Variational principles for continuum dynamics on geometric rough paths.

研究的背景包括潜在的影响。该项目将在连续介质动力学的变换理论和多时间均匀化的基础上，通过发展几何粗糙路径(GRP)理论，为计算流体力学中的不确定性量化和减少不确定性开发一种新的方法。在我们之前的结果中，随机流体速度分解结果表明，变换理论和多时间均匀化的原理使得将流体传输速度分解为漂移部分和随机部分的方法具有物理意义、数据驱动和数学基础。这种方法可以直接应用于确定运动是基于几何粗糙路径上定义的基本变分原理的连续统流。*目的和目标：该项目旨在将连续介质动力学中的拉格朗日轨迹分解为快和慢，或可解析和不可解析的分量，并将我们的随机建模经验扩展到几何粗糙流领域，作为量化先验不确定性的基础，然后使用数据同化方法(例如，粒子滤波)来减少不确定性。*研究方法的新颖性：到目前为止，几何粗糙路径上连续介质动力学的变分原理领域只有一篇论文写过，那就是达里尔·霍姆及其合作者。*EPSRC的战略目标是数学科学*研究领域是连续体力学*目前还没有公司参与。合作者包括帝国理工学院数学系的几位同事

Context of the research including potential impact. The project will develop a new methodology for uncertainty quantification and reduction of uncertainty in computational fluid dynamics by developing the theory of Geometric Rough Paths (GRP) based in the foundations of transformation theory and multi-time homogenisation for continuum dynamics. In our earlier results, the stochastic fluid velocity decomposition results of show that the principles of transformation theory and multi-time homogenisation enable a physically meaningful, data-driven and mathematically-based approach for decomposing the fluid transport velocity into its drift and stochastic parts. This approach can be applied immediately to the class of continuum flows whose deterministic motion is based on fundamental variational principles defined on geometric rough paths. * Aims and objectives: The project aims to decompose the Lagrangian trajectories in continuum dynamics into their fast and slow, or resolvable and unresolvable, components and extend our stochastic modelling experience to the realm of geometric rough flows as a basis for quantifying a priori uncertainty and then using data assimilation methods (e.g., particle filtering) for reducing the uncertainty. * Novelty of the research methodology: The field of variational principles for continuum dynamics on geometric rough paths has only one paper written about it so far, namely by Darryl Holm and his collaborators. * EPSRC strategic them is Mathematical Sciences * Research area is continuum mechanics * No companies involved yet. Collaborators involve several colleagues in the Imperial College Mathematics department