Burman, Erik; Gillissen, Jurriaan JJ; Oksanen, Lauri; (2023) Stability estimate for scalar image velocimetry. Journal of Inverse and Ill-posed Problems 10.1515/jiip-2020-0107. (In press).

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Abstract

In this paper, we analyze 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 ψ by minimizing a cost functional that penalizes the difference between the reconstructed scalar field ϕ and the measured scalar field ψ, under the constraint that ϕ 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 ψ to the reconstructed velocity field u, on any interior subset, is Hölder continuous. </m:math> {{u}} from images of a passive scalar field ψ by minimizing a cost functional that penalizes the difference between the reconstructed scalar field ϕ and the measured scalar field ψ, under the constraint that ϕ is advected by the reconstructed velocity field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> </m:math> {{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 ψ to the reconstructed velocity field u, on any interior subset, is Hölder continuous.

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