A conjugate gradient like method for p-norm minimization in functional spaces

We develop an iterative algorithm to recover the minimum p -norm solution of the functional linear equation A x = b , where A : X ⟶ Y is a continuous linear operator between the two Banach spaces X = L p , 1 < p < 2 , and Y = L r , r > 1 , with x ∈ X and b ∈ Y . The algorithm is conceived w...

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Vydáno v:	Numerische Mathematik Ročník 137; číslo 4; s. 895 - 922
Hlavní autoři:	Estatico, Claudio, Gratton, Serge, Lenti, Flavia, Titley-Peloquin, David
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2017 Springer Nature B.V Springer Verlag
Témata:	Algorithms Banach spaces Computer Arithmetic Computer Science Conjugate gradient method Descent Hilbert space Iterative algorithms Iterative methods Linear equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Queuing theory Regularization Restoration Robustness (mathematics) Simulation Smoothness Theoretical 65F10 47A52 65J20 Conjugate gradient method Iterative regularization Banach spaces
ISSN:	0029-599X, 0945-3245
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Abstract	We develop an iterative algorithm to recover the minimum p -norm solution of the functional linear equation A x = b , where A : X ⟶ Y is a continuous linear operator between the two Banach spaces X = L p , 1 < p < 2 , and Y = L r , r > 1 , with x ∈ X and b ∈ Y . The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311–329, 2006 ). Indeed, the algorithm is based on using, at the n -th iteration, a linear combination of the steepest current “descent functional” A ∗ J b - A x n and the previous descent functional, where J denotes a duality map of the Banach space Y . In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p -norm solution of the functional linear equation A x = b and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of L p spaces, numerical experiments show that the method is fast, robust in terms of both restoration accuracy and stability, promotes sparsity and reduces the over-smoothness in reconstructing edges and abrupt intensity changes.
AbstractList	We develop an iterative algorithm to recover the minimum p -norm solution of the functional linear equation A x = b , where A : X ⟶ Y is a continuous linear operator between the two Banach spaces X = L p , 1 < p < 2 , and Y = L r , r > 1 , with x ∈ X and b ∈ Y . The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311–329, 2006 ). Indeed, the algorithm is based on using, at the n -th iteration, a linear combination of the steepest current “descent functional” A ∗ J b - A x n and the previous descent functional, where J denotes a duality map of the Banach space Y . In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p -norm solution of the functional linear equation A x = b and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of L p spaces, numerical experiments show that the method is fast, robust in terms of both restoration accuracy and stability, promotes sparsity and reduces the over-smoothness in reconstructing edges and abrupt intensity changes. We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation Ax=b, where A:X⟶Y is a continuous linear operator between the two Banach spaces X=Lp, 1<p<2, and Y=Lr, r>1, with x∈X and b∈Y. The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311–329, 2006). Indeed, the algorithm is based on using, at the n-th iteration, a linear combination of the steepest current “descent functional” A∗J(b−Axn) and the previous descent functional, where J denotes a duality map of the Banach space Y. In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p-norm solution of the functional linear equation Ax=b and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of Lp spaces, numerical experiments show that the method is fast, robust in terms of both restoration accuracy and stability, promotes sparsity and reduces the over-smoothness in reconstructing edges and abrupt intensity changes. We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation A x = b , where A : X ⟶ Y is a continuous linear operator between the two Banach spaces X = L p , 1 < p < 2 , and Y = L r , r > 1 , with x ∈ X and b ∈ Y . The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311–329, 2006). Indeed, the algorithm is based on using, at the n-th iteration, a linear combination of the steepest current “descent functional” A ∗ J b - A x n and the previous descent functional, where J denotes a duality map of the Banach space Y . In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p-norm solution of the functional linear equation A x = b and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of L p spaces, numerical experiments show that the method is fast, robust in terms of both restoration accuracy and stability, promotes sparsity and reduces the over-smoothness in reconstructing edges and abrupt intensity changes.
Author	Estatico, Claudio Gratton, Serge Titley-Peloquin, David Lenti, Flavia
Author_xml	– sequence: 1 givenname: Claudio surname: Estatico fullname: Estatico, Claudio organization: Dipartimento di Matematica, Università degli Studi di Genova – sequence: 2 givenname: Serge surname: Gratton fullname: Gratton, Serge organization: Université de Toulouse, INP-IRIT – sequence: 3 givenname: Flavia surname: Lenti fullname: Lenti, Flavia email: flavialenti@gmail.com organization: Institut de Recherche en Informatique de Toulouse – sequence: 4 givenname: David surname: Titley-Peloquin fullname: Titley-Peloquin, David organization: Department of Bioresource Engineering, McGill University
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Cites_doi	10.1109/TGRS.2015.2458014 10.1088/0266-5611/25/1/015013 10.1088/0266-5611/20/5/005 10.1007/s10589-012-9527-2 10.1109/TGRS.2015.2411854 10.1887/0750304359 10.1088/0266-5611/26/11/115014 10.1016/0022-247X(91)90144-O 10.1155/AAA/2006/84919 10.1088/0266-5611/21/4/007 10.1088/0266-5611/24/5/055008 10.1002/cpa.20042 10.1080/10556788.2010.549231 10.1088/0266-5611/22/1/017 10.1016/0041-5553(67)90040-7 10.1090/S0002-9904-1967-11678-1 10.1137/1.9780898719697 10.1007/978-3-662-53294-2_8 10.1007/978-94-009-2121-4 10.1515/jiip-2016-0027 10.1515/9783110255720
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Snippet	We develop an iterative algorithm to recover the minimum p -norm solution of the functional linear equation A x = b , where A : X ⟶ Y is a continuous linear... We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation A x = b , where A : X ⟶ Y is a continuous linear... We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation Ax=b, where A:X⟶Y is a continuous linear operator...
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SubjectTerms	Algorithms Banach spaces Computer Arithmetic Computer Science Conjugate gradient method Descent Hilbert space Iterative algorithms Iterative methods Linear equations Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Numerical Analysis Numerical and Computational Physics Queuing theory Regularization Restoration Robustness (mathematics) Simulation Smoothness Theoretical
Title	A conjugate gradient like method for p-norm minimization in functional spaces
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