Algorithms and Numerical Methods for Optimization with Partial Differential Equation Constraints

带约束的优化问题是科学和工程中普遍存在的问题。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，最大限度地从油井中回收石油，设计可以使用添加剂制造的新材料，以及用于解决逆问题的机器学习。这些问题中的许多具有内在的非线性和非光滑性，这使得算法的开发和分析极具挑战性。该项目旨在创造新的优化算法，以克服这些挑战，并得到广泛应用。精确的靶向应用包括，磁性药物靶向，量子自旋链，调和映射，结构设计，以及使用机器学习解决反问题。将创建开源软件，并与从业人员进行合作，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组(PDE)建模。这些偏微分方程组是几何(调和映射)、非局部(分数)、多物理(磁性药物输送)、具有未知区域的多尺度，即自由边界问题(FBPs)。本课题的目标是研究具有偏微分方程约束的优化问题，即偏微分方程约束优化问题。具体地说，它的目标是创建基于深度学习和增广拉格朗日框架的新的优化方法，以解决目前难以解决的几个优化问题，例如，在磁靶向药物传递中出现的受平流控制(也限制传输方程)的问题。这些问题都是非线性、非凸性、非光滑性的。新的优化算法将为研究非凸非光滑问题提供新的思路。特别是对于具有状态或梯度约束的优化问题，其中通常需要来自集值分析的概念。深度学习工作将有助于开创新的研究方向。癌细胞只吸收少量的药物，磁性药物靶向已显示出在不损害重要器官的情况下提高这种吸收速度。这项研究还将提高我们对磁性流体的理解，并将建立对控制守恒定律的数学理解。非局部问题在科学和工程中越来越重要。例如，它们导致了更好的量子自旋链、心脏电反应、添加剂制造(材料科学)、图像去噪和相分离的模型。将创建开放源码软件。这不仅将使优化、FBP和非局部问题的科学家受益，也将使非线性偏微分方程组和数据科学的科学家受益。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。将为学生设立阅读研讨会。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.