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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Popis
Shrnutí:	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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0029-599X 0945-3245
DOI:	10.1007/s00211-017-0893-7