Methods and Applications for Optimization with Differential Equations

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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.