Making Cubature on Wiener Space Work

求积(或高维求积)是计算面积的经典方法，历史上与积分学的发展有关。在它的现代形式中，它可以追溯到高斯的工作，指的是在有限数量的精心选择的点上，通过函数值的加权和来逼近函数的定积分。Lyons和Victoir的工作将这一基本思想与现代随机分析机制相结合，并将其应用于无限维路空间。这导致了一种新的粒子方法，可以用来跟踪一大类随机系统的演化。与经典方法(Euler)不同，当粒子沿着允许的轨迹演化时，该近似迅速收敛并且是健壮的。此外，虽然潜在的想法是概率的，但近似值是确定性的。在过滤问题中，我们的目标是基于对系统的部分观察，对复杂现象的演化做出合理的推断。这样的问题是自然的，几乎有各种形状和大小：从移动跟踪移动对象的摄像头的焦点，到医院现代核磁共振扫描仪产生的成像，到通过超级计算机预测下周的天气。这项拟议的研究的目的是帮助将维纳空间上的立方体从一种有希望的新颖的数值积分方法转变为一种强大的方法，这种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此类问题。这项研究将结合概率、数值分析和代数的思想，对路径空间上立方体的构造有一个更系统的理解。这些立方体导致了高效的粒子方法，该方法结合了快速收敛和对演化度量的粒子描述的复杂性的透明界限。作为这个项目的一部分，我们希望降低其他在学术界和工业界工作的研究人员采用我们的想法的门槛。因此，我们建议开发高效和可访问的数值方法的C++实现，并将它们贡献给现有的开源计算粗路径库。

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.