论文标题
凸化数值算法，用于带反向散射数据的2D反向散射问题
Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data
论文作者
论文摘要
本文涉及反向散射问题，该问题旨在确定从与入射平面波的单个方向相关的多频反向散射数据的2D Helmholtz方程的空间分布的介电常数系数。我们提出了一种全局收敛的凸构数值算法来解决这个非线性和不良反向问题。与常规优化方法相比，我们方法的关键优势在于，它不需要对解决方案进行良好的第一猜测。首先，我们使用变量更改从Helmholtz方程式消除了系数。接下来，使用截断的扩展相对于特殊的傅立叶，我们将逆问题重新加密为准线性椭圆PDES系统，可以通过加权的准可逆性方法在数值上求解。加权准可逆性方法的成本功能是构建的，是涉及卡尔曼重量函数的类似Tikhonov的功能。我们的数值研究表明，使用梯度下降方法的版本，可以找到这种Tikhonov的最小化功能，而无需任何高级\ emph {a a a先验知识。
This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced \emph{a priori} knowledge about it.
