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 a priori knowledge about it.

本文关注的是逆散射问题，该问题旨在从与入射平面波的单个方向相关的多频反向散射数据中确定二维亥姆霍兹方程的空间分布介电常数系数。我们提出了一种全局收敛凸化数值算法来解决这个非线性和不适定的逆问题。与传统优化方法相比，我们的方法的主要优势在于它不需要对解决方案进行良好的初步猜测。首先，我们使用变量的变化从亥姆霍兹方程中消除系数。接下来，使用关于特殊傅里叶基的截断展开，我们将逆问题近似地重新表述为拟线性椭圆 PDE 系统，这可以通过加权准可逆方法进行数值求解。加权准可逆方法的成本函数构造为涉及卡尔曼权重函数的类 Tikhonov 函数。我们的数值研究表明，使用梯度下降法的一种版本，无需任何高级方法就可以找到这个类 Tikhonov 泛函的极小值。关于它的先验知识。