Data Assimilation, Noise Models, and Dimensional Reduction, with Applications

在科学中有许多问题，人们必须根据不完美的模型加上噪声和/或不完整的数据来得出结论和做出预测；例如天气预报、气候预报、经济学和材料科学。该项目的目标是使执行这些任务更加高效和准确，并将方法扩展到新的情况。在前面的工作中，我们展示了如何优化寻找最佳结论的任务。在新的建议下，我们将寻找新的、更有效的方法来估计误差幅度，并使用误差估计来改进结论。同样的技术可以用来在数据的基础上改进模型本身；我们将尝试将其扩展到数据是定性的问题，例如，设计最终将允许工程师将关于发动机平稳运行的观察转化为可用于设计的精确定量信息的方法。最后，我们将把这些方法扩展到数学模型不可靠的问题，不是因为知识不完整或实验错误，而是因为可用于求解它们的计算能力不足；在这种情况下，我们的方法将使使用统计方法来提高计算精度成为可能。(1)进一步改进数据同化工具的主要障碍之一是难以得出现实的噪声模型，特别是因为噪声取决于信号，因此很难与之分离。人们建议通过使用统计物理学的工具来实现这种分离，特别是最近对Mori-zwanzig形式主义的推广。(2)噪声模式中噪声的估计问题和复杂动力学的简化模式的推导问题非常相似；这一点可用于发展定性数据的数据同化方法。这需要在模型中的数据集和参数集之间创建映射，这通过使用简化的动力学变得可行，该映射将由本提议中的方法导出。(Iii)在欠分辨数值动力学中，人们可以将解的缺失分量解释为附加噪声。这表明，这些缺失的分量可以从数值或实验数据中估计出来，以提高数值解的精度。有人建议开发这样做的方法。(4)建议对科学计算的各种领域进行简化说明。这项工作的应用将包括来自地球物理、随机控制、机器人、燃烧和流体力学的问题。

There are many problems in science where one must draw conclusions and make predictions on the basis of imperfect models supplemented by noisy and/or incomplete data; examples are weather forecasting, climate prediction, economics, and materials science. The goal of the project is to make carrying out these task more efficient and accurate, and extend the methodology to new situations. In earlier work we showed how to optimize the task of finding the best conclusions. Under the new proposal we will look for new, more efficient ways to estimate margins of error, and also use the estimates of the error to improve the conclusions. The same technology can be used to improve the models themselves on the basis of data; we will try to expand this to problems where the data are qualitative, for example, devise methods that will eventually allow an engineer to translate observations about when an engine runs smoothly into precise quantitative information that can be used for design. Finally, we will extend these methods to problems where the mathematical models are unreliable, not because of incomplete knowledge or of experimental error, but because the computing power available to solve them is insufficient; in such cases, our methods would make it possible to use statistical methods to enhance the accuracy of the computations.The proposal has several thrusts. (i) One of the main obstacles to further improvement of data assimilation tools is the difficulty in deriving realistic noise models, in particular because the noise depends on the signal and is therefore hard to separate from it. It is proposed to perform this separation by using tools from statistical physics, in particular recent generalizations of the Mori-Zwanzig formalism. (ii) The problem of estimating the noise in noisy models and the problem of deriving reduced models of complex dynamics are very similar; this remark can be used to develop data assimilation methods for qualitative data. This requires creating maps between data sets and parameter sets in the models, which becomes feasible through the use of reduced dynamics, to be derived by the methods in the present proposal. (iii) One can interpret the missing components of the solution in underresolved numerical dynamics as an added noise. This remark suggests that these missing components can be estimated from numerical or experimental data, so as to enhance the accuracy of the numerical solutions. It is proposed to develop methods for doing so. (iv) It is proposed to develop reduced descriptions in a variety of areas of scientific computation. The applications of this work will include problems from geophysics, stochastic control, robotics, combustion, and fluid mechanics.