作者: Löhner, Rainald;Antil, Harbir;Gimenez, Juan Marcelo;Idelsohn, Sergio;Oñate, Eugenio
• 通讯作者: Oñate, Eugenio