Alternating step size method for solving ill-posed linear operator equations in energetic space

Abstract

An alternating step size method for solving ill-posed linear operator equations is introduced in energetic space, a subspace of the Hilbert space. Focus is on the case of a positive bounded self-conjugated operator under the assumption that the error for the right hand side of the equation is available. The advantage of the energetic norm is that does not require smoothness of the exact solution x. Convergence of the new method is discussed and shown to be faster for similar error estimates. Furthermore, the conditions for convergence in the energetic norm are derived which also imply convergence in the original Hilbert space norm. The new method is demonstrated on several numerical examples of the integral equation in L20,1 space, taken from the class of inverse problems in potential theory, and convergence speed compared with that of the fixed step size method. Finally, a novel real-word application of the integral equation is proposed: deblurring averages of trial-to-trial jittered EEG responses. When applying the Kosambi–Hilbert Torsion (KTH) to align these responses, the jitter is more effectively reduced and alignment improves with the deblurred average as a reference than with the original average.