Abstract
Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. Regularization aims to reduce this sensitivity. Typically, regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term, and a q-norm to measure the regularization term, has received considerable attention. The relative importance of these terms is determined by a regularization parameter. This paper discussed how the latter parameter can be determined with the aid of the discrepancy principle. We primarily focus on the situation when \(p=2\) and \(0<q\le 2\), where we note that when \(0<q<1\), the minimization problem may be non-convex.
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Brill, M., Schock, E.: Iterative solution of ill-posed problems—a survey, in Model Optimization in Exploration Geophysics, ed. A. Vogel, Vieweg, Braunschweig, pp. 17–37 (1987)
Buccini, A.: Regularizing preconditioners by non-stationary iterated Tikhonov with general penalty term. Appl. Numer. Math. 116, 64–81 (2017)
Cai, J.-F., Chan, R.H., Shen, L., Shen, Z.: Simultaneously inpainting in image and transformed domains. Numer. Math. 112, 509–533 (2009)
Cai, J.-F., Chan, R.H., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmonic Anal. 24, 131–149 (2008)
Cai, J.-F., Chan, R.H., Shen, Z.: Linearized Bregman iterations for frame-based image deblurring. SIAM J. Imaging Sci. 2, 226–252 (2009)
Cai, J.-F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8, 337–369 (2009)
Chan, R.H., Liang, H.X.: Half-quadratic algorithm for \(\ell _p\)-\(\ell _q\) problems with applications to TV-\(\ell _1\) image restoration and compressive sensing. In: Proceedings of Efficient Algorithms for Global Optimization Methods in Computer Vision, Lecture Notes in Comput. Sci. # 8293, pp. 78–103. Springer, Berlin (2014)
Daniel, J.W., Gragg, W.B., Kaufman, L., Stewart, G.W.: Reorthogonalization and stable algorithms for updating the Gram–Schmidt QR factorization. Math. Comput. 30, 772–795 (1976)
Donatelli, M., Hanke, M.: Fast nonstationary preconditioned iterative methods for ill-posed problems with application to image deblurring. Inverse Probl. 29, 095008 (2013)
Donatelli, M., Huckle, T., Mazza, M., Sesana, D.: Image deblurring by sparsity constraint on the Fourier coefficients. Numer. Algorithms 72, 341–361 (2016)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)
Estatico, C., Gratton, S., Lenti, F., Titley-Peloquin, D.: A conjugate gradient like method for \(p\)-norm minimization in functional spaces. Numer. Math. 137, 895–922 (2017)
Gazzola, S., Nagy, J.G.: Generalized Arnoldi–Tikhonov method for sparse reconstruction. SIAM J. Sci. Comput. 36, B225–B247 (2014)
Gazzola, S., Novati, P., Russo, M.R.: On Krylov projection methods and Tikhonov regularization. Electron. Trans. Numer. Anal. 44, 83–123 (2015)
Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3, 253–315 (1993)
Hanke, M., Groetsch, C.W.: Nonstationary iterated Tikhonov regularization. J. Optim. Theory Appl. 98, 37–53 (1998)
Hansen, P.C.: Rank Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998)
Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM, Philadelphia (2006)
Hiriart-Urruty, J.-P., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, New York (2004)
Huang, G., Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: Majorization-minimization generalized Krylov subspace methods for \(\ell _p-\ell _q\) optimization applied to image restoration. BIT Numer. Math. 57, 351–378 (2017)
Huang, G., Reichel, L., Yin, F.: Projected nonstationary iterated Tikhonov regularization. BIT Numer. Math. 56, 467–487 (2016)
Huang, J., Donatelli, M., Chan, R.H.: Nonstationary iterated thresholding algorithms for image deblurring. Inverse Probl. Imaging 7, 717–736 (2013)
Lampe, J., Reichel, L., Voss, H.: Large-scale Tikhonov regularization via reduction by orthogonal projection. Linear Algebra Appl. 436, 2845–2865 (2012)
Lanza, A., Morigi, S., Reichel, L., Sgallari, F.: A generalized Krylov subspace method for \(\ell _p-\ell _q\) minimization. SIAM J. Sci. Comput. 37, S30–S50 (2015)
Rodríguez, P., Wohlberg, B.: Efficient minimization method for a generalized total variation functional. IEEE Trans. Image Process. 18, 322–332 (2009)
Serra-Capizzano, S.: A note on antireflective boundary conditions and fast deblurring models. SIAM J. Sci. Comput. 25, 1307–1325 (2004)
Wolke, R., Schwetlick, H.: Iteratively reweighted least squares: algorithms, convergence analysis, and numerical comparisons. SIAM J. Sci. Stat. Comput. 9, 907–921 (1988)
Acknowledgements
The authors would like to thank the referees for their insightful comments that improved the presentation. The first author is a member of the INdAM Research group GNCS and his work is partially founded by the group. The second author is supported in part by NSF Grants DMS-1720259 and DMS-1729509.
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Buccini, A., Reichel, L. An \(\ell ^2-\ell ^q\) Regularization Method for Large Discrete Ill-Posed Problems. J Sci Comput 78, 1526–1549 (2019). https://doi.org/10.1007/s10915-018-0816-5
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DOI: https://doi.org/10.1007/s10915-018-0816-5