论文标题
Lippmann-Schwinger方程在强烈散射的声学介质中的迭代解决方案通过预先调节器的随机结构
Iterative solution of the Lippmann-Schwinger equation in strongly scattering acoustic media by randomized construction of preconditioners
论文作者
论文摘要
在这项工作中，Lippmann-Schwinger方程用于建模强烈散射的声学介质中的地震波。我们考虑Helmholtz方程，它是频域中的标量波方程，其恒定密度和可变速度，并将其转换为Lippmann-Schinginger类型的积分方程。为了直接解决矩阵反转的离散问题是耗时的，因此我们使用迭代方法。 Born系列是一个众所周知的散射系列，它以相对较小的成本为解决方案提供了有限的用途，因为它仅收敛于小散射电位。还有其他带有预调节器的散射系列，这些系列已被证明会收敛任何对比度，但是对于具有高对比度的模型，这些方法可能需要许多迭代。在这里，我们基于随机矩阵近似值和层次矩阵开发新的预调节器，这些矩阵可以使散射系列与较少数量的迭代形成鲜明对比。我们描述了两个不同的预处理。一种最适合较低的频率，另一个适用于较高的频率。我们在构造前进器和迭代解决方案中使用快速的傅立叶变换，这使方法有效。通过在两个2D模型上进行数值实验来说明这些方法的性能。
In this work the Lippmann-Schwinger equation is used to model seismic waves in strongly scattering acoustic media. We consider the Helmholtz equation, which is the scalar wave equation in the frequency domain with constant density and variable velocity, and transform it to an integral equation of the Lippmann-Schwinger type. To directly solve the discretized problem with matrix inversion is time-consuming, therefore we use iterative methods. The Born series is a well-known scattering series which gives the solution with relatively small cost, but it has limited use as it only converges for small scattering potentials. There exist other scattering series with preconditioners that have been shown to converge for any contrast, but the methods might require many iterations for models with high contrast. Here we develop new preconditioners based on randomized matrix approximations and hierarchical matrices which can make the scattering series converge for any contrast with a low number of iterations. We describe two different preconditioners; one is best for lower frequencies and the other for higher frequencies. We use the fast Fourier transform both in the construction of the preconditioners and in the iterative solution, and this makes the methods efficient. The performance of the methods are illustrated by numerical experiments on two 2D models.
