A structured L-BFGS method and its application to inverse problems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916153930022912 |
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| author | Mannel, Florian Aggrawal, Hari Om Modersitzki, Jan |
| author_facet | Mannel, Florian Aggrawal, Hari Om Modersitzki, Jan |
| contents | Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an LBFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka--Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an LBFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured LBFGS methods and classical LBFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical LBFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07296 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A structured L-BFGS method and its application to inverse problems Mannel, Florian Aggrawal, Hari Om Modersitzki, Jan Optimization and Control 65J22 65K05 65K10 90C06 90C26 90C30 90C48 90C53 90C90 Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an LBFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka--Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an LBFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured LBFGS methods and classical LBFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical LBFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available. |
| title | A structured L-BFGS method and its application to inverse problems |
| topic | Optimization and Control 65J22 65K05 65K10 90C06 90C26 90C30 90C48 90C53 90C90 |
| url | https://arxiv.org/abs/2310.07296 |