A new step size rule for the superiorization method and its application in computerized tomography

In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the val...

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Veröffentlicht in:	Numerical algorithms Jg. 90; H. 3; S. 1253 - 1277
Hauptverfasser:	Nikazad, T., Abbasi, M., Afzalipour, L., Elfving, T.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.07.2022 Springer Nature B.V
Schlagworte:	Algebra Algorithms Computed tomography Computer Science Conjugate gradient method Convergence Iterative methods Least squares method Numeric Computing Numerical Analysis Original Paper Regularization Theory of Computation Tomography Strictly quasi-nonexpansive operator Convex feasibility problem Computed tomography 65F10 Convex optimization Sequential block method Perturbation resilient iterative method 47J25 Superiorization 47H14
ISSN:	1017-1398, 1572-9265, 1572-9265
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Abstract	In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T ). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known “fast iterative shrinkage-thresholding algorithm” (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.
AbstractList	In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T ). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known “fast iterative shrinkage-thresholding algorithm” (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods. In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped with a new step size rule which uses subgradient projections. The superiorization method is a two-step method where one step reduces the value of the penalty term and the other step reduces the residual of the underlying linear system (using an algorithmic operator T). For the new step size rule, we present a convergence analysis for the case when T belongs to a large subclass of strictly quasi-nonexpansive operators. To examine our algorithm numerically, we consider box constraints and use the total variation (TV) functional as a regularization term. The specific test cases are chosen from computed tomography using both noisy and noiseless data. We compare our algorithm with previously used parameters in superiorization. The T operator is based on sequential block iteration (for which our convergence analysis is valid), but we also use the conjugate gradient method (without theoretical support). Finally, we compare with the well-known “fast iterative shrinkage-thresholding algorithm” (FISTA). The numerical results demonstrate that our new step size rule improves previous step size rules for the superiorization methodology and is competitive with, and in several instances behaves better than, the other methods.
Author	Afzalipour, L. Abbasi, M. Elfving, T. Nikazad, T.
Author_xml	– sequence: 1 givenname: T. orcidid: 0000-0002-9704-2893 surname: Nikazad fullname: Nikazad, T. email: tnikazad@iust.ac.ir organization: School of Mathematics, Applied Mathematics Department, Iran University of Science and Technology – sequence: 2 givenname: M. surname: Abbasi fullname: Abbasi, M. organization: Department of Mathematics, University of Qom – sequence: 3 givenname: L. surname: Afzalipour fullname: Afzalipour, L. organization: School of Mathematics, Applied Mathematics Department, Iran University of Science and Technology – sequence: 4 givenname: T. surname: Elfving fullname: Elfving, T. organization: Department of Mathematics, Linköping University
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Cites_doi	10.1080/10556788.2016.1209500 10.1088/1361-6420/aa5d79 10.1137/S089547980138705X 10.1118/1.3504603 10.1088/0266-5611/28/3/035005 10.1137/070711712 10.1109/JSTSP.2007.910263 10.1007/978-1-84628-723-7 10.1137/050639399 10.1137/S0036144593251710 10.1088/1361-6420/33/4/044002 10.1137/1.9781611971484 10.1088/1361-6420/33/4/044001 10.1088/1361-6420/aaa49b 10.1088/0266-5611/25/11/115011 10.1137/14095724X 10.1137/1.9780898718836 10.1007/s11075-017-0430-x 10.1088/1361-6420/33/4/044006 10.1016/0022-5193(70)90109-8 10.1137/080716542 10.1118/1.4745566 10.1515/auom-2015-0046 10.1109/42.959302 10.1007/s10957-013-0408-3 10.1080/10556788.2018.1437159 10.1088/0266-5611/26/6/065008 10.1137/S1052623499362111 10.1515/jiip-2015-0082 10.1137/1.9780898719284 10.1007/978-3-642-30901-4
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Copyright	The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021 The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021.
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Keywords	Strictly quasi-nonexpansive operator Convex feasibility problem Computed tomography 65F10 Convex optimization Sequential block method Perturbation resilient iterative method 47J25 Superiorization 47H14
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Snippet	In this paper, we consider a regularized least squares problem subject to convex constraints. Our algorithm is based on the superiorization technique, equipped...
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SubjectTerms	Algebra Algorithms Computed tomography Computer Science Conjugate gradient method Convergence Iterative methods Least squares method Numeric Computing Numerical Analysis Original Paper Regularization Theory of Computation Tomography
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Title	A new step size rule for the superiorization method and its application in computerized tomography
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