论文标题
在存在较低的随机性的情况下，色散PDE的近似值
Approximations of dispersive PDEs in the presence of low-regularity randomness
论文作者
论文摘要
我们介绍了一类新的数值方案，这些方案允许较低的规律性近似值$ \ mathbb {e}（| u_ {k {k}（τ，τ，v^^η）|^2）$，其中$ u_k $表示$ k $ - the of the solution $ u $ u $ u $ u $ u $ u usterive $ u usterive和$ v v^^^usitifation和$ v v^^^usation $ usational（x）。该数量在物理学中起着重要的作用，特别是在波湍流的研究中，人们需要采用统计方法才能深入了解解决方案对分散方程的通用长期行为。我们的新计划基于Wick的定理和Feynman图，以及基于共振的离散化（请参阅Arxiv：2005.01649），设置在更一般的环境中：我们介绍了一种新颖的组合结构，称为配对的装饰森林，它们是两种装饰的树木在叶子上的装饰树。该方案的特征通过规则性结构从对奇异随机部分微分方程的处理中汲取灵感。与古典方法相反，我们不会离散PDE本身，而是其期望。这使我们能够在有限维（离散）级别上大量利用最佳共振结构和基本增益。
We introduce a new class of numerical schemes which allow for low regularity approximations to the expectation $ \mathbb{E}(|u_{k}(τ, v^η)|^2)$, where $u_k$ denotes the $k$-th Fourier coefficient of the solution $u$ of the dispersive equation and $ v^η(x) $ the associated random initial data. This quantity plays an important role in physics, in particular in the study of wave turbulence where one needs to adopt a statistical approach in order to obtain deep insight into the generic long-time behaviour of solutions to dispersive equations. Our new class of schemes is based on Wick's theorem and Feynman diagrams together with a resonance based discretisation (see arXiv:2005.01649) set in a more general context: we introduce a novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair. The character of the scheme draws its inspiration from the treatment of singular stochastic partial differential equations via Regularity Structures. In contrast to classical approaches, we do not discretize the PDE itself, but rather its expectation. This allows us to heavily exploit the optimal resonance structure and underlying gain in regularity on the finite dimensional (discrete) level.
