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

基本信息

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

项目摘要

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.

带有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油率、设计可使用增材制造制造的新材料以及用于解决逆问题的机器学习。其中许多问题本质上是非线性和非平滑的，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来克服这些挑战并广泛适用。精确的目标应用包括磁性药物靶向、量子自旋链、调和图、结构设计以及使用机器学习解决逆问题。将创建开源软件，并与从业者进行合作，以最大限度地发挥工作的影响。上述应用程序可以使用偏微分方程（PDE）进行建模。这些偏微分方程是几何（调和图）、非局部（分数）、多物理场（磁性药物输送）、具有未知域的多尺度问题，即自由边界问题（FBP）。该项目的目标是研究具有 PDE 约束的优化问题，即 PDE 约束优化。具体来说，它的目标是创建基于深度学习和增强拉格朗日框架的新优化方法，以解决目前几个棘手的优化问题，例如磁性靶向药物输送中出现的平流主导（也限制传输方程）所限制的问题。所有这些问题本质上都是非线性、非凸和非光滑的。新颖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。癌细胞仅吸收少量药物，磁性药物靶向已被证明可以提高吸收率而不损害重要器官。这项研究还将增进我们对磁性流体的理解，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变得越来越重要。例如，它们为量子自旋链、心电反应、增材制造（材料科学）、图像去噪和相分离提供了更好的模型。将创建开源软件。这不仅有利于优化、FBP 和非局部问题领域的科学家，也有利于非线性偏微分方程和数据科学领域的科学家。结果将通过专题课程、研究出版物和演讲来传播。两名博士生将获得博士学位。将为学生举办阅读研讨会。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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

期刊论文数量（20）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

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.
影响因子：
0
作者：
Harbir Antil;P. Dondl;Ludwig Striet
通讯作者：
Harbir Antil;P. Dondl;Ludwig Striet

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
通讯作者：
Tianyi Shi;Harbir Antil;D. Kouri

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
影响因子：
0
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