波动方程逆散射问题的域导数与形状重构
Domain derivative and shape reconstruction for an inverse backscattering problem for the wave equation
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中文总结 AI 辅助
针对三维无界自由空间中含紧支可穿透散射障碍物的时谐声波逆后向散射问题，研究人员建立时谐远场模式关于障碍物形状的Fréchet可微性，结合卷积求积与边界元方法实现正则化高斯-牛顿法，数值示例验证了算法的潜力与局限性。
中文摘要 AI 辅助
我们考虑三维无界自由空间中含紧支可穿透散射障碍物的时谐声波逆后向散射问题。假设对应少数时谐入射平面波的散射波动态后向散射远场数据可用，目标是重构散射障碍物的形状。我们证明了时谐远场模式关于散射障碍物形状的Fréchet可微性，并通过其拉普拉斯变换刻画了相关的时间域导数。该刻画随后被用于结合卷积求积与边界元方法的正则化高斯-牛顿法的高效实现，以求解逆后向散射问题。数值示例展示了该算法的潜力与局限性。
英文摘要
We consider an inverse backscattering problem for time-dependent acoustic waves with a compactly supported penetrable scattering obstacle in unbounded three-dimensional free space. Assuming that dynamic backscattering far field data of scattered waves corresponding to a few time-dependent incident plane waves are available, the goal is to reconstruct the shape of the scattering obstacle. We establish Fréchet differentiability of the time-dependent far field pattern with respect to the shape of the scattering obstacle and give a characterization of the associated temporal domain derivative in terms of its Laplace transform. This characterization is then utilized in an efficient implementation of a regularized Gauß-Newton method for the inverse backscattering problem using convolution quadrature and a boundary element method. Numerical examples demonstrate potentials and limitations of the algorithm.
