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涉及识别非线性波动方程（如描述超声传播的Westervelt方程）中的系数问题。本文将问题转换至频域，使Westervelt方程被带有二次非线性的耦合亥姆霍兹方程组替代。针对待求非线性系数为未知区域（可能非连通）$D$的特征函数的情形，我们设计并测试了一种基于加权点源近似与牛顿法相结合的重建算法。在更抽象的框架下，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，证明了正则化牛顿型方法在该反问题中的收敛性。