In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton’s method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton’s method for its solution.

中文翻译：

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频域振动声学近轴模型中声速和非线性参数的同步重建

在本文中，我们考虑振动声学的逆问题，这是一种利用非线性效应增强超声成像的技术。它相当于确定偏微分方程系统中的两个空间可变系数，描述两个定向声束的传播以及它们非线性相互作用产生的波。为了证明使用牛顿法解决这个反问题的合理性，一方面，我们验证了对应于两个版本的偏微分方程模型的前向算子的良好定义性和可微性；另一方面，我们考虑反问题的一次性表述，并证明其求解的牛顿法的收敛性。