A standard solution technique for linear operator equations of first kind is the Landweber scheme which is an iterative method that uses the negative gradient of the current residual as search direction, which is then also called the Landweber direction. Though this method proves to be stable with respect to noisy data, it is known to be numerically slow for problems in Hilbert spaces and this behavior shows to be even worse in some Banach space settings. This is why the idea came up to use several search directions instead of the Landweber direction only which has led to the development of Sequential Subspace Optimization (SESOP) methods. This idea is related to Conjugate Gradient (CG) techniques that are known to be amongst the most effective methods to solve linear equations in Hilbert spaces. SESOP methods in Banach spaces do not share the conjugacy property with CG methods. In this talk we present the concept of generalized (g-)orthogonality in Banach spaces and apply metric projections to orthogonalize the current Landweber direction with respect to the search space of the last iteration. This leads to an accelerated SESOP method which is confirmed by various numerical experiments.
(joined work with Frederik Heber and Frank Schoepfer)