论文标题
弥散和标量波的反向散射系列的范围依赖性收敛和稳定性
Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves
论文作者
论文摘要
这项工作分析了基于Helmholtz方程的标量波和从时间无关的扩散方程中的弥漫性波的前向散射序列，它们在各种应用中都是重要的PDE。与以前的工作不同的是，在$ l^p $ norms下研究了稳定性和反向散射系列的收敛半径，稳定性以及该系列的近似误差，我们在Sobolev $ h^s $ Norm中研究了这些数量，该数量与$ l^2 $的函数空间相关联。 $ h^s $ norm具有基于其在傅立叶域中的定义的自然频谱偏差：$ s <0 $偏向较低频率，而情况$ s> 0 $偏向较高的频率。我们在参数和数据域中使用不同的$ H^s $规范比较稳定性估计，并为实践反演过程中的频率加权技术提供了理论上的理由。我们还提供数值反转示例，以证明不同度量空间下收敛的反向散射半径的差异。
This work analyzes the forward and inverse scattering series for scalar waves based on the Helmholtz equation and the diffuse waves from the time-independent diffusion equation, which are important PDEs in various applications. Different from previous works, which study the radius of convergence for the forward and inverse scattering series, the stability, and the approximation error of the series under the $L^p$ norms, we study these quantities under the Sobolev $H^s$ norm, which associates with a general class of $L^2$-based function spaces. The $H^s$ norm has a natural spectral bias based on its definition in the Fourier domain: the case $s<0$ biases towards the lower frequencies, while the case $s>0$ biases towards the higher frequencies. We compare the stability estimates using different $H^s$ norms for both the parameter and data domains and provide a theoretical justification for the frequency weighting techniques in practical inversion procedures. We also provide numerical inversion examples to demonstrate the differences in the inverse scattering radius of convergence under different metric spaces.
