Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with either partial boundary or interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.

在本文中，我们从理论和数值两个方面研究了双曲型方程含时二阶系数中空间分量的确定问题。通过一般双曲算子的卡尔曼估计以及一个辅助卡尔曼估计，我们在某些几何条件下，利用部分边界测量或内部测量建立了局部赫尔德稳定性。对于数值重构，我们最小化一个惩罚未知函数梯度的吉洪诺夫泛函。基于所得的变分方程，我们设计了一种迭代方法，该方法在每一步通过求解一个泊松方程进行更新。一维原型示例说明了所提出迭代方法的数值性能。