Dynamic mode decomposition (DMD) is an emerging methodology that has recently attracted computational scientists working on nonintrusive reduced order modeling. One of the major strengths that DMD possesses is having ground theoretical roots from the Koopman approximation theory. Indeed, DMD may be viewed as the data-driven realization of the famous Koopman operator. Nonetheless, the stable implementation of DMD incurs computing the singular value decomposition of the input data matrix. This, in turn, makes the process computationally demanding for high dimensional systems. In order to alleviate this burden, we develop a framework based on sketching methods, wherein a sketch of a matrix is simply another matrix which is significantly smaller, but still sufficiently approximates the original system. Such sketching or embedding is performed by applying random transformations, with certain properties, on the input matrix to yield a compressed version of the initial system. Hence, many of the expensive computations can be carried out on the smaller matrix, thereby accelerating the solution of the original problem. We conduct numerical experiments conducted using the spherical shallow water equations as a prototypical model in the context of geophysical flows. The performance of several sketching approaches is evaluated for capturing the range and co-range of the data matrix. The proposed sketching-based framework can accelerate various portions of the DMD algorithm, compared to classical methods that operate directly on the raw input data. This eventually leads to substantial computational gains that are vital for digital twinning of high dimensional systems.
翻译:动态模式分解( DMD) 是一个新兴的方法,最近吸引了从事非侵入性减少秩序模型的计算学家。 DMD拥有的主要优势之一是从Koopman近似理论中产生基础理论根基。 事实上, DMD可以被视为著名的Koopman 操作者的数据驱动实现。 然而, DMD的稳定实施可以计算输入数据矩阵的单值分解。这反过来又使得高维系统在计算过程中需要高维系统。为了减轻这一负担,我们开发了一个基于素描方法的框架,其中矩阵的草图只是另一个非常小但仍然足够接近原始系统的矩阵。这种草图或嵌入是通过随机转换(具有某些特性的)投入矩阵实现的。因此,许多昂贵的计算可以在较小的矩阵上进行,从而加速解决原始问题。为了减轻这一负担,我们用球质浅水方程式进行数字实验,作为基础的原始模型在地球物理结构流动中进行,但仍然足够接近原始系统。这些矩阵的素描图或嵌嵌化,通过随机转换模型进行,从而将各种关键数据递增数据模型进行。