Methods and Applications for Optimization with Differential Equations

随着优化方法在工程、基础科学、金融和数据科学中的重要性得到越来越广泛的认识，对先进优化软件工具的需求急剧增加。计算能力的急剧增加，以及建模语言和自动微分等支持技术的改进，推动了对优化算法的需求，这些算法可以解决越来越多的计算挑战问题，包括具有微分方程约束和/或离散变量的非线性优化问题。本项目主要研究具有微分方程约束的优化问题的并行隐式求解所涉及的几个基本计算问题。这样的问题出现在工程和科学计算的许多环境中，因为物理现实通常通过涉及常微分方程和偏微分方程式的模型来表达。微分方程约束的精确离散化导致了非常大的结构化约束优化问题，其中大部分结构反映了离散化。一个具体的目标是开发现代算法，这些算法非常适合在高级计算平台(如具有多核和GPU架构的平台)上实施，可与相关领域的高性能软件(如线性代数)互操作，并可针对特定的重要应用程序轻松定制。该项目的一个主要部分涉及软件的开发及其在制造业、工程界和科学界的传播。作为该项目的一部分开发的软件将提供一种有效的技术转让方法，并将扩大调查人员开发的现有代码PLTMG、MC和SNOPT的范围和效力。微分方程式很方便地描述了科学和工程中许多复杂系统的物理规律。它们也是用于模拟和预测这些系统行为的数学模型的核心。需要优化这类系统的性能是实际应用的共同特点，这些应用涉及大型神经生物网络模型的设计、引力波的数值建模以及航天器和无人驾驶飞行器(UAV)的轨迹规划。在该项目的支持下开发的软件将为工程师和科学家提供即时访问最先进的方法，以建模和优化涉及微分方程约束的复杂系统。由此带来的这些模型在效率、准确性和稳健性方面的改进，将在对美国全球竞争力至关重要的制造和工程领域产生重大影响。

The demand for advanced optimization software tools is increasing sharply as the importance of optimization methodology in engineering, basic science, finance and data science is becoming more widely recognized. The dramatic increase in computing power, and the improvements in supporting technologies such as modeling languages and automatic differentiation are fueling demand for optimization algorithms that solve more and more computationally challenging problems, including nonlinear optimization problems with differential equation constraints and/or discrete variables. This project focuses on several fundamental computational issues involved in the parallel implicit solution of optimization problems with differential equation constraints. Such problems arise in many contexts in engineering and scientific computation, since physical reality is often expressed through models involving ordinary and partial differential equations. Accurate discretizations of differential equation constraints lead to very large structured constrained optimization problems, where much of the structure reflects the discretization. A specific goal is the development of modern algorithms that are well-suited to implementation on advanced computing platforms (such as those with multicore and GPU architectures), interoperable with high-performance software in related areas (such as linear algebra), and readily customizable for particular important applications. A major part of the project involves the development of software and its dissemination within the manufacturing, engineering and scientific community. Software developed as part of the project will provide an effective method of technology transfer and will extend the scope and effectiveness of the existing codes PLTMG, MC and SNOPT developed by the investigators. Differential equations conveniently characterize the physical laws of many complex systems occurring in science and engineering. They also lie at the heart of the mathematical models used to simulate and predict the behavior of these systems. The need to optimize the performance of such systems is the common feature of practical applications that range over such diverse areas as the design of large neurobiological network models, the numerical modeling of gravitational waves, and trajectory planning for spacecraft and unmanned autonomous vehicles (UAVs). Software developed under the auspices of this project will provide engineers and scientists with instant access to state-of-the-art methods for the modeling and optimization of complex systems involving differential equation constraints. The resulting improvements in the efficiency, accuracy and robustness of these models will have a substantial impact in areas of manufacturing and engineering that are vital to US global competitiveness.