The iterated Golub-Kahan-Tikhonov method
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914379037933568 |
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| author | Bianchi, Davide Donatelli, Marco Furchì, Davide Reichel, Lothar |
| author_facet | Bianchi, Davide Donatelli, Marco Furchì, Davide Reichel, Lothar |
| contents | The Golub-Kahan-Tikhonov method is a popular solution technique for large linear discrete ill-posed problems. This method first applies partial Golub-Kahan bidiagonalization to reduce the size of the given problem and then uses Tikhonov regularization to compute a meaningful approximate solution of the reduced problem. It is well known that iterated variants of this method often yield approximate solutions of higher quality than the standard non-iterated method. Moreover, it produces more accurate computed solutions than the Arnoldi method when the matrix that defines the linear discrete ill-posed problem is far from symmetric.
This paper starts with an ill-posed operator equation in infinite-dimensional Hilbert space, discretizes the equation, and then applies the iterated Golub-Kahan-Tikhonov method to the solution of the latter problem. An error analysis that addresses all discretization and approximation errors is provided. Additionally, a new approach for choosing the regularization parameter is described. This solution scheme produces more accurate approximate solutions than the standard (non-iterated) Golub-Kahan-Tikhonov method and the iterated Arnoldi-Tikhonov method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The iterated Golub-Kahan-Tikhonov method Bianchi, Davide Donatelli, Marco Furchì, Davide Reichel, Lothar Numerical Analysis The Golub-Kahan-Tikhonov method is a popular solution technique for large linear discrete ill-posed problems. This method first applies partial Golub-Kahan bidiagonalization to reduce the size of the given problem and then uses Tikhonov regularization to compute a meaningful approximate solution of the reduced problem. It is well known that iterated variants of this method often yield approximate solutions of higher quality than the standard non-iterated method. Moreover, it produces more accurate computed solutions than the Arnoldi method when the matrix that defines the linear discrete ill-posed problem is far from symmetric. This paper starts with an ill-posed operator equation in infinite-dimensional Hilbert space, discretizes the equation, and then applies the iterated Golub-Kahan-Tikhonov method to the solution of the latter problem. An error analysis that addresses all discretization and approximation errors is provided. Additionally, a new approach for choosing the regularization parameter is described. This solution scheme produces more accurate approximate solutions than the standard (non-iterated) Golub-Kahan-Tikhonov method and the iterated Arnoldi-Tikhonov method. |
| title | The iterated Golub-Kahan-Tikhonov method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.12307 |