Increasing the efficiency of numerical methods for estimating the state of a partially observed system. High order methods for solving parabolic PDEs

提高估计部分观测系统状态的数值方法的效率。

• 批准号: EP/H000550/1
• 负责人: Dan Crisan
• 金额: $ 40.13万
• 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: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH000550%2F1
• 关键词: Increasing; efficiency; numerical; methods; estimating

Many aspects of phenomena critical to our lives on this planet such as the global climate, the state of the economy, the evolution of the human foetus are not available for direct measurements. Fortunately models of these phenomena, together with more limited observations frequently allow us to make reasonable inferences about the state of the systems that affect us. The process of using partial observations and a stochastic model to make inferences about an evolving system is known as stochastic filtering. The scoop of applications of stochastic filtering is huge and ranges from non-invasive methods to identify tumours to digital recording. The practical implementation of this process to concrete classes of models raises many important mathematical questions. One key question is how to approximate to the true description of the state of the system in an optimal, computationally feasible way. Under certain conditions, the distribution of the hidden'' model solves a non-linear stochastic PDE. It is desirable to find efficient numerical methods to handle this nonlinear PDE. Particle approximations are some of the most successful methods, especially for moderate and high-dimensional models (for example, for satellite tracking one needs to solve a six-dimensional stochastic PDE). A particle approximation uses a cloud of particles that evolve in the underlying state space. The choice of the particles' trajectories has a crucial influence on the properties of the ensuing approximations. The cubature method recently introduced by Lyons and Victoir produces particle approximations for linear/deterministic PDEs. In this case the particle evolve along admissible trajectory (unlike those produced by classical methods such us the Euler methods) and branch at pre-determined time intervals. The proposed research aims to extend the cubature results and the methods to produce high order approximations for the nonlinear stochastic PDE governing the solution of the filtering problem. Moreover we aim to tackle the increase in the complexity of the computation with time. We aim to study two methods for doing this: The first method consists in the addition of a randomized selection by which the particles that follow the right paths are multiplied and those drifting away from the plausible signal trajectories are rapidly removed. The second method consists in a recombination procedure by which the population of particles is divided into subsets. Then each subset of existing particles is replaced by a single particle which inherits the position of one of the particles in the original subset. The particles are recombined in a way that keeps the accuracy of the approximation unchanged.

对我们在这个星球上的生活至关重要的现象的许多方面，例如全球气候、经济状况、人类胎儿的进化，都无法直接测量。幸运的是，这些现象的模型以及更有限的观察经常使我们能够对影响我们的系统的状态做出合理的推断。使用部分观察和随机模型对不断演化的系统进行推断的过程称为随机过滤。随机滤波的应用范围非常广泛，从识别肿瘤的非侵入性方法到数字记录。这一过程在具体模型类别中的实际实现提出了许多重要的数学问题。一个关键问题是如何以一种最佳的、计算上可行的方式逼近系统状态的真实描述。在某些条件下，隐藏模型的分布求解非线性随机偏微分方程。人们希望找到有效的数值方法来处理这种非线性偏微分方程。粒子近似是最成功的方法之一，特别是对于中维和高维模型（例如，对于卫星跟踪，需要求解六维随机偏微分方程）。粒子近似使用在底层状态空间中演化的粒子云。粒子轨迹的选择对随后的近似值的性质具有至关重要的影响。 Lyons 和 Victoir 最近引入的立方法可生成线性/确定性偏微分方程的粒子近似值。在这种情况下，粒子沿着可接受的轨迹演化（与欧拉方法等经典方法产生的轨迹不同）并以预定的时间间隔分支。所提出的研究旨在扩展体积结果和方法，以产生控制滤波问题解决方案的非线性随机偏微分方程的高阶近似值。此外，我们的目标是解决计算复杂性随时间增加的问题。我们的目标是研究两种方法来实现这一点：第一种方法包括添加随机选择，通过该随机选择，遵循正确路径的粒子会成倍增加，而那些偏离合理信号轨迹的粒子会被快速移除。第二种方法包括重组过程，通过该过程将粒子群划分为子集。然后，现有粒子的每个子集都被单个粒子替换，该粒子继承原始子集中粒子之一的位置。粒子以保持近似精度不变的方式重新组合。

项目成果

Paris-Princeton Lectures on Mathematical Finance 2013: Editors: Vicky Henderson, Ronnie Sircar

巴黎-普林斯顿数学金融讲座 2013：编辑：Vicky Henderson、Ronnie Sircar

• 发表时间: 2013
• 作者: Benth Fred Espen

A high order time discretization of the solution of the non-linear filtering problem

非线性滤波问题解的高阶时间离散化

• DOI: 10.1007/s40072-019-00157-3
• 发表时间: 2019
• 期刊: Analysis and Computations
• 作者: Crisan D

ON THE STABILITY OF SEQUENTIAL MONTE CARLO METHODS IN HIGH DIMENSIONS

• DOI: 10.1214/13-aap951
• 发表时间: 2014-08-01
• 期刊: ANNALS OF APPLIED PROBABILITY
• 影响因子: 1.8
• 作者: Beskos, Alexandros;Crisan, Dan;Jasra, Ajay
• 通讯作者: Jasra, Ajay

Kusuoka-Stroock gradient bounds for the solution of the filtering equation

滤波方程解的 Kusuoka-Stroock 梯度界限

• DOI: 10.1016/j.jfa.2014.12.009
• 发表时间: 2015
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Crisan D

Numerical solution for a class of SPDEs over bounded domains

• DOI: 10.1080/17442508.2013.819510
• 发表时间: 2014-05
• 期刊: Stochastics An International Journal of Probability and Stochastic Processes
• 作者: D. Crisan;J. Xiong