Comparing RSVD and Krylov methods for linear inverse problems
In this work we address regularization parameter estimation for ill-posed linear inverse problems with an ℓ2 penalty. Regularization parameter selection is of utmost importance for all of inverse problems and estimating it generally relies on the experience of the practitioner. For regularization wi...
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| Published in: | Computers & geosciences Vol. 137; p. 104427 |
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01.04.2020
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| Abstract | In this work we address regularization parameter estimation for ill-posed linear inverse problems with an ℓ2 penalty. Regularization parameter selection is of utmost importance for all of inverse problems and estimating it generally relies on the experience of the practitioner. For regularization with an ℓ2 penalty there exist a lot of parameter selection methods that exploit the fact that the solution and the residual can be written in explicit form. Parameter selection methods are functionals that depend on the regularization parameter where the minimizer is the desired regularization parameter that should lead to a good solution. Evaluation of these parameter selection methods still requires solving the inverse problem multiple times. Efficient evaluation of the parameter selection methods can be done through model order reduction. Two popular model order reduction techniques are Lanczos based methods (a Krylov subspace method) and the Randomized Singular Value Decomposition (RSVD). In this work we compare the two approaches. We derive error bounds for the parameter selection methods using the RSVD. We compare the performance of the Lanczos process versus the performance of RSVD for efficient parameter selection. The RSVD algorithm we use is based on the Adaptive Randomized Range Finder algorithm which allows for easy determination of the dimension of the reduced order model. Some parameter selection also require the evaluation of the trace of a large matrix. We compare the use of a randomized trace estimator versus the use of the Ritz values from the Lanczos process. The examples we use for our experiments are two model problems from geosciences.
•Overview of some parameter selection methods with references to the relevant literature.•Comparison of Krylov based methods versus the RSVD for parameter selection.•Error bounds for TSVD and RSVD for parameter selection methods.•Comparison of trace estimators for different types of inverse problems. |
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| AbstractList | In this work we address regularization parameter estimation for ill-posed linear inverse problems with an ℓ2 penalty. Regularization parameter selection is of utmost importance for all of inverse problems and estimating it generally relies on the experience of the practitioner. For regularization with an ℓ2 penalty there exist a lot of parameter selection methods that exploit the fact that the solution and the residual can be written in explicit form. Parameter selection methods are functionals that depend on the regularization parameter where the minimizer is the desired regularization parameter that should lead to a good solution. Evaluation of these parameter selection methods still requires solving the inverse problem multiple times. Efficient evaluation of the parameter selection methods can be done through model order reduction. Two popular model order reduction techniques are Lanczos based methods (a Krylov subspace method) and the Randomized Singular Value Decomposition (RSVD). In this work we compare the two approaches. We derive error bounds for the parameter selection methods using the RSVD. We compare the performance of the Lanczos process versus the performance of RSVD for efficient parameter selection. The RSVD algorithm we use is based on the Adaptive Randomized Range Finder algorithm which allows for easy determination of the dimension of the reduced order model. Some parameter selection also require the evaluation of the trace of a large matrix. We compare the use of a randomized trace estimator versus the use of the Ritz values from the Lanczos process. The examples we use for our experiments are two model problems from geosciences.
•Overview of some parameter selection methods with references to the relevant literature.•Comparison of Krylov based methods versus the RSVD for parameter selection.•Error bounds for TSVD and RSVD for parameter selection methods.•Comparison of trace estimators for different types of inverse problems. In this work we address regularization parameter estimation for ill-posed linear inverse problems with an ℓ2 penalty. Regularization parameter selection is of utmost importance for all of inverse problems and estimating it generally relies on the experience of the practitioner. For regularization with an ℓ2 penalty there exist a lot of parameter selection methods that exploit the fact that the solution and the residual can be written in explicit form. Parameter selection methods are functionals that depend on the regularization parameter where the minimizer is the desired regularization parameter that should lead to a good solution. Evaluation of these parameter selection methods still requires solving the inverse problem multiple times. Efficient evaluation of the parameter selection methods can be done through model order reduction. Two popular model order reduction techniques are Lanczos based methods (a Krylov subspace method) and the Randomized Singular Value Decomposition (RSVD). In this work we compare the two approaches. We derive error bounds for the parameter selection methods using the RSVD. We compare the performance of the Lanczos process versus the performance of RSVD for efficient parameter selection. The RSVD algorithm we use is based on the Adaptive Randomized Range Finder algorithm which allows for easy determination of the dimension of the reduced order model. Some parameter selection also require the evaluation of the trace of a large matrix. We compare the use of a randomized trace estimator versus the use of the Ritz values from the Lanczos process. The examples we use for our experiments are two model problems from geosciences. |
| ArticleNumber | 104427 |
| Author | van Leeuwen, Tristan Luiken, Nick |
| Author_xml | – sequence: 1 givenname: Nick surname: Luiken fullname: Luiken, Nick email: n.a.luiken@uu.nl – sequence: 2 givenname: Tristan surname: van Leeuwen fullname: van Leeuwen, Tristan email: t.vanleeuwen@uu.nl |
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| Cites_doi | 10.1137/090771806 10.1016/0041-5553(84)90253-2 10.1016/j.cageo.2007.02.003 10.1080/00401706.1979.10489751 10.1137/0914086 10.1190/geo2017-0386.1 10.1023/A:1022383005969 10.1137/0904012 10.1137/S0895479899345960 10.1088/0266-5611/24/3/034006 10.1007/BF01937276 10.1016/j.cageo.2018.09.005 10.1016/j.matcom.2011.01.016 10.1088/0266-5611/22/5/021 10.1088/0266-5611/29/8/085008 10.1080/10618600.1997.10474725 10.1016/0377-0427(96)00018-0 10.1080/03610919008812866 10.1137/16M1104974 10.1137/15M1030200 10.1137/1034115 10.1016/S0098-3004(01)00009-7 10.1016/j.cam.2011.09.039 10.1137/S1064827593252672 10.1002/nla.802 |
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| Keywords | Model order reduction Randomized singular value decomposition Krylov subspaces Tikhonov regularization Regularization parameter |
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| SubjectTerms | algorithms computers geostatistics Krylov subspaces Model order reduction model validation Randomized singular value decomposition Regularization parameter spatial data Tikhonov regularization |
| Title | Comparing RSVD and Krylov methods for linear inverse problems |
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