Improving Particle Filter Performance in Spatially-Extended Problems Using Generalized Random Field Likelihoods

模型模拟集合被用于各种领域来估计和预测降雨、浮游生物繁殖或油井压力等情况。集成用于为预测提供不确定性量化：人们不仅想知道最可能的估计，还想知道该估计的可能性有多大，以及是否可能有任何其他结果。由于历史原因，支持严格使用系综进行不确定性量化的主要数学框架被称为“粒子滤波器”。每个集合成员都是一个“粒子”。不幸的是，粒子滤波器不能很好地解决高维度的问题——这里的“维度”可以被宽松地理解为可以获得观测数据的位置，而不是空间和时间的维度——因为它们需要天文数字般数量的集合成员。该项目将开发方法来提高粒子滤波器在空间范围问题（例如天气预报）中的性能。改进是通过平滑观察来降低有效维度来实现的。例如，每天对大气和海洋进行数百万次卫星观测；该项目以与压缩图像类似的方式降低了数据的维度。由于粒子滤波器所需的集合大小对系统的有效维度呈指数敏感，因此即使数据的小压缩也可以导致粒子滤波器性能的巨大改进。众所周知，具有重采样的顺序重要性采样粒子滤波器在无限集合大小的限制下收敛于动力系统过滤问题的贝叶斯后验（在温和的假设下）。不幸的是，收敛速度很慢：所需的集合大小是系统有效维度的指数。这对于有效维度巨大的天气预报等空间扩展问题来说是令人望而却步的。像集成卡尔曼滤波器这样的替代方法在实践中非常成功，但是没有对集成成员代表的分布和真实的贝叶斯后验进行严格的分析。该项目旨在通过降低空间扩展问题系统的有效维数来提高粒子滤波器的性能。通过平滑观测来改变表示观测数据和系统状态之间关系的真实似然。这降低了系统的有效维数，相当于将观测误差建模为广义随机场。尽管粒子滤波器收敛得更快，但它收敛到的分布不是真正的贝叶斯后验。然而，真实后验和近似后验之间的误差特征是已知的，并且可以进行控制以平衡精度和成本，这与集成卡尔曼滤波器不同，其中集成分布和真实后验之间的差异是未知且不受控制的。该项目的主要技术目标是开发可应用于笛卡尔坐标或球体上的分散空间数据的平滑算子。这些运算符需要具有计算效率，并且允许平滑程度可调。将基于数据的径向基函数插值开发快速方法，然后快速应用平滑积分算子，并使用多分辨率高斯原子进行近似。该方法将应用于气象数据，以建立关于平滑程度如何影响后验的直觉。如有必要，该方法将与其他方法相结合，以提高粒子过滤器性能，例如隐式采样或最佳传输。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Ensembles of model simulations are used in a variety of fields to estimate and predict things like rain, plankton blooms, or oil well pressure. Ensembles are used to provide uncertainty quantification to the predictions: one wants to know not just the most likely estimate but also how likely this estimate is, and whether any other outcomes are likely. The main mathematical framework that underpins the rigorous use of ensembles for uncertainty quantitfication is called, for historical reasons, the 'particle filter.' Each ensemble member is a 'particle.' Unfortunately particle filters do not work well for problems in high dimensions -- a `dimension' can be loosely understood here as a location where observational data is available, not the dimensions of space and time -- since they require an astronomically large number of ensemble members. This project will develop methods to improve the performance of particle filters in problems with spatial extent, like weather forecasting. The improvement comes by reducing the effective dimensionality by smoothing the observations. For example, millions of satellite observations of the atmosphere and oceans are taken each day; the project reduces the dimensionality of this data in a manner qualitatively similar to compressing an image. Since the required ensemble size for a particle filter is exponentially sensitive to the effective dimension of the system, even a small compression of the data can lead to enormous improvements in the performance of the particle filter.The sequential importance sampling particle filter with resampling is known to converge, in the limit of infinite ensemble size, to the Bayesian posterior of the filtering problem for dynamical systems (under mild assumptions). Unfortunately the rate of convergence is slow: the required ensemble size is exponential in the effective dimension of the system. This is prohibitive for spatially-extended problems like weather forecasting, where the effective dimension is enormous. Alternative methods like the ensemble Kalman filters are very successful in practice, but there is no rigorous analysis relating the distribution that the ensemble members represent and the true Bayesian posterior. This project aims to improve particle filter performance by reducing the effective dimensionality of the system for spatially extended problems. The true likelihood representing the relationship between the observational data and the system state is altered by smoothing the observations. This reduces the effective dimensionality of the system and is equivalent to modeling the observation error as a generalized random field. Although the particle filter converges more rapidly, it converges to a distribution that is not the true Bayesian posterior. However, the character of the error between the true and approximate posteriors is known and can be controlled to balance accuracy and cost, unlike the ensemble Kalman filters where the difference between the ensemble distribution and the true posterior is unknown and uncontrolled. The main technical goal of the project is to develop smoothing operators that can be applied to scattered spatial data in Cartesian coordinates or on the sphere. These operators need to be computationally efficient, and to allow the degree of smoothing to be tunable. Fast methods will be developed based on radial basis function interpolation of the data, followed by the fast application of a smoothing integral operator, approximated using multi-resolution Gaussian atoms. The method will be applied to meteorological data to build intuition on how the degree of smoothing impacts the posterior. If necessary, the method will be combined with other methods for improving particle filter performance, like implicit sampling or optimal transport.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.