摘要
近年来,由于工程的实际需要,波动方程反问题的研究越来越受到重视。在实际应用中,往往需要反演一些用波动方程所描述的数学模型中的参数。例如,地震层析成像中的参数反演问题广泛应用于构建地底结构或寻找矿藏;根据直杆的波动方程,用直杆的横截面积作为反演参数模拟管道损伤;通过反演二维非均匀粘弹介质中波传播的计算公式,可以实现对地震波场的模拟等等。
在研究波动方程反演问题的方法中,已经有一些比较有效的方法,例如Tikhonov正则化方法,Landweber迭代法,以及一些基于具体问题给出的反演策略。这些方法典型的缺点是反演结果精度不高,解对初值的依赖性比较强。随着波动方程的广泛应用,针对波动方程的参数反演研究需要提出高精度的算法。
本文主要针对H 1正则化方法在二维弹性波方程参数识别中的运用进行理论研究。受H 1正则化方法在抛物型系统参数识别运用中的启发,本文将H 1正则化方法用来识别二维弹性波方程参数。文章给出了用H 1正则化方法和最小二乘法联合处理二维弹性波方程得到的约束优化问题,证明了该优化问题的解的存在性。本文对得到的优化问题给出了具体算法-Armijo算法。最后用具体的算例,进行数值实验,结果表明了H 1正则化方法识别二维弹性波方程参数具有反演结果精细,解对初值的依赖比较弱的优点。
Recently, due to the practical applications in engineering, the investigation of wave equation inversion becomes more and more important. We need to recover parameters for wave equation which is used as the mathematical models. For instance, the seismic tomography is widely applied for determining the structure of the Earth’s deep interior or detecting the mineral. The cross-area of the pipe is considered as the parameter of the non-destruction investigation determined while corroding or others in the pipe. Applying the computing formula of 2D viscoelastic wave equation, we can realize seismic simulations on computer.
Some efficient methods have presented for wave equation inversion, such as the Tikhonov-type regularization, Landweber iteration and some inversion strategies for the practical application. But these methods have some short-comings. The results are not enough refined and the solutions strongly depend on the initial values.
We investigate the H 1-regularization of parameters identification for two-dimensional elastic wave equation. And in the heuristic of the H 1-regularization used for parameter identification in parabolic systems, we attempt to applied H1 -regularization for parameters identification for two-dimensional elastic wave equation. The identification problem is formulated as a constrained minimization one using the output least squares approach with the H 1-regularization and the existence of minimizer has been proved. We use the optimal algorithm- Armijo algorithm to solve this problem. Numerical experiments show the efficiency of the proposed method. And numerical results demonstrate H1 -regularization has such advantages in parameter idenfication for two-dimensional elastic wave equation: the solutions are refined and slightly depend on the initial values.
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