A Projection‐Based Approach to General‐Form Tikhonov Regularization

We present a projection-based iterative algorithm for computing general-form Tikhonov regularized solutions to the problem $\min_x\{\| Ax-b \|_2^2+\lambda^2\| Lx \|_2^2\}$, where the regularization matrix $L$ is not the identity. Our algorithm is designed for the common case where $\lambda$ is not k...

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Bibliographic Details
Published inSIAM journal on scientific computing Vol. 29; no. 1; pp. 315 - 330
Main Authors Kilmer, Misha E., Hansen, Per Christian, Español, Malena I.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2007
Subjects
Algorithms
Decomposition
Iterative methods
Noise
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ISSN1064-8275
1095-7197
DOI10.1137/050645592

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Summary:We present a projection-based iterative algorithm for computing general-form Tikhonov regularized solutions to the problem $\min_x\{\| Ax-b \|_2^2+\lambda^2\| Lx \|_2^2\}$, where the regularization matrix $L$ is not the identity. Our algorithm is designed for the common case where $\lambda$ is not known a priori. It is based on a joint bidiagonalization algorithm and is appropriate for large-scale problems when it is computationally infeasible to transform the regularized problem to standard form. By considering the projected problem, we show how estimates of the corresponding optimal regularization parameter can be efficiently obtained. Numerical results illustrate the promise of our projection-based approach.
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ISSN:1064-8275
1095-7197
DOI:10.1137/050645592