Volume 43, pp. 223-243, 2014-2015.

Matrix decompositions for Tikhonov regularization

Lothar Reichel and Xuebo Yu


Tikhonov regularization is a popular method for solving linear discrete ill-posed problems with error-contaminated data. This method replaces the given linear discrete ill-posed problem by a penalized least-squares problem. The choice of the regularization matrix in the penalty term is important. We are concerned with the situation when this matrix is of fairly general form. The penalized least-squares problem can be conveniently solved with the aid of the generalized singular value decomposition, provided that the size of the problem is not too large. However, it is impractical to use this decomposition for large-scale penalized least-squares problems. This paper describes new matrix decompositions that are well suited for the solution of large-scale penalized least-square problems that arise in Tikhonov regularization with a regularization matrix of general form.

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Key words

ill-posed problem, matrix decomposition, generalized Krylov method, Tikhonov regularization.

AMS subject classifications

65F22, 65F10, 65R30

Links to the cited ETNA articles

[14]Vol. 38 (2011), pp. 233-257 Stefan Kindermann: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems
[15]Vol. 40 (2013), pp. 58-81 Stefan Kindermann: Discretization independent convergence rates for noise level-free parameter choice rules for the regularization of ill-conditioned problems

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