Collaborative Research: Efficient High Order Methods for Deterministic and Stochastic Problems in Flow Analysis and Control

为了分析和控制可压缩流体非定常流动中的大规模确定性和随机性问题，将开发高效的高阶谱元算法。重点将集中在空间、时间、多项式混沌和基于伴随的最优控制算法，这些算法可以用于复杂的依赖于时间的流动的主动控制，特别是那些包括空气动力噪声的产生和传播的流动。对于时间和空间近似，我们将发展间断Galerkin空间近似和高阶优化隐式/显式(IMEX)方法，以最小化相位和耗散误差。将使用基于伴随的流量控制方法，并开发适当使用不连续Galerkin近似的控制策略。由于实际流中存在不确定性，因此我们将包括有效的随机配置近似。最后，我们将允许控制是确定性的或随机的，并开发出在存在不确定性的情况下控制流量的策略。这些方法处理的这类问题具有非常重要的实际意义，需要使用高效和高阶方法进行非常大的计算密集型模拟。控制噪声产生的能力对于从风车农场到喷气发动机的各种应用都很重要。在具有概率参数、不确定初始条件、不确定输入情况和基于不完全知识的模型的科学、工程和技术问题中，不确定性条件下的最优性问题经常出现。工程设计、供应分配、生产计划与调度、交通运输、库存网络、金融、能源系统、环境保护、模式识别、军事物流等大量问题要求在存在不确定性的情况下进行决策。不确定性支配着燃料的价格、电力的供应以及对化学品的需求。生活中的许多事情都要求我们在不确定的情况下做出最优选择，即在不确定的情况下从一些可选的行动方案中选择最优的。显然，在有和没有不确定性的情况下，最优控制的数学和计算方法的发展在数学内部和外部都产生了广泛的影响。

Efficient, high order, spectral element algorithms will be developed to analyze and control large-scale deterministic and stochastic problems in unsteady compressible fluid flow. The focus will be on spatial, temporal, polynomial chaos, and adjoint based optimal control algorithms that can be used for active control of complex time-dependent flows, particularly those that include the generation and propagation of aerodynamic noise. For the time and space approximations, we will develop discontinuous Galerkin spatial approximations and high order, optimized implicit/explicit (IMEX) methods that minimize phase and dissipation errors. Adjoint based methods for flow control will be used, and control strategies developed for appropriate use with the discontinuous Galerkin approximation. Since uncertainty is present in real flows, we will include efficient stochastic collocation approximations. Finally, we will allow controls to be deterministic or stochastic, and develop strategies to control flows in the presence of uncertainty.The class of problems addressed by these methods are of great practical importance and require very large, computationally intensive simulations with efficient and high order methods. The ability to control acoustic noise generation is important to a wide variety applications, from windmill farms to jet engines. Problems of optimality under uncertainty occur frequently in a wide variety of problems in science, engineering and technology that have probabilistic parameters, nondeterministic initial conditions, uncertain input situations, and models based on incomplete knowledge. A large number of problems such as engineering design, supply allocation, production planning and scheduling, transportation, inventory networks, finance, energy systems, environmental protection, pattern recognition, and military logistics require that decisions be made in the presence of uncertainty. Uncertainty governs the prices of fuels, the availability of electricity, and the demand for chemicals. Much of life requires us to make optimal choices under uncertainty, i.e., to choose the optimum from some set of optional courses of action in uncertain situations. Clearly, the development of the mathematics and computational methods for optimal control with and without uncertainty has broad impacts both inside and outside mathematics.