×

zbMATH — the first resource for mathematics

IR tools: a MATLAB package of iterative regularization methods and large-scale test problems. (English) Zbl 1415.65003
Summary: This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called IR TOOLS, serves two related purposes: we provide implementations of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test problems in the form of discretizations of 2D linear inverse problems. The solvers include iterative regularization methods where the regularization is due to the semi-convergence of the iterations, Tikhonov-type formulations where the regularization is explicitly formulated in the form of a regularization term, and methods that can impose bound constraints on the computed solutions. All the iterative methods are implemented in a very flexible fashion that allows the problem’s coefficient matrix to be available as a (sparse) matrix, a function handle, or an object. The most basic call to all of the various iterative methods requires only this matrix and the right hand side vector; if the method uses any special stopping criteria, regularization parameters, etc., then default values are set automatically by the code. Moreover, through the use of an optional input structure, the user can also have full control of any of the algorithm parameters. The test problems represent realistic large-scale problems found in image reconstruction and several other applications. Numerical examples illustrate the various algorithms and test problems available in this package.

MSC:
65-04 Software, source code, etc. for problems pertaining to numerical analysis
65F10 Iterative numerical methods for linear systems
65F22 Ill-posedness and regularization problems in numerical linear algebra
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Andersen, MS; Hansen, PC, Generalized row-action methods for tomographic imaging, Numer. Algorithms, 67, 121-144, (2014) · Zbl 1302.65285
[2] Andrews, H., Hunt, B.: Digital image restoration. Prentice-Hall, Englewood cliffs NJ (1977)
[3] Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imag. Sci., 2, 183-202, (2009) · Zbl 1175.94009
[4] Bertero, M., Boccacci, P.: Introduction to inverse problems in imaging. IOP Publishing Ltd., London (1998) · Zbl 0914.65060
[5] Bortolotti, V.; Brown, RJS; Fantazzini, P.; Landi, G.; Zama, F., Uniform penalty inversion of two-dimensional NMR relaxation data, Inverse Prob., 33, 015003, (2016) · Zbl 1361.65098
[6] Buzug, T.M.: Computed tomography. Springer, Berlin (2008)
[7] Calvetti, D.; Landi, G.; Reichel, L.; Sgallari, F., Non-negativity and iterative methods for ill-posed problems, Inverse Prob., 20, 1747-1758, (2004) · Zbl 1077.65041
[8] Calvetti, D.; Lewis, B.; Reichel, L., GMRES-Type methods for inconsistent systems, Linear Algebra Appl., 316, 157-169, (2000) · Zbl 0963.65042
[9] Calvetti, D.; Morigi, S.; Reichel, L.; Sgallari, F., Tikhonov regularization and the L-curve for large discrete ill-posed problems, J. Comput. Appl. Math., 123, 423-446, (2000) · Zbl 0977.65030
[10] Calvetti, D.; Reichel, L.; Shuibi, A., Enriched Krylov subspace methods for ill-posed problems, Lin. Alg. Appl., 362, 257-273, (2003) · Zbl 1017.65025
[11] Chung, J.: Numerical approaches for large-scale Ill-posed inverse problems. PhD Thesis, Emory University, Atlanta (2009)
[12] Chung, J., Knepper, S., Nagy, J.G.: Large-scale inverse prob. in imaging. In: Scherzer, O. (ed.) Handbook of mathematical methods in imaging. Springer, Heidelberg (2011) · Zbl 1259.94014
[13] Chung, J.; Nagy, JG; O’Leary, DP, A weighted-GCV method for Lanczos-hybrid regularization, Electron. Trans. Numer. Anal., 28, 149-167, (2008) · Zbl 1171.65029
[14] Dong, Y.; Zeng, T., A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imag. Sci., 6, 1598-1625, (2013) · Zbl 1283.52012
