This note is concerned with inverse wave scattering in one and two dimensional domains. It seeks to recover an unknown function based on measurements collected at the boundary of the domain. For one-dimensional problem, only one point of the domain is assumed to be accessible. For the two dimensional domain, the outer boundary is assumed to be accessible. It develops two iterative algorithms, in which an assumed initial guess for the unknown function is updated. The first method uses a set of sampling functions to formulate a moment problem for the correction to the assumed value. This method is applied to both one-dimensional and two dimensional domains. For two dimensional Helmholtz equation, it relies on a new effective filtering technique which is another contribution of the present work. The second method uses a direct formulation to recover the correction term. This method is only developed for the one-dimensional case. For all cases presented here, the correction to the assumed value is obtained by solving an over-determined linear system through the use of least-square minimization. Tikhonov regularization is also used to stabilize the least-square solution. A number of numerical examples are used to show their applicability and robustness to noise.

本说明与一维和二维域中的逆波散射有关。它试图根据在域边界处收集的测量值来恢复未知函数。对于一维问题，假定仅该域的一个点是可访问的。对于二维域，假定外边界是可访问的。它开发了两个迭代算法，其中对未知函数的假定初始猜测进行了更新。第一种方法使用一组采样函数来表达矩问题用于校正假设值。此方法适用于一维和二维域。对于二维亥姆霍兹方程，它依赖于一种新的有效滤波技术，这是当前工作的另一项贡献。第二种方法使用直接公式来恢复校正项。仅针对一维情况开发此方法。对于此处介绍的所有情况，通过使用最小二乘最小化求解超定线性系统，即可获得对假定值的校正。Tikhonov正则化也用于稳定最小二乘解。许多数值示例用于显示其对噪声的适用性和鲁棒性。