Collaborative Research: Determining Forms and Data Assimilation with Stochastic Data

使准确的天气预报成为一项挑战的众多因素之一是初始化。分布在地球和大气上的有限数量的数据收集仪器只能对任何时刻的天气状况作出不完整的描述。然而，在过去很长一段时间里，有大量这样的数据，当与某些数学模型结合起来时，可以提供更完整的描述。一般的过程称为数据同化。它为计算机模拟未来天气提供了更准确的起始值。该奖项资助研究一种新的数据同化方法，这种方法足够灵活，可以与各种模拟技术相结合。这项工作将涉及到该方法的实施，以及不可避免地发生的数据测量误差的影响。同化过程产生了另一种数学模型，可以用来清除过去数据中的这种噪声。在第二个方向上，目标不是为将来开始模拟获得更高的分辨率条件，而是在原始数据的时间段内重建高分辨率版本。这在语音和模式识别方面有自然的应用。研究小组将在几个重要的物理系统中实现基于反馈控制的最新数据同化算法。该方法可以应用于各种确定参数，如节点值和有限体积单元。该团队还将使用基于反馈控制的新确定形式进行计算。这个常微分方程的稳态恰好是全局吸引子中轨迹的有限维投影。该小组将从计算和分析两方面研究数据中随机扰动的影响。在数据同化算法的情况下，计划是量化和控制噪声对高分辨率状态的影响，以便在随后的直接数值模拟中使用。在决定形式的情况下，我们的想法是利用它的进化过程来消除数据本身的噪音。

Among the numerous factors that make accurate weather forecasting a challenge is initialization. The finite number of data collecting instruments distributed about the globe and atmosphere give only an incomplete description of the state of the weather at any instant. Yet there is a wealth of such data over an extended time period in the past, which when combined with certain mathematical models can provide a more complete description. The general procedure for this is called data assimilation. It provides more accurate starting values for computer simulations of the weather going into the future. This award funds research into a new method of data assimilation that is flexible enough to be combined with a variety of simulation techniques. The work will concern both the implementation of this method, and the effect of measurement errors in the data, which inevitably occur. The assimilation process leads to another mathematical model that can be used to cleanse the past data of such noise. In this second direction, the goal is not to obtain a higher resolution condition for starting a simulation for the future, but to reconstruct a highly resolved version over the time period of the original data. This has natural applications in voice and pattern recognition. The research team will implement a recent data assimilation algorithm based on feedback control for several important physical systems. The approach can be applied with a variety of determining parameters, such as nodal values, and finite volume elements. The team will also carry out computations with a new determining form based on feedback control. The steady states of this ordinary differential equation are precisely the finite-dimensional projections of trajectories in the global attractor. The team will study both computationally and analytically the effects of stochastic perturbations in the data. In the case of the data assimilation algorithm, the plan is to quantify and control the effect of the noise on the highly resolved state to be used in a subsequent direct numerical simulation. In the case of the determining form, the idea is to use its evolutionary process to remove the noise from the data itself.