关于Westervelt方程中非线性系数与声速的同时辨识
On the simultanenous identification of the nonlinearity coefficient and the sound speed in the Westervelt equation
AI总结：
本文针对Westervelt方程，研究从时域边界测量同时辨识空间依赖的非线性参数与波速的问题，设计了基于平方慢度和组合系数重构的冻结牛顿法并证明收敛性，通过数值算例验证了方法的有效性与局限性。
摘要：
本文研究非线性声学中应用最广泛的模型之一Westervelt方程，旨在从时域边界测量值中恢复两个具有物理意义的空间依赖参数，具体为声学文献中常被称为$B/A$的非线性参数$κ(x)$以及波速$c_0(x)$。确定这些量的空间变化可作为一种成像手段。本文针对这类应用中相关的单次或两次边界测量场景，研究参数的可辨识性。针对以平方慢度$\boldsymbol{\frak{s}}=1/c_0^2$和组合系数$η=\frac{B/A+2}{\rho_0 c_0^4}$重构的问题，本文设计了一种冻结牛顿法并证明了其收敛性。数值算例验证了该迭代格式的有效性与局限性。
英文摘要：
This paper considers the Westervelt equation, one of the most widely used models in nonlinear acoustics, and seeks to recover two spatially-dependent parameters of physical importance from time-trace boundary measurements. Specifically, these are the nonlinearity parameter $κ(x)$ often referred to as $B/A$ in the acoustics literature and the wave speed $c_0(x)$. The determination of the spatial change in these quantities can be used as a means of imaging. We consider identifiability from one or two boundary measurements as relevant in these applications. For a reformulation of the problem in terms of the squared slowness $\mathfrak{s}=1/c_0^2$ and the combined coefficient $η=\frac{B/A+2}{\varrho_0 c_0^4}$ we devise a frozen Newton method and prove its convergence. The effectiveness (and limitations) of this iterative scheme are demonstrated by numerical examples.
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