Nonlinearity parameter tomography leads to the problem of identifying a coefficient in a nonlinear wave equation (such as the Westervelt equation) modeling ultrasound propagation. In this paper we transfer this into frequency domain, where the Westervelt equation gets replaced by a coupled system of Helmholtz equations with quadratic nonlinearities. For the case of the to-be-determined nonlinearity coefficient being a characteristic function of an unknown, not necessarily connected domain $D$, we devise and test a reconstruction algorithm based on weighted point source approximations combined with Newton's method. In a more abstract setting, convergence of a regularised Newton type method for this inverse problem is proven by verifying a range invariance condition of the forward operator and establishing injectivity of its linearisation.

翻译：非线性参数层析成像问题源于识别超声传播建模中非线性波动方程（如Westervelt方程）的系数。本文将问题转换至频率域，其中Westervelt方程被替换为具有二次非线性的Helmholtz方程耦合系统。针对待求非线性系数为未知且非必然连通区域$D$的特征函数这一情形，我们设计并测试了一种基于加权点源逼近结合牛顿法的重建算法。在更抽象的理论框架下，通过验证正算子的值域不变性条件并建立其线性化算子的单射性，证明了该反问题正则化牛顿型方法的收敛性。