Next generation particle filters for stochastic partial differential equations

用于随机偏微分方程的下一代粒子滤波器

• 批准号: EP/W016125/1
• 负责人: Colin Cotter
• 金额: $ 38.83万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW016125%2F1
• 关键词: Next; generation; particle; filters; stochastic

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.

许多方面对我们的生活至关重要（例如，可再生能源资源的可用性，人心的电气模式以及全球大流行的演变）无法直接测量。将模型与测量数据混合，使我们可以对系统状态做出合理的推论。当使用随机模型的部分观测来对不断发展的系统进行严格的推断时，我们称此过程随机过滤。国家模型的选择在任何给定随机过滤方法的适用性中都起着至关重要的作用。拟议的研究将将随机部分微分方程作为国家模型集中，因为它们是在人类努力的各个领域应用的最通用的模型：物理学，生物学，化学，化学，天气预测，金融，可再生能源和制造等。实际实施了随机过滤的实际实施，用于由现有零件平等模拟的现象模型。一个关键的问题是如何以计算可行的方式近似于系统状态的“真实”描述。粒子过滤器（PFS）是解决过滤问题的一些最成功的方法，为有关隐藏系统状态的推论提供了理论上合理的方法。 PFS涉及“粒子”集：状态模型的不同实现。定期使用部分和嘈杂的观测值校正颗粒的云。用数据同化（DA）的语言，将粒子演变为模型的实现是预测步骤，而使用数据进行校正是分析步骤。事实证明，PF在工程应用中取得了巨大成功（例如），前提是状态模型的规模较小至中等。它们通过顺序处理数据来成功：在分析步骤中，仅使用新的观测值，而无需重新审视过去的观察。在当前时间之前，通常无法处理可用的整个数据集。许多具有挑战性的现实世界问题具有较大的模型状态，每个分析步骤都可以使用大量的观察结果。在数字天气预测中，每个分析时间都会发生数以千计的数据测量值。这使得每个单独的分析步骤几乎与非顺序处理数据一样困难。最近，PI，CO-I及其同事研究了可以减轻此问题的方法，但肯定不会克服它。我们需要一个新的范式来为这些具有挑战性的问题开发有效的PFS。当前的范式将DA机理分为顺序的预测和分析步骤，以方便而教学的方式描述了PF。特别是，分析步骤之间粒子的演变忽略了即将到来的数据。在我们的新范式中，我们没有使用随机部分微分方程的数值近似来发展颗粒，而是提倡“观察到了”的轨迹，其中颗粒在适当选择的方向上“轻轻”。目前，在数据同化中，将粒子向观察的过程是临时方法论，理论上没有理由。相比之下，我们将开发出滤波问题的近似值一致的方法。我们的轻推将在观察数据的情况下扰动粒子的轨迹，以最大程度地提高其位置的可能性。这引入了通过明智的粒子选择消除的偏差。从流体动力学，反应扩散方程，Allen-Cahn等，新方法将针对随机局部微分方程进行优化。我们的项目将为我们的范式提供完整的管道，从理论分析到可以在大型平行计算机上运行的高性能软件实现，从而对提出挑战的基础标记进行了绩效评估。