Optimization, Differential Equations and Applications

研究人员研究了计算科学中的几个基本主题，包括优化方法，数值最优控制，参数估计，逆问题，带约束的演化偏微分方程以及并行自适应有限元方法。所有这些主题的共同特征是具有常微分方程和偏微分方程约束的优化问题的并行隐式解。许多基本的物理系统是用（通常是非线性的）有约束的常微分或偏微分运动方程（通常也是非线性的）来描述的，因为运动方程的精确解总是满足一组给定的约束。例子包括麦克斯韦方程、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。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，其中许多结构反映了离散化。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。受微分方程约束的函数优化在工程和科学计算的许多情况下都会出现，因为物理现实通常是通过包含常微分方程和偏微分方程的模型来表达的。研究者的研究计划受到工程和科学中计算科学的一些具有挑战性的应用的激励和指导。这些应用包括大型神经生物学网络模型的设计、海上石油勘探、引力波的数值模拟、航天器和无人驾驶汽车的轨迹规划。此外，研究人员还积极参与计算机软件的开发和传播，这些软件体现了作为他们研究计划的一部分而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让方法，使美国科学家和工程师能够立即获得最先进的方法。在计算数学中心，研究人员提供了一个教学和研究项目，强调计算科学在不同和不断变化的领域的问题的表述、建模和解决中的作用。研究人员进行的研究计划有助于吸引高级研究生进入计算科学领域，这在研究制造，工程和自然科学中的系统中起着至关重要的作用。

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.