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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Volume 22, Issue 3


A nonlinear multigrid method for inversion of two-dimensional acoustic wave equation

Jingjun Zhao / Tao Liu / Guofeng Feng
Published Online: 2014-01-14 | DOI: https://doi.org/10.1515/jip-2012-0060


We investigate the problem of estimating the velocity in a two-dimensional acoustic wave equation, which plays an important role in geological survey. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. To reduce the computational burden, a nonlinear multigrid method is applied to solve this inverse problem. In the multigrid inversion process, in order to make the objective functionals at different scales compatible, they are dynamically adjusted. In this way, the necessary condition of “the optimal solution should be the fixed point of multigrid inversion” can be met. The stable and fast regularized Gauss–Newton method is applied to each grid. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation, improve the inversion results, and have the anti-noise ability.

Keywords: Nonlinear multigrid method; inverse problems; Tikhonov regularization; acoustic wave equation; geological exploration

MSC: 35R30; 35R05; 65M55; 86A60

About the article

Received: 2012-08-31

Published Online: 2014-01-14

Published in Print: 2014-06-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11101109

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11271102

Funding Source: Natural Science Foundation of Hei-long-jiang Province of China

Award identifier / Grant number: A201107

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 22, Issue 3, Pages 429–448, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jip-2012-0060.

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