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-definedeness 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.

翻译：本文研究振动声成像技术的逆问题，该技术通过利用非线性效应增强超声成像效果。其核心在于确定描述两束定向声束传播及其非线性相互作用产生的波形的偏微分方程组中两个空间可变系数。一方面，为论证使用牛顿法求解该逆问题的合理性，我们验证了对应于偏微分方程模型两个版本的正演算子的适定性与可微性；另一方面，我们采用全同时公式化处理该逆问题，并证明了牛顿法求解该问题的收敛性。