Optimization, Differential Equations and Applications

研究人员对计算科学中的几个基本问题进行了研究，包括优化方法、数值最优控制、参数估计、反问题、带约束的发展偏微分方程组和并行自适应有限元方法。这些问题的一个共同特点是具有常微分方程和偏微分方程组约束的最优化问题的并行隐式解。许多基本物理系统由(通常是非线性的)约束的常微分方程组或偏微分方程组(通常也是非线性的)描述，因为运动方程的精确解总是满足给定的一组约束。例如麦克斯韦方程、不可压缩的纳维-斯托克斯方程、杨-米尔斯方程和爱因斯坦方程等等。微分方程约束的精确离散化导致了非常大的稀疏约束优化问题，其中大部分结构反映了离散化。这些问题是高度非线性的，优化的可靠性是制定成功方法的主要因素。在工程和科学计算中，受微分方程约束的函数的优化在许多情况下都会出现，因为物理现实通常通过涉及常微分方程和偏微分方程的模型来表示。研究人员的研究计划是由计算科学在工程和科学中的一些具有挑战性的应用程序推动和指导的。这些应用包括大型神经生物网络模型的设计，近海石油勘探，引力波的数值模拟，以及航天器和无人驾驶飞行器的轨迹规划。此外，研究人员还积极开发和传播计算机软件，这些软件体现了作为研究计划一部分开发的先进计算技术。调查人员开发的软件提供了一种有效的技术转移方法，并为美国科学家和工程师提供了即时访问最先进方法的途径。在计算数学中心内，研究人员提供一个教学和研究计划，强调计算科学在制定、建模和解决来自不同和不断变化的领域的问题方面的作用。研究人员开展的研究计划有助于吸引高级研究生进入计算科学领域，该领域在制造业、工程学和自然科学中产生的系统研究中发挥着至关重要的作用。

The investigators conduct research on several fundamental topics in computational science, which include optimization methods, numerical optimal control, parameter estimation, inverse problems, evolution partial differential equations with constraints, and parallel adaptive finite element methods. A feature common to all of these topics is the parallel implicit solution of optimization problems with ordinary and partial differential equations constraints.Many fundamental physical systems are described by (generally nonlinearly) constrained ordinary or partial differential equations of motion (also generally nonlinear) in the sense that exact solutions to the equations of motion always satisfy a given set of constraints. Examples include Maxwell's equations, the incompressible Navier-Stokes equations, the Yang-Mills equations, and Einstein's equations, among others. Accurate discretizations of differential equation constraints lead to very large sparse constrained optimization problems, where much of the structure reflects the discretization. The problems are highly nonlinear, with reliability of the optimization being the dominant factor in the formulation of successful methods.The optimization of functions subject to differential equation constraints arises in many contexts in engineering and scientific computation, since physical reality is often expressed through models involving ordinary and partial differential equations. The investigator's research program is motivated and guided by some challenging applications of computational science in engineering and science. These applications include the design of large neurobiological network models, off-shore petroleum exploration, the numerical modeling of gravitational waves, and trajectory planning for spacecraft and unmanned autonomous vehicles.In addition, the investigators are active in the development and dissemination of computer software that embodies the advanced computation techniques developed as part of their research program. Software developed by the investigators provides an effective method of technology transfer and provides US scientists and engineers with instant access to state-of-the-art methods. Within the Center for Computational Mathematics, the Investigators offer a program of instruction and research that emphasizes the role of computational science in the formulation, modeling, and solution of problems from diverse and changing areas. The program of research conducted by the investigators helps to attract advanced graduate students into the area of computational science, which plays a vital role in the study of systems arising in manufacturing, engineering and the natural sciences.