当前位置: X-MOL 学术SIAM J. Numer. Anal. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Carleman-Based Reconstruction Algorithm for Waves
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2021-04-19 , DOI: 10.1137/20m1315798
Lucie Baudouin , Maya de Buhan , Sylvain Ervedoza , Axel Osses

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 998-1039, January 2021.
We present a globally convergent numerical algorithm based on global Carleman estimates to reconstruct the speed of wave propagation in a bounded domain with Dirichlet boundary conditions from a single measurement of the boundary flux of the solutions in a finite time interval. The global convergence of the proposed algorithm naturally arises from the proof of the Lipschitz stability of the corresponding inverse problem for both sufficiently large observation time and boundary using global Carleman inequalities. The speed of propagation is supposed to be independent of time but varying in space with a trace and normal derivative known at the boundary and belonging to a certain admissible set that limits the speed fluctuations with respect to a given exterior point $x_0$. In order to recover the speed, we also require a single experiment with null initial velocity and initial deformation having some monotonicity properties in the direction of $x-x_0$. We perform numerical simulations in the discrete setting in order to illustrate and to validate the feasibility of the algorithm in both one and two dimensions in space. As proved theoretically, we verify that the numerical reconstruction is achieved for any admissible initial guess, even in the presence of small random disturbances on the measurements.


中文翻译:

基于Carleman的波浪重构算法

SIAM数值分析学报,第59卷,第2期,第998-1039页,2021年1月。
我们提出了一种基于全局Carleman估计的全局收敛数值算法,用于通过在有限时间间隔内对溶液边界通量的一次测量来重建Dirichlet边界条件在有界域中的波传播速度。所提出的算法的全局收敛性自然来自于相应的反问题的Lipschitz稳定性的证明,对于足够大的观测时间和边界,使用全局Carleman不等式既可以证明Lipschitz稳定性。传播速度应该与时间无关,但是在空间中会随着边界处已知的迹线和法线导数变化,并且属于某个可允许的集合,该集合限制了相对于给定外部点$ x_0 $的速度波动。为了恢复速度,我们还需要一个单独的实验,该实验的初始速度为零,初始变形为零,且在$ x-x_0 $方向上具有一些单调性。我们在离散环境中进行数值模拟,以说明和验证该算法在空间的一维和二维中的可行性。正如理论上所证明的,我们验证了即使在测量中存在小的随机干扰的情况下,对于任何可接受的初始猜测都可以实现数值重构。
更新日期:2021-04-20
down
wechat
bug