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 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.