Data assimilation in scientific computing

在这个项目中，研究人员和他们的同事们深化和扩展了他们对存在测量误差和随机扰动的非线性偏微分方程组的数据同化算法的渐近分析。这项研究包括发展无限维Banach空间中系综和粒子滤波器的随机收敛理论，相应的有限维渐近统计理论，以及对这些滤波器的时空协方差结构的谱逼近的行为的研究。特别重要的是小波和FFT谱近似在超高维非高斯情况下的计算效率和统计收敛。利用Sobolev空间上的概率度量、空间统计学中的随机场和随机谱展开之间的联系，有效地降低了系统的维度。我们的目标是开发和演示非常快速且节省内存的算法，这些算法可以证明在适当的随机意义上收敛到正确的答案。利用高性能计算设施，研究人员的野火蔓延模型被表示为由真实世界或蒙特卡罗数据驱动的天气-火灾耦合系统，作为评估各种数据同化算法和近似方法的计算效率和统计收敛特性的主要试验台。数据同化是将到达的实时信息纳入运行的复杂模拟中的艺术和科学，以使模拟以稳健和合理的方式调整和适应新数据。这个项目最感兴趣的主题领域是跟踪野地火灾，因为它们在包括森林、草原和人类社区在内的大片地形上蔓延。为此，研究人员维护了一个在高性能计算设施中运行的荒地火灾和天气模拟器，该模拟器嵌入了数据同化框架，以便模型可以在传入数据(高空照片、气象站数据等)到达时进行自我修正。数据同化的一般方法是许多科学领域感兴趣的，因此，关键是所采用的算法在计算上非常有效和在统计上稳健，并且可以证明它们在适当的极限内收敛到正确的答案。该项目致力于此类算法的开发和研究，并对收敛和效率进行严格的分析证明。参与该项目的博士生和博士后助理将获得高度跨学科的接触，学习计算和应用数学、统计学、气象学、大气物理学和火灾科学。

In this project, the investigators and their colleagues deepen and extend their asymptotic analysis of data assimilation algorithms for systems of nonlinear partial differential equations in the presence of measurement error and stochastic perturbation. This research includes the development of a stochastic convergence theory for ensemble and particle filters in infinite-dimensional Banach spaces, a corresponding statistical theory for finite-dimensional asymptotics, and a study of the behavior of spectral approximations for the spatio-temporal covariance structures of those filters. Of special importance is the computational efficiency and statistical convergence of wavelet and FFT spectral approximations in very-high-dimensional non-Gaussian cases. Connections between probability measures on Sobolev spaces, random fields in spatial statistics, and stochastic spectral expansions are exploited to effectively reduce the dimensionality of the system. The goal is to develop and demonstrate very fast and memory-efficient algorithms that provably converge in a suitable stochastic sense to the correct answer. Using high-performance computing facilities, the investigators' model of the spread of wildland fires, expressed as a coupled weather-fire system driven by real-world or Monte Carlo data, serves as the primary testbed for assessing the computational efficiency and statistical convergence properties of a wide variety of data assimilation algorithms and approximation methods.Data assimilation is the art and science of incorporating real-time information, as it arrives, into a running complex simulation, in such a way that the simulation adjusts and adapts in a robust and reasonable way to the new data. The subject area of greatest interest to this project is the tracking of wildland fires as they spread across extended terrains that can include forests, grasslands, and human communities. For this purpose the investigators maintain a wildland fire-and-weather simulator that runs in a high-performance computing facility, embedded in a data assimilation framework so that the model can correct itself in response to incoming data (overhead photographs, weather station data, etc) as it arrives. The general methodology of data assimilation is of interest to many areas of science, and so it is critical that the algorithms employed be computationally very efficient and statistically robust, and that they can be proven to converge in the appropriate limit to the right answer. This project is dedicated to the development and study of such algorithms, and to rigorous analytical proofs of convergence and efficiency. Doctoral students and post-doctoral associates involved in the project receive a highly interdisciplinary exposure to computational and applied mathematics, statistics, meteorology, atmospheric physics, and fire science.