Next generation particle filters for stochastic partial differential equations

许多对我们生活至关重要的方面(例如，可再生能源的可用性、人类心脏的电模式，以及全球流行病的演变)都无法直接测量。将模型与测量数据混合在一起，可以让我们对系统的状态做出合理的推断。当使用随机模型的部分观测来对一个演化系统做出严格的推断时，我们称之为随机过滤。状态模型的选择对任何给定的随机滤波方法的适用性都起着至关重要的作用。拟议的研究将把随机偏微分方程作为状态模型，因为它们是应用于人类工作的不同领域的一些最通用的模型：物理、生物、化学、天气预报、金融、可再生能源和制造业等。对由随机偏微分方程建模的现象的实际实现仍然是一个突出的挑战。一个关键问题是如何以一种在计算上可行的方式接近系统状态的“真实”描述。粒子滤波(PFS)是解决滤波问题的最成功的方法之一，它提供了一种理论上合理的方法来推断隐藏系统的状态。PFS涉及多组“粒子”：状态模型的不同实现。每隔一定的时间间隔，使用局部观测和噪声观测来修正粒子云。在数据同化语言(DA)中，作为模式实现的演化粒子是预测步骤，而使用数据进行校正是分析步骤。PFS在工程应用中已被证明是非常成功的(例如)，只要状态模型具有小到中等大小。它们通过按顺序处理数据而取得成功：在分析步骤中，只使用新的观测数据，而不需要重新查看过去的观测数据。通常不可能处理到当前时间为止可用的整个数据集。许多具有挑战性的现实世界问题都有很大的模型状态，在每个分析步骤中都有大量的观察数据。在数值天气预报中，每个分析时间都会出现数千万次数据测量。这使得每个单独的分析步骤(几乎)都像无序处理数据一样困难。最近，PI、Co-I和他的同事们研究了可以缓解这个问题的方法，但肯定不能克服它。我们需要一个新的范式来开发有效的PFS来解决这些具有挑战性的问题。目前的范式将DA机制分为连续的预测和分析步骤，以一种方便和有教育意义的方式描述了PF。特别是，分析步骤之间粒子的演化忽略了即将到来的数据。在我们的新范式中，我们不是使用随机偏微分方程式的数值近似来演化粒子，而是提倡“观察通知”的轨迹，在这种轨迹中，粒子在适当的选择方向上被“轻推”。目前在数据同化中，将粒子移动到观测中的程序是特别的方法，在理论上是不合理的。相反，我们将开发对过滤问题进行可证明的一致近似的方法。我们的轻推将扰乱粒子的轨迹，以最大限度地提高给定观测数据的位置的可能性。这引入了一种通过明智地选择粒子来消除的偏差。新方法将针对流体动力学、反应扩散方程、Allen-Cahn等随机偏微分方程式进行优化。我们的项目将为我们的范例提供完整的管道，从理论分析到可在大型并行计算机上运行的高性能软件实施，从而能够在具有挑战性的基准上进行性能评估。

Many aspects critical to our lives (e.g. availability of renewable energy resources, electrical patterns in the human heart, and the evolution of global pandemics) are not available for direct measurement. Blending models with measured data lets us make reasonable inferences about the state of a system. When using partial observations with a stochastic model to make rigorous inferences about an evolving system, we call this process stochastic filtering. The choice of the state model plays a crucial role in the applicability of any given stochastic filtering methodology. The proposed research will focus stochastic partial differential equations as state models, as they are some of the most versatile models applied in diverse areas of human endeavour: physics, biology, chemistry, weather prediction, finance, renewable energy and manufacturing, etc. Practical implementation of stochastic filtering for phenomena modelled by stochastic partial differential equations remains an outstanding challenge. One key question is how to approximate to the "true" description of the state of the system in a computationally feasible way. Particle filters (PFs) are some of the most successful methods for solving the filtering problem, offering a theoretically justified approach to inferences about the state of hidden systems. PFs involve sets of "particles": different realizations of the state model. At regular intervals, the cloud of particles is corrected using partial and noisy observations. In the language of Data Assimilation (DA), evolving particles as realizations of the model is the forecast step, whilst correction using data is the analysis step. PFs have proved immensely successful in engineering applications (for example) provided the state model has small to moderate size. They succeed by processing data sequentially: at the analysis step only the new observations are used, without the need to revisit past observations. It is frequently impossible to process the entire set of data available up to the current time. Many challenging real world problems have large model states, with large quantities of observations becoming available at each analysis step. In numerical weather prediction, tens of millions of data measurements occur at each analysis time. This makes each individual analysis step (almost) as hard as processing data nonsequentially. Recently, the PI, Co-I and colleagues have researched methodologies that can alleviate this problem, but certainly not overcome it.We need a new paradigm for developing efficient PFs for these challenging problems. The current paradigm separates the DA mechanism into sequential forecast and analysis steps, describing the PF in a convenient and pedagogical manner. In particular, the evolution of the particles between analysis steps ignores the forthcoming data. In our new paradigm, rather than evolving the particles using a numerical approximation of the stochastic partial differential equation, we advocate "observation informed" trajectories, where the particles are "nudged" in suitably chosen directions. Currently in data assimilation, procedures that move the particles towards observations are ad-hoc methodologies are not theoretically justified. In contrast, we will develop methodologies that are provably consistent approximations of the filtering problem. Our nudges will perturb the trajectories of the particles to maximise the likelihood of their positions given the observation data. This introduces a bias that is eliminated through the judicious selection of particles. The new methodology will be optimized for stochastic partial differential equations, from fluid dynamics, reaction-diffusion equations, Allen-Cahn etc. Our project will deliver the complete pipeline for our paradigm, from theoretical analysis to high performance software implementations that can run on large parallel computers, enabling performance evaluations on challenging benchmarks.