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

对我们在这个星球上的生活至关重要的许多现象，如全球气候、经济状况、人类胎儿的进化等，都无法进行直接测量。幸运的是，这些现象的模型，加上更有限的观察，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。使用部分观测和随机模型对一个演化系统进行推断的过程称为随机滤波。随机过滤的应用范围是巨大的，从识别肿瘤的非侵入性方法到数字记录。这一过程在具体的模型类中的实际实施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优的、在计算上可行的方式接近系统状态的真实描述。在一定条件下，隐模型的分布可解一个非线性随机偏微分方程组。寻找有效的数值方法来处理这种非线性偏微分方程解是人们所期望的。粒子近似是最成功的方法之一，特别是对于中高维模型(例如，对于卫星跟踪，需要求解一个六维随机PDE)。粒子近似使用在基础状态空间中演化的粒子云。粒子轨迹的选择对随后的近似性质有至关重要的影响。Lyons和Victoir最近提出的体积法产生了线性/确定性偏微分方程组的粒子近似。在这种情况下，粒子沿着允许的轨迹(不同于由经典方法，如欧拉方法产生的轨迹)演化，并以预定的时间间隔分支。本文的研究目的是推广非线性随机偏微分方程解的体积结果和产生高阶近似的方法。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这样做的方法：第一种方法是增加随机选择，通过这种方法，沿着正确路径的粒子被相乘，而那些偏离合理信号轨迹的粒子被快速移除。第二种方法是重组过程，通过重组过程将粒子群分成多个子集。然后将现有粒子的每个子集替换为继承原始子集中其中一个粒子的位置的单个粒子。粒子以保持近似精度不变的方式进行重新组合。

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.