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 $$Ax = b,$$Ax=b, where $$A: \mathcal {X}\longrightarrow \mathcal {Y}\,$$A:X⟶Y is a continuous linear operator between the two Banach spaces $$\mathcal {X}= L^p$$X=Lp, $$1< p < 2$$1 1$$r>1, with $$x \in \mathcal {X}$$x∈X and $$b \in \mathcal {Y}$$b∈Y. The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311–329, 2006). Indeed, the algorithm is based on using, at the n-th iteration, a linear combination of the steepest current “descent functional” $$A^* J \left( b - A x_n \right) $$A∗Jb-Axn and the previous descent functional, where J denotes a duality map of the Banach space $$\mathcal {Y}$$Y. In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p-norm solution of the functional linear equation $$Ax = b$$Ax=b and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of $$L^p$$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.