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，以估计和预测降雨、浮游生物水华或油井压力等。集成用于为预测提供不确定性量化：人们不仅想知道最有可能的估计，还想知道这个估计的可能性，以及是否有其他结果可能。由于历史原因，支持对不确定性量化严格使用集合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，粒子滤波器在高维问题上不起作用--这里的‘维’可以粗略地理解为观测数据可用的位置，而不是空间和时间的维--因为它们需要天文学上大量的系综成员。该项目将开发方法来提高粒子滤波器在空间范围问题中的性能，如天气预报。通过平滑观测值来降低有效维度，从而实现改进。例如，每天对大气和海洋进行数百万次卫星观测;该项目以类似于压缩图像的方式减少了这些数据的维度。由于粒子滤波器所需的集合规模对系统的有效维数是指数敏感的，因此即使是很小的数据压缩也会导致粒子滤波器性能的巨大改善。已知具有重采样的序列重要性采样粒子滤波器在无限集合规模的限制下收敛，到动态系统的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：所需的系综大小在系统的有效维度上是指数的。这对于像天气预报这样的空间扩展问题是禁止的，其中有效维度是巨大的。其他方法，如集合卡尔曼滤波器在实践中是非常成功的，但没有严格的分析相关的分布，集合成员代表和真正的贝叶斯后验。该项目旨在通过减少空间扩展问题的系统的有效维数来提高粒子滤波器的性能。表示观测数据和系统状态之间的关系的真实似然通过平滑观测值来改变。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。虽然粒子滤波收敛更快，但它收敛到的分布不是真正的贝叶斯后验分布。然而，真实和近似后验之间的误差的特性是已知的，并且可以被控制以平衡精度和成本，这与集合卡尔曼滤波器不同，在集合卡尔曼滤波器中，集合分布和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系或球面上的散乱空间数据的平滑算子。这些算子需要计算效率高，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开发，然后快速应用平滑积分算子，使用多分辨率高斯原子近似。该方法将被应用于气象数据，以建立直观的平滑程度如何影响后验。如有必要，该方法将与其他方法相结合，以提高粒子滤波器的性能，如隐式采样或优化transport.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.