Conjugate Gradient Method for the Solution of Inverse Problems: Application in Linear Seismic Tomography

Authors

  • Tuanny Elyz Brandeleiro Brufati UFPR
  • Saulo Pomponet Oliveira UFPR
  • Amin Bassrei UFBA

DOI:

https://doi.org/10.5540/tema.2015.016.03.0185

Keywords:

Seismic tomography, Conjugate gradient method, Truncated iteration

Abstract

We consider the conjugate gradient method for the normal equations in the solution of discrete ill-posed problems arising from seismic tomography. We use a linear approach of traveltime tomography that is characterized by an ill-conditioned linear system whose unknowns are the slownesses in each block of the computational domain. The algorithms considered in this work regularize the linear system by stopping the conjugate gradient method in an early iteration. They do not depend on the singular-value decomposition and represent an attractive and economic alternative for large-scale problems.  We review two recently proposed stopping criteria and propose a modified stopping criterion that takes into account the oscillations in the approximate solution.

References

A. H. Andersen and A. C. Kak. Digital ray tracing in two-dimensional refractive fields. Journal of the Acoustic Society of America, 72(5):1593-1606, 1982.

F. S. V. Bazán. Fixed-point iterations in determining the Tikhonov regularization parameter. Inverse Problems, 24(3):035001, 2008.

F. S. V. Bazán, M. C. C. Cunha, and L. S. Borges. Extension of GKB-FP algorithm to large-scale general-form Tikhonov regularization. Numerical Linear Algebra with Applications, 21(3):316-339, 2014.

R. Chan, J. Nagy, and R. Plemmons. FFT-based preconditioners for

Toeplitz-block least squares problems. SIAM Journal on Numerical Analysis, 30(6):1740-1768, 1993.

H. Fleming. Equivalence of regularization and truncated iteration in the solution of ill-posed image reconstruction problems. Linear Algebra and its Applications, 130:133-150, 1990.

P. C. Hansen. Rank-defficient and discrete ill-posed problems. SIAM, Philadelphia, 1998.

S. Ivansson. Seismic borehole tomography - theory and computational methods. Proceedings of the IEEE, 74(2):328-338, 1986.

E.-J. Lee, H. Huang, J. Dennis, P. Chen, and L. Wang. An optimized parallel LSQR algorithm for seismic tomography. Computers and Geosciences, 61:184-197, 2013.

G. Nolet. Solving or resolving inadequate and noisy tomographic systems. Journal of Computational Physics, 61(3):463-482, 1985.

C. Paige and M. Saunders. LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Transactions on Mathematical Software, 8(1):43-71, 1982.

E. Santos and A. Bassrei. L- and Theta-curve approaches for the selection of regularization parameter in geophysical diffraction tomography. Computers and Geosciences, 33(5):618 - 629, 2007.

E. T. F. Santos, A. Bassrei, and J. Costa. Evaluation of L-curve and Theta-curve approaches for the selection of regularization parameter in anisotropic traveltime tomography. Journal of Seismic Exploration, 15:245-272, 2006.

R. J. Santos. Equivalence of regularization and truncated iteration for general ill-posed problems. Linear Algebra and its Applications, 236:25-33, 1996.

R. J. Santos. Preconditioning conjugate gradient with symmetric algebraic reconstruction technique (ART) in computerized tomography. Applied Numerical Mathematics, 47(2):255 - 263, 2003.

R. J. Santos and A. de Pierro. The effect of the nonlinearity on GCV applied to conjugate gradients in computerized tomography. Computational and Applied Mathematics, 25:111-128, 2006.

J. Scales. Tomographic inversion via the conjugate gradient method. Geophysics, 52(2):179-185, 1987.

H. A. Shots. Well-to-well and well-to-surface seismic tomography using direct waves (in Portuguese). M.Sc. Dissertation, Universidade Federal da Bahia, 1990.

D. M. Titterington. General structure of regularization procedures in image reconstruction. Astronomy and Astrophysics, 144:381-387, 1985.

A. van der Sluis and H. van der Vorst. SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems. Linear Algebra and its Applications, 130:257-303, 1990.

J. VanDecar and R. Snieder. Obtaining smooth solutions to large, linear, inverse problems. Geophysics, 59(5):818-829, 1994.

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Published

2016-01-28

How to Cite

Brufati, T. E. B., Oliveira, S. P., & Bassrei, A. (2016). Conjugate Gradient Method for the Solution of Inverse Problems: Application in Linear Seismic Tomography. Trends in Computational and Applied Mathematics, 16(3), 185. https://doi.org/10.5540/tema.2015.016.03.0185

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Section

Original Article