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

模型模拟的集合被用于各种领域，以估计和预测诸如降雨、浮游生物繁殖或油井压力之类的事情。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，而且想知道这个估计的可能性有多大，以及是否有其他可能的结果。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”。每个系综成员都是一个“粒子”。不幸的是，粒子过滤器不适用于高维问题——这里的“维度”可以粗略地理解为可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数量的集合成员。这个项目将开发方法来提高粒子过滤器在空间范围问题上的性能，比如天气预报。改进是通过平滑观测值来降低有效维数。例如，每天对大气和海洋进行数百万次卫星观测；该项目以一种类似于压缩图像的定性方式降低了这些数据的维度。由于粒子滤波器所需的集合尺寸对系统的有效维数呈指数级敏感，因此即使对数据进行很小的压缩也可以极大地改善粒子滤波器的性能。已知具有重采样的顺序重要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滤波问题的贝叶斯后验（在温和假设下）。不幸的是，收敛速度很慢：所需的集合大小在系统的有效维数中是指数级的。对于像天气预报这样的空间扩展问题来说，这是令人望而却步的，因为它的有效维度是巨大的。替代方法，如集合卡尔曼滤波器在实践中非常成功，但没有严格的分析有关的分布，集合成员代表和真正的贝叶斯后验。该项目旨在通过降低系统的有效维数来提高粒子滤波性能，以解决空间扩展问题。通过对观测数据进行平滑处理，可以改变表征观测数据与系统状态之间关系的真似然。这降低了系统的有效维数，相当于将观测误差建模为一个广义随机场。虽然粒子滤波收敛得更快，但它收敛到的分布并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，并且可以控制以平衡精度和成本，不像集成卡尔曼滤波器，其中集成分布与真实后验之间的差异是未知的和不受控制的。该项目的主要技术目标是开发可应用于笛卡尔坐标或球面上分散空间数据的平滑算子。这些运算符需要计算效率高，并允许平滑程度可调。快速方法将基于数据的径向基函数插值，然后快速应用平滑积分算子，使用多分辨率高斯原子逼近。该方法将应用于气象数据，以建立对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，以提高粒子滤波性能，如隐式采样或最优传输。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.