Making Cubature on Wiener Space Work

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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.