Abstract
We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation \(Ax = b,\) where \(A: \mathcal {X}\longrightarrow \mathcal {Y}\,\) is a continuous linear operator between the two Banach spaces \(\mathcal {X}= L^p\), \(1< p < 2\), and \(\mathcal {Y}= L^r\), \(r > 1\), with \(x \in \mathcal {X}\) and \(b \in \mathcal {Y}\). 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” \(A^* J \left( b - A x_n \right) \) and the previous descent functional, where J denotes a duality map of the Banach space \(\mathcal {Y}\). 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 \(L^p\) 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.
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Acknowledgements
The work of C. Estatico is partly supported by PRIN 2012 2012MTE38N and GNCS-INdAM. The work of F. Lenti is supported by the AVENUE Project of the RTRA STAE foundation.
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Estatico, C., Gratton, S., Lenti, F. et al. A conjugate gradient like method for p-norm minimization in functional spaces. Numer. Math. 137, 895–922 (2017). https://doi.org/10.1007/s00211-017-0893-7
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DOI: https://doi.org/10.1007/s00211-017-0893-7