Data assimilation in scientific computing

在这个项目中，研究人员和他们的同事加深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系统的数据同化算法的渐近分析。本研究包括发展了无限维Banach空间中系综和粒子滤波器的随机收敛理论，有限维渐近的相应统计理论，以及这些滤波器的时空协方差结构的谱近似行为的研究。特别重要的是计算效率和统计收敛的小波和FFT谱近似在非常高维的非高斯情况。利用Sobolev空间上的概率测度、空间统计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数。目标是开发和演示非常快速和内存高效的算法，这些算法可以证明在合适的随机意义上收敛到正确答案。使用高性能计算设施，研究者的野火蔓延模型，表示为由现实世界或蒙特卡罗数据驱动的耦合天气-火灾系统，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测试平台。数据同化是一门艺术和科学，它将实时信息整合到运行中的复杂模拟中，使模拟能够以稳健和合理的方式调整和适应新数据。该项目最感兴趣的主题领域是跟踪野火在包括森林、草原和人类社区在内的扩展地形上的蔓延。为此，研究人员维护了一个在高性能计算设备中运行的野外火灾和天气模拟器，该模拟器嵌入在数据同化框架中，以便模型能够在传入数据（架空照片、气象站数据等）到达时对其进行自我校正。数据同化的一般方法是许多科学领域感兴趣的，因此所采用的算法在计算上非常有效和统计上健壮是至关重要的，并且它们可以被证明在适当的限制下收敛到正确的答案。本项目致力于此类算法的开发和研究，并对收敛性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计算和应用数学、统计学、气象学、大气物理学和火灾科学等高度跨学科的机会。

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.