A conjugate gradient like method for p-norm minimization in functional spaces - Estatico, Claudio; Gratton, Serge; Lenti, Flavia; Titley-Peloquin, David (Numerische Mathematik, 2017)
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.