×

A Golub-Kahan-type reduction method for matrix pairs. (English) Zbl 1329.65082

Summary: We describe a novel method for reducing a pair of large matrices \(\{A,B\}\) to a pair of small matrices \(\{H,K\}\). The method is an extension of Golub-Kahan bidiagonalization to matrix pairs, and simplifies to the latter method when \(B\) is the identity matrix. Applications to Tikhonov regularization of large linear discrete ill-posed problems are described. In these problems the matrix \(A\) represents a discretization of a compact integral operator and \(B\) is a regularization matrix.

MSC:

65F22 Ill-posedness and regularization problems in numerical linear algebra
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arioli, M., Orban, D.: Iterative methods for symmetric quasi-definite linear systems. Part I: theory, Cahier du GERAD G-2013-32, GERAD (2013) · Zbl 1267.65045
[2] Belge, M., Kilmer, M.E., Miller, E.L.: Efficient selection of multiple regularization parameters in a generalized L-curve framework. Inverse Probl. 28, 1161-1183 (2002) · Zbl 1018.65073
[3] Brezinski, C., Redivo-Zaglia, M., Rodriguez, G., Seatzu, S.: Multi-parameter regularization techniques for ill-conditioned linear systems. Numer. Math. 94, 203-224 (2003) · Zbl 1024.65036
[4] Brezinski, C., Rodriguez, G., Seatzu, S.: Error estimates for the regularization of least squares problems. Numer. Algorithms 51, 61-76 (2009) · Zbl 1166.65331
[5] 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
[6] Calvetti, D., Reichel, L.: Tikhonov regularization of large linear problems. BIT 43, 263-283 (2003) · Zbl 1038.65048
[7] Eldén, L.: A weighted pseudoinverse, generalized singular values, and constrained least squares problems. BIT 22, 487-501 (1982) · Zbl 0509.65019
[8] Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996) · Zbl 0859.65054
[9] Gazzola, S., Novati, P.: Multi-parameter Arnoldi-Tikhonov methods. Electron. Trans. Numer. Anal. 40, 452-475 (2013) · Zbl 1288.65084
[10] Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston (1984) · Zbl 0545.65034
[11] Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia (1998) · Zbl 0890.65037
[12] Hansen, P.C.: Regularization tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189-194 (2007) · Zbl 1128.65029
[13] Hochstenbach, M.E., Reichel, L.: An iterative method for Tikhonov regularization with a general linear regularization operator. J. Integral Equ. Appl. 22, 465-482 (2010) · Zbl 1210.65092
[14] Hoffnung, L., Li, R.-C., Ye, Q.: Krylov type subspace methods for matrix polynomials. Linear Algebra Appl. 415, 52-81 (2006) · Zbl 1104.65032
[15] Kilmer, M.E., Hansen, P.C., Español, M.I.: A projection-based approach to general-form Tikhonov regularization. SIAM J. Sci. Comput. 29, 315-330 (2007) · Zbl 1140.65030
[16] Kindermann, S.: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems. Electron. Trans. Numer. Anal. 38, 233-257 (2011) · Zbl 1287.65043
[17] Li, R.-C., Ye, Q.: A Krylov subspace method for quadratic matrix polynomials with application to constrained least squares problems. SIAM J. Matrix Anal. Appl. 25, 405-428 (2003) · Zbl 1050.65038
[18] Lu, S., Pereverzev, S.V.: Multi-parameter regularization and its numerical realization. Numer. Math. 118, 1-31 (2011) · Zbl 1221.65128
[19] Morigi, S., Reichel, L., Sgallari, F.: Orthogonal projection regularization operators. Numer. Algorithms 44, 99-114 (2007) · Zbl 1124.65043
[20] Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8, 43-71 (1982) · Zbl 0478.65016
[21] Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639 (1990)
[22] Phillips, D.L.: A Technique for the numerical solution of certain integral equations of the first kind. J. ACM 9, 84-97 (1962) · Zbl 0108.29902
[23] Reichel, L., Rodriguez, G.: Old and new parameter choice rules for discrete ill-posed problems. Numer. Algorithms 63, 65-87 (2013) · Zbl 1267.65045
[24] Reichel, L., Sgallari, F., Ye, Q.: Tikhonov regularization based on generalized Krylov subspace methods. Appl. Numer. Math. 62, 1215-1228 (2012) · Zbl 1246.65068
[25] Reichel, L., Yu, X.: Tikhonov regularization via flexible Arnoldi reduction. BIT Numer. Math. doi:10.1007/s10543-014-0542-9 · Zbl 1332.65058
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.