[15] Elfving, T.; Hansen, PC; Nikazad, T., Semi-convergence properties of Kaczmarz’s method, Inverse Prob., 30, 055007, (2014) · Zbl 1296.65054
[16] Gazzola, S.; Nagy, JG, Generalized Arnoldi-Tikhonov method for sparse reconstruction, SIAM J. Sci. Comput., 36, b225-b247, (2014) · Zbl 1296.65061
[17] Gazzola, S.; Novati, P., Automatic parameter setting for Arnoldi-Tikhonov methods, J. Comput. Appl. Math., 256, 180-195, (2014) · Zbl 1314.65061
[18] Gazzola, S.; Novati, P.; Russo, MR, On Krylov projection methods and Tikhonov regularization, Electron. Trans. Numer. Anal., 44, 83-123, (2015) · Zbl 1312.65065
[19] Gazzola, S.; Wiaux, Y., Fast nonnegative least squares through flexible Krylov subspaces, SIAM J. Sci. Comput., 39, a655-a679, (2017) · Zbl 1365.65161
[20] Golub, GH; Heath, MT; Wahba, G., Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21, 215-223, (1979) · Zbl 0461.62059
[21] Guo, L.; Meng, X.; Shi, L., Gridding aeromagnetic data using inverse interpolation, Geophys. J. Int., 189, 1353-1360, (2012)
[22] Hansen, P.C.: Discrete inverse problems: insight and algorithms. SIAM Philadelphia (2010) · Zbl 1197.65054
[23] Hansen, PC, Regularization Tools version 4.0 for Matlab 7.3, Numer. Algorithms, 46, 189-194, (2007) · Zbl 1128.65029
[24] Hansen, PC; Jensen, TK, Noise propagation in regularizing iterations for image deblurring, Electron. Trans. Numer. Anal., 31, 204-220, (2008) · Zbl 1171.65032
[25] Hansen, P. C., Jørgensen, J. S.: AIR Tools II: Algebraic iterative reconstruction methods, improved implementation. Numer. Algor. 1-31. https://doi.org/10.1007/s11075-017-0430-x (2017)
[26] Hansen, PC; Nagy, JG; Tigkos, K., Rotational image deblurring with sparse matrices, BIT Numer. Math., 54, 649-671, (2014) · Zbl 1302.65055
[27] Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring Images: Matrices, Spectra and Filtering. SIAM, Philadelphia PA (2006)
[28] Kilmer, ME; Hansen, PC; Español, MI, A projection-based approach to general-form Tikhonov regularization. SIAM, J. Sci. Comput., 29, 315-330, (2007) · Zbl 1140.65030
[29] Lagendijk, R.L., Biemond, J.: Iterative Identification and Restoration of Images. Kluwer Academic Publishers, Boston/Dordrecht/London (1991) · Zbl 0752.68093
[30] Min, T.; Geng, B.; Ren, J., Inverse estimation of the initial condition for the heat equation, Intl. J. Pure Appl. Math., 82, 581-593, (2013) · Zbl 1302.35448
[31] Mitchell, J.; Chandrasekera, TC; Gladden, LF, Numerical estimation of relaxation and diffusion distributions in two dimensions, Prog. Nucl. Magn. Reson. Spectrosc., 62, 34-50, (2012)
[32] Nagy, JG; Palmer, K.; Perrone, L., Iterative methods for image deblurring: a Matlab object oriented approach, Numer. Algorithms, 36, 73-93, (2004) · Zbl 1048.65039
[33] Nagy, J.G., Strakoš, Z.: Enforcing nonnegativity in image reconstruction algorithms. In: Wilson, D. C. (Ed.): Mathematical Modeling, Estimation, and Imaging. Proceedings of SPIE 4121 182—190 (2000)
[34] Novati, P.; Russo, MR, A GCV-based Arnoldi-Tikhonov regularization methods, BIT Numerical Mathematis, 54, 501-521, (2014) · Zbl 1317.65104
[35] Roggemann, M.C., Welsh, B.: Imaging through Turbulence. CRC Press, Boca Raton (1996)
[36] Rodríguez, P., Wohlberg, B.: An efficient algorithm for sparse representations with \(ℓ\)\(p\) data fidelity term. Proc. 4th IEEE Andean Technical Conference (ANDESCON) (2008)
[37] Sauer, K.; Bouman, C., A local upyear strategy for iterative reconstruction from projections, IEEE Trans. Signal Proc., 41, 534-548, (1993) · Zbl 0825.92085
[38] Vogel, C.R.: Computational methods for inverse problems. SIAM Philadelphia (2002) · Zbl 1008.65103
[39] Zhdanov, M.: Geophysical Inverse Theory and Regularization Problems. Elsevier, Amsterdam (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.