We propose a regularization method based on the iterative conjugate gradient method for the solution of a Cauchy problem for the wave equation in one dimension. This linear but ill-posed Cauchy problem consists of finding the displacement and flux on a hostile and inaccessible part of the medium boundary from Cauchy data measurements of the same quantities on the remaining friendly and accessible part of the boundary. This inverse boundary value problem is recast as a least-squares minimization problem that is solved by using the conjugate gradient method whose iterations are stopped according to the discrepancy principle for obtaining stable reconstructions. The objective functional associated is proved Fréchet differentiable and a formula for its gradient is derived. The well-posed direct, adjoint and sensitivity problems present in the conjugate gradient method are solved by using a finite-difference method. Two numerical examples to illustrate the accuracy and stability of the proposed numerical procedure are thoroughly presented and discussed.

我们提出了一种基于迭代共轭梯度法的正则化方法，用于求解一维波动方程的柯西问题。这个线性但不适定的柯西问题包括从边界的剩余友好和可访问部分的相同数量的柯西数据测量中找到介质边界的敌对和不可访问部分的位移和通量。这个逆边值问题被重铸为最小二乘最小化问题，通过使用共轭梯度方法解决，该方法根据差异原理停止迭代以获得稳定的重建。证明了相关的目标泛函 Fréchet 可微分，并推导出了其梯度公式。恰到好处的直接，共轭梯度法中存在的伴随和敏感性问题通过使用有限差分法来解决。详细介绍和讨论了两个数值示例，以说明所提出的数值程序的准确性和稳定性。