Making Cubature on Wiener Space Work

求积（或在更高维度的求积）是计算面积的经典方法，历史上与积分的发展有关。在其现代形式，它可以追溯到高斯的工作，是指近似的定积分的一个功能的加权总和的功能值在有限数量的精心挑选的点。工作的里昂和维克托结合了这一基本思想与机械的现代随机分析和应用它的无限维路径空间。这导致了一种新的粒子方法，可用于跟踪一大类随机系统的演变。近似收敛速度快，是强大的，因为粒子的演变，不像在经典的方法（欧拉）沿着容许的轨迹。此外，虽然基本思想是概率性的，但近似是确定性的。在过滤问题中，我们的目标是根据系统的部分观测结果对复杂现象的演化做出合理的推断。这些问题是自然的，几乎有各种各样的形式和大小：从跟踪移动物体的移动的相机的焦点，到医院现代核磁共振扫描仪产生的图像，再到通过超级计算机预测下周的天气。拟议研究的目的是帮助将维纳空间上的体积从一种有前途的新方法转化为“实验室”中的数值积分，成为一种强大的方法，可以很容易地被从业者采用，以帮助解决影响我们生活的问题。这项研究将把概率论、数值分析和代数学的思想结合起来，以更系统地理解路径空间上的立体图的构造。这些cubatures导致高效的粒子方法，联合收割机结合快速收敛与透明的边界上的粒子描述的复杂性的不断发展的措施。作为该项目的一部分，我们希望降低学术界和工业界其他研究人员采用我们想法的障碍。因此，我们建议开发有效的和可访问的C++实现的数值方法，并将其贡献给现有的开源计算粗糙路径库。

Quadrature (or in higher dimensions cubature) is a classical method for calculating areas and historically related to the development of the integral calculus. In its modern form it goes back to the work of Gauss and refers to the approximation of the definite integral of a function by a weighted sum of function values at a finite number of carefully chosen points. The work of Lyons and Victoir has combined this fundamental idea with the machinery of modern stochastic analysis and applied it on infinite dimensional path spaces. This has resulted in a novel particle method that can be used to track the evolution of a large class of random systems. The approximation convergences rapidly and is robust as the particles evolve unlike in classical methods (Euler) along admissible trajectories. Moreover, while the underlying ideas are probabilistic the approximation is deterministic. In filtering problems, we aim to make reasonable inferences about the evolution of complex phenomena based on partial observations of the system. Such problems are natural and come in virtually all shapes and sizes: from the focus of a camera in a mobile tracking a moving object, via the imaging produced by a modern MRI scanner in hospital, to the prediction of next week's weather by means of a supercomputer. The aim of the proposed research is to help to transform cubature on Wiener space from a promising and novel approach to numerical integration "in the lab" to a powerful method that can easily be adopted by practitioners to help solve such problems that impact our lives. The proposed research will bring together ideas from probability, numerical analysis and algebra to gain a more systematic understanding of the construction of cubatures on path space. These cubatures result in highly efficient particle methods that combine rapid convergence with transparent bounds on the complexity of the particle descriptions of the evolving measures. As part of this project we want to lower the hurdle for other researchers working in academia and industry to adopt our ideas. Hence, we propose to develop efficient and accessible C++ implementations of the numerical methods and to contribute them to the existing open source computational rough path library.