Stability estimate for scalar image velocimetry
| Autor: | Erik Burman, Jurriaan J. J. Gillissen, Lauri Oksanen |
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| Rok vydání: | 2023 |
| Předmět: | |
| Zdroj: | Journal of Inverse and Ill-posed Problems. |
| ISSN: | 1569-3945 0928-0219 |
| DOI: | 10.1515/jiip-2020-0107 |
| Popis: | 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 {{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 {{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. |
| Databáze: | OpenAIRE |
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