Reducing data resolution for better super-resolution: Reconstructing turbulent flows from noisy observation

Abstract

A super-resolution (SR) method for the reconstruction of Navier-Stokes (NS) flows from noisy observations is presented. In the SR method, first the observation data is averaged over a coarse grid to reduce the noise at the expense of losing resolution and, then, a dynamic observer is employed to reconstruct the flow field by reversing back the lost information. We provide a theoretical analysis, which indicates a chaos synchronization of the SR observer with the reference NS flow. It is shown that, even with noisy observations, the SR observer converges toward the reference NS flow exponentially fast, and the deviation of the observer from the reference system is bounded. Counter-intuitively, our theoretical analysis shows that the deviation can be reduced by increasing the lengthscale of the spatial average, i.e., making the resolution coarser. The theoretical analysis is confirmed by numerical experiments of two-dimensional NS flows. The numerical experiments suggest that there is a critical lengthscale for the spatial average, below which making the resolution coarser improves the reconstruction.

Figures (3)

• Figure 1: Snapshots of the $x_1$-component of the velocity for $\nu=0.0015$. From the left to the right columns, the plots denote the ground truth $u_1(\bm{x},t)$, noisy observation $y_1(\bm{x},t)$, low-resolution observation, $(\mathcal{C}y_1)(\bm{x},t)$, and the estimated state, $z_1(\bm{x},t)$. The low-resolution observation and the state estimation are for $c=32$, i.e., the subdomain size of $h=32 \delta_x$. The noise level is $\zeta = 1.6$, indicating the standard deviation of the noise is 1.6 times larger than TKE. The top row shows the initial condition and the middle row denotes the velocity at $t=8$. The bottom row shows the velocity profiles at the middle of the domain, i.e., $u_1(x_1,\pi,t)$. In the plots, the spatial coordinate is scaled by $\pi$.
• Figure 2: Fig. A and B show the temporal evolution of the error for a range of $c$ for $\nu=0.006$ and $0.003$, respectively. The noise magnitude is $\zeta=1.6$ and $L$ is set to 5. Fig. C shows the temporal changes of the error in terms of $L$ for $\nu=0.0015$, $c=32$ and $\zeta=1.6$.
• Figure 3: Changes of the relative error with respect to $c$ for $\nu=0.009$ (A), $0.003$ (B), and $0.0015$ (C). $c$ is related to the subdomain for the spatial averaging $|\Omega_i| = (c \delta_x)^2$. $L$ is fixed ($L=10$). The dashed line indicates $\sim c^{-0.5}$.