A Compressive Landweber Iteration for Solving Ill-Posed Inverse Problems

In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear ill-posed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for many applications the implementation of Landweber’s method is numerically very intensive. Therefore we propose an adaptive variant of Landweber’s iteration that significantly may reduce the computational expense, i.e. leading to a compressed version of Landweber’s iteration. We lend the concept of adaptivity that was primarily developed for well-posed operator equations (in particular, for elliptic PDE’s) essentially exploiting the concept of wavelets (frames), Besov regularity, best N -term approximation and combine it with classical iterative regularization schemes. As the main result of this paper we define an adaptive variant of Landweber’s iteration. In combination with an adequate refinement/stopping rule (a-priori as well as a-posteriori principles) we prove that the proposed procedure is an regularization method which converges in norm for exact and noisy data. The proposed approach is verified in the field of computerized tomography imaging. MSC: 15A29, 47A52, 68U10, 94A08, 94A40