Stability estimate for scalar image velocimetry

TL;DR

This paper analyzes the stability of scalar image velocimetry, combining numerical reconstruction techniques with mathematical proofs that show unique determination and stability of velocity fields from scalar measurements in 2D turbulence.

Contribution

It provides a mathematical proof of uniqueness and a stability estimate for velocity reconstruction from scalar fields in 2D turbulence, advancing the theoretical understanding of scalar image velocimetry.

Findings

Reconstruction method is stable in synthetic 2D turbulence data.

Unique determination of velocity from scalar field in 2D Navier-Stokes.

Conditional Hölder stability estimate established.

Abstract

In this paper we analyse the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors ${u}$ from images of a passive scalar field $\psi$ by minimising a cost functional, that penalises the difference between the reconstructed scalar field $\phi$ and the measured scalar field $\psi$, under the constraint that $\phi$ is advected by the reconstructed velocity field ${u}$, which again is governed by the Navier-Stokes equations. We investigate the stability of the reconstruction by applying this method to synthetic scalar fields in two-dimensional turbulence, that are generated by numerical simulation. Then we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two dimensional case, smooth solutions of the Navier-Stokes equations are uniquely determined by the measured scalar field. We also prove a conditional stability estimate showing that the map from the measured scalar field $\psi$ to the reconstructed velocity field $u$, on any interior subset, is H\"older continuous.