Oversmoothing Tikhonov regularization in Banach spaces

Abstract

This paper develops a Tikhonov regularization theory for nonlinear ill-posed operator equations in Banach spaces. As the main challenge, we consider the so-called oversmoothing state in the sense that the Tikhonov penalization is not able to capture the true solution regularity and leads to the infinite penalty value in the solution. We establish a vast extension of the Hilbertian convergence theory through the use of invertible sectorial operators from the holomorphic functional calculus and the prominent theory of interpolation scales in Banach spaces. Applications of the proposed theory involving ${\ell }^1$, Bessel potential spaces, and Besov spaces are discussed.

References in corpus (4)

• Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales
• Paley-Littlewood decomposition for sectorial operators and interpolation spaces
• Convergence Rates of Tikhonov Regularizations for Elliptic and Parabolic Inverse Radiativity Problems
• A new approach to nonlinear constrained Tikhonov regularization