论文标题
标量图像测得的稳定性估计值
Stability estimate for scalar image velocimetry
论文作者
论文摘要
在本文中，我们分析了偏微分方程系统建模标量图像速度法的稳定性。 We first revisit a successful numerical technique to reconstruct velocity vectors ${u}$ from images of a passive scalar field $ψ$ by minimising a cost functional, that penalises 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由Navier-Stokes方程。我们通过将此方法应用于二维湍流中的合成标量字段，研究了重建的稳定性，这些湍流是由数值模拟产生的。然后，我们对非线性耦合问题进行了数学分析，并证明在两个维度的情况下，Navier-Stokes方程的平滑解决方案由测量的标量场唯一地确定。我们还证明了有条件的稳定性估计，表明从测量标量字段$ψ$到重建的速度字段$ U $（在任何内部子集中）的地图是连续的。
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 $ψ$ by minimising a cost functional, that penalises 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.
