Inverse Problems for the Family of Wave Equations

研究了介质不连续性的识别，如夹杂、孔洞、裂纹等，以及介质的物理性质。为了识别介质的不连续性，我们改进并采用了探针法和封闭法。特别地，我们研究了探针法所必需的反射解的性质和唯一的连续性，实现了探针法。对于封闭方法，我们通过引入振荡衰减解来代替复杂的几何光学解，从而扩大了它的应用范围。证明了反边值问题的重构方法，如探针法、奇异源法、无响应检验法等统一为无响应检验法，探针法和奇异源法是同一种方法。对于逆散射问题，我们提出了两种新的重建方法，解决了线性采样方法的困难。此外，我们还成功地建立了一维抛物型方程的探测方法，为Shirota的波动方程系数不连续性识别的计算方法提供了理论框架;在介质物性识别方面，我们研究了钢-混凝土连接梁的残余应力和损伤识别的两个反问题。给出了瑞利波速度的频散公式，并将其应用于前一反问题。对于后一个问题，我们建立了从实测数据的频响函数来识别损伤的方法。我们还研究了非线性波动方程的系数识别，并成功地观察到，我们可以通过线性化Dirichlet到Neumann映射来识别系数的线性和二次部分。

We studied identifying the discontinuity of the medium such as inclusions, cavities, cracks and the physical property of the medium. For identifying the discontinuity for the medium, we improved and adopted the probe method and enclosure method. Especially, we studied the behavior of the reflected solution and the unique continuation property which are essential for the probe method, and we accomplished the probe method. As for the enclosure method, we enlarged its application by replacing the complex geometric optic solution which is difficult to construct and localize by introducing the osciallating-decaying solution. We also showed that the reconstruction methods for the inverse boundary value problem such as the probe method, singular source method, no response test are unified into the no response test, and the probe method and singular source method are the same methods. For the inverse scattering problem, we solved the difficulty of the linear sampling method by proposing two new reconstruction methods. Moreover, we succeeded in establishing the probe method for the one space dimensional parabolic equation and giving the theoretical frame work for Shirota's computational method for identifying the discontinuity of the coefficient for the wave equation.As for identifying the physical property of the medium, we studied two inverse problems for identifying the residual stress and the damage of steel-concrete connected beam. We gave the dispersion formula of the speed of the Rayleigh wave and applied it for the former inverse problem. For the latter problem, we established identifying the damage from the frequency response function which is a practical measured data. We also studied identifying the coefficient for the nonlinear wave equation and succeeded in observing that we can identify the linear and the quadratic part of the coefficient by linearizing the Dirichlet to Neumann map.