Data-driven discovery with Limited Data Acquisition for fluid flow
  across cylinder
Dr. Himanshu Singh
One of the central challenge for extracting governing principles of dynamical
system via Dynamic Mode Decomposition (DMD) is about the limit data
availability or formally called as Limited Data Acquisition in the present
paper. In the interest of discovering the governing principles for a dynamical
system with limited data acquisition, we provide a variant of Kernelized
Extended DMD (KeDMD) based on the Koopman operator which employ the notion of
Gaussian random matrix to recover the dominant Koopman modes for the standard
fluid flow across cylinder experiment. It turns out that the traditional kernel
function, Gaussian Radial Basis Function Kernel, unfortunately, is not able to
generate the desired Koopman modes in the scenario of executing KeDMD with
limited data acquisition. However, the Laplacian Kernel Function successfully
generates the desired Koopman modes when limited data is provided in terms of
data-set snapshot for the aforementioned experiment and this manuscripts serves
the purpose of reporting these exciting experimental insights. This paper also
explores the functionality of the Koopman operator when it interacts with the
reproducing kernel Hilbert space (RKHS) that arises from the normalized
probability Lebesgue measure
$d\mu_{\sigma,1,\mathbb{C}^n}(z)=(2\pi\sigma^2)^{-n}\exp\left(-\frac{\|z\|_2}{\sigma}\right)dV(z)$
when it is embedded in $L^2-$sense for the holomorphic functions over
$\mathbb{C}^n$, in the aim of determining the Koopman modes for fluid flow
across cylinder experiment. We explore the operator-theoretic characterizations
of the Koopman operator on the RKHS generated by the normalized Laplacian
measure $d\mu_{\sigma,1,\mathbb{C}^n}(z)$ in the $L^2-$sense. In doing so, we
provide the compactification & closable characterization of Koopman operator
over the RKHS generated by the normalized Laplacian measure in the $L^2-$sense.