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现象的许多方面，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，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。使用部分观测值和随机模型对演化系统进行推断的过程称为随机滤波。随机滤波的应用范围很广，从非侵入性方法识别肿瘤到数字记录。这一过程在具体模型类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最优的、计算上可行的方式近似于系统状态的真实描述。在一定条件下，隐模型的分布解出了一个非线性随机偏微分方程。因此，需要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，特别是对于中高维模型（例如，对于卫星跟踪，需要求解六维随机PDE）。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的性质有着至关重要的影响。容积法最近推出的里昂和Victoir产生的线性/确定性偏微分方程的粒子近似。在这种情况下，粒子沿沿着容许的轨迹（不像由诸如欧拉方法的经典方法产生的轨迹）并且以预定的时间间隔分支演化。本研究的目的是扩展容积结果和方法，以产生高阶近似的非线性随机偏微分方程的过滤问题的解决方案。此外，我们的目标是解决随着时间的推移计算的复杂性的增加。我们的目标是研究两种方法来做到这一点：第一种方法包括增加一个随机选择，通过该选择，遵循正确路径的粒子被倍增，而那些远离看似合理的信号轨迹的粒子被迅速删除。第二种方法包括一个重组过程，通过该过程，粒子群被划分为子集。然后，现有粒子的每个子集被单个粒子替换，该单个粒子继承原始子集中的一个粒子的位置。粒子以保持近似精度不变的方式重新组合。

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.