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Multidirectional subspace expansion for one-parameter and multiparameter Tikhonov regularization. (English) Zbl 1370.65021

The authors present a new method for large scale Tikhonov regularization problems which combines a multidirectional subspace expansion with optional truncation to produce a higher quality search space. The multidirectional expansion generates a richer search space whereas the truncation ensures moderate growth. The authors present lower and upper bounds on the regularization parameter when the discrepancy principle is applied to one-parameter regularization, and use numerical results to illustrate that their method can yield more accurate results or faster convergence.

MSC:

65F22 Ill-posedness and regularization problems in numerical linear algebra
65F08 Preconditioners for iterative methods
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[1] Belge, M., Kilmer, M.E., Miller, E.L.: Efficient determination of multiple regularization parameters in a generalized l-curve framework. Inverse Probl. 18(4), 1161-1183 (2002) · Zbl 1018.65073
[2] Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Numer. Math. 94(2), 203-228 (2003) · Zbl 1024.65036
[3] Calvetti, D., Reichel, L.: Tikhonov regularization of large linear problems. BIT 43(2), 263-283 (2003) · Zbl 1038.65048
[4] Fong, D., Saunders, M.A.: LSMR: an iterative algorithm for sparse least-squares problems. SIAM J. Sci. Comput. 33(5), 2950-2971 (2011) · Zbl 1232.65052
[5] Fornasier, M., Naumova, V., Pereverzyev, S.V.: Parameter choice strategies for multipenalty regularization. SIAM J. Numer. Anal. 52(4), 1770-1794 (2014) · Zbl 1304.47012
[6] Gazzola, S., Novati, P.: Multi-parameter Arnoldi-Tikhonov methods. Electron. Trans. Numer. Anal. 40, 452-475 (2013) · Zbl 1288.65084
[7] Gazzola, S., Reichel, L.: A new framework for multi-parameter regularization. BIT, 1-31 (2015) · Zbl 1353.65032
[8] Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996) · Zbl 0865.65009
[9] Groetsch, C.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman Publishing, Boston (1984) · Zbl 0545.65034
[10] Hansen, P.C.: Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems. Numer. Algorithms 6, 1-35 (1994) · Zbl 0789.65029
[11] Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia (1998) · Zbl 0890.65037
[12] Hochstenbach, M.E., Reichel, L.: An iterative method for Tikhonov regularization with a general linear regularization operator. J. Integral Equ. Appl. 22(3), 465-482 (2010) · Zbl 1210.65092
[13] Hochstenbach, M.E., Reichel, L., Yu, X.: A Golub-Kahan-type reduction method for matrix pairs. J. Sci. Comput. 65(2), 767-789 (2015) · Zbl 1329.65082
[14] Ito, K., Jin, B., Takeuchi, T.: Multi-parameter Tikhonov Regularization. arXiv:1102.1173v2 [math.NA] (2011). Preprint · Zbl 1285.65032
[15] Kilmer, M.E., Hansen, P., Español, M.: A projection-based approach to general-form Tikhonov regularization. SIAM J. Sci. Comput. 29(1), 315-330 (2007) · Zbl 1140.65030
[16] Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM J. Imaging Sci. 6(2), 938-983 (2013) · Zbl 1280.49053
[17] Lampe, J., Reichel, L., Voss, H.: Large-scale Tikhonov regularization via reduction by orthogonal projection. Linear Algebra Appl. 436(8), 2845-2865 (2012) · Zbl 1241.65044
[18] Li, R.C., Ye, Q.: A Krylov subspace method for quadratic matrix polynomials with applications to constrained least squares problems. SIAM J. Matrix Anal. Appl. 25(2), 405-528 (2003) · Zbl 1050.65038
[19] Lu, S., Pereverzyev, S.V.: Multi-parameter regularization and its numerical realization. Numer. Math. 118(1), 1-31 (2011) · Zbl 1221.65128
[20] Lu, S., Pereverzyev, S.V., Shao, Y., Tautenhahn, U.: Discrepancy curves for multi-parameter regularization. J. Inverse Ill-Posed Probl. 18(6), 655-676 (2010) · Zbl 1280.47016
[21] Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8(1), 43-71 (1982) · Zbl 0478.65016
[22] Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629-639 (1990)
[23] Reichel, L., Sgallari, F., Ye, Q.: Tikhonov regularization based on generalized Krylov subspace methods. Appl. Numer. Math. 62(9), 1215-1228 (2012) · Zbl 1246.65068
[24] Reichel, L., Yu, X.: Matrix decompositions for Tikhonov regularization. Electron. Trans. Numer. Anal. 43, 223-243 (2015) · Zbl 1327.65076
[25] Reichel, L., Yu, X.: Tikhonov regularization via flexible Arnoldi reduction. BIT 55(4), 1145-1168 (2015) · Zbl 1332.65058
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