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Identification of the forcing term in hyperbolic equations
International Journal of Computer Mathematics ( IF 1.8 ) Pub Date : 2020-12-10 , DOI: 10.1080/00207160.2020.1854744
M. Alosaimi 1, 2 , D. Lesnic 1 , Dinh Nho Hào 3
Affiliation  

We investigate the problem of recovering the possibly both space and time-dependent forcing term along with the temperature in hyperbolic systems from many integral observations. In practice, these average weighted integral observations can be considered as generalized interior point measurements. This linear but ill-posed problem is solved using the Tikhonov regularization method in order to obtain the closest stable solution to a given a priori known initial estimate. We prove the Fréchet differentiability of the Tikhonov regularization functional and derive a formula for its gradient. This minimization problem is solved iteratively using the conjugate gradient method. The numerical discretization of the well-posed problems, that are: the direct, adjoint and sensitivity problems that need to be solved at each iteration is performed using finite-difference methods. Numerical results are presented and discussed for one and two-dimensional problems.



中文翻译:

双曲方程中强迫项的识别

我们研究了从许多积分观测中恢复双曲系统中可能同时存在空间和时间相关的强迫项以及温度的问题。在实践中,这些平均加权积分观测可以被视为广义内点测量。使用 Tikhonov 正则化方法解决了这个线性但不适定的问题,以获得最接近给定先验的稳定解已知的初始估计。我们证明了 Tikhonov 正则化泛函的 Fréchet 可微性并推导出其梯度的公式。这个最小化问题是使用共轭梯度法迭代解决的。适定问题的数值离散化,即:在每次迭代中需要解决的直接、伴随和敏感性问题是使用有限差分方法进行的。给出并讨论了一维和二维问题的数值结果。

更新日期:2020-12-10
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