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

#### Abstract

#### Keywords

#### Full Text:

PDF#### 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.

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

#### Article Metrics

_{Metrics powered by PLOS ALM}

### Refbacks

- There are currently no refbacks.

**Trends in Computational and Applied Mathematics**

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

Indexed in: