Variational principles for continuum dynamics on geometric rough paths.

研究背景，包括潜在影响。该项目将开发一种新的方法，用于不确定性量化和减少计算流体动力学中的不确定性，方法是在连续动力学的变换理论和多时间均匀化的基础上开发几何粗糙路径（GRP）理论。在我们早期的结果中，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一个物理上有意义的，数据驱动的和基于几何的方法分解成它的漂移和随机部分的流体传输速度。这种方法可以立即应用到一类连续流的确定性运动是基于基本变分原理定义的几何粗糙路径。* 宗旨和目标：该项目旨在将连续动力学中的拉格朗日轨迹分解为快速和慢速，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）以减少不确定性。* 新奇的研究方法：该领域的变分原理连续动力学的几何粗糙路径只有一个文件写的，迄今为止，即由达里尔霍尔姆和他的合作者。* EPSRC的战略是数学科学 * 研究领域是连续介质力学 * 还没有公司参与。合作者涉及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