Convergence analysis of a two-point gradient method for nonlinear ill-posed problems

Abstract

We perform a convergence analysis of a two-point gradient method which is based on Landweber iteration and on Nesterov's acceleration scheme. Additionally, we show the usefulness of this method via two numerical example problems based on a nonlinear Hammerstein operator and on the nonlinear inverse problem of single photon emission computed tomography.

Regularization of inverse problems by two-point gradient methods in Banach spaces

“…In this paper we will consider (3.3) with λ δ n satisfying suitable conditions to be specified later. Note that our method (3.3) requires the use of the previous two iterations at every iteration step, which follows the spirit from [14]; on the other hand, our method allows the use of a general p-convex penalty function Θ, which could be non-smooth, to reconstruct solutions with special features such as sparsity and discontinuities.…”

“…Therefore, it is necessary to find out other strategy for generating λ δ n such that (3.15) and (3.27) hold. We will adapt the discrete backtracking search (DBTS) algorithm introduced in [14] to our situation. To this end, we take a function q : N → N that is non-increasing and ∞ i=0 q(i) < ∞.…”

“…We note also that, when both X and Y are Hilbert spaces and Θ(x) = x 2 /2, our method becomes the two-point gradient methods introduced in [14] for solving inverse problems in Hilbert spaces. Unlike the method in [14], our method not only works for inverse problems in Banach spaces, but also allows the use of general convex penalty functions including the L 1 and the total variation like functions. Due to the possible non-smoothness of Θ and the non-Hilbertian structures of X and Y, we need to use tools from convex analysis and subdifferential calculus to carry out the convergence analysis.…”

“…Under certain conditions on the combination parameters {λ δ n }, we obtain a convergence result on our method. In order to find nontrivial λ δ n , we adapt the discrete backtracking search (DBTS) algorithm in [14] to our situation and provide a complete convergence analysis of the corresponding method by showing a uniform convergence result for the noise-free counterpart with the combination parameters chosen in a certain range. Our analysis in fact improves the convergence result in [14] by removing the technical conditions on {λ δ n } chosen by the DBTS algorithm.…”

Second order asymptotical regularization methods for inverse problems in partial differential equations

“…In recent years, there has been increasing evidence to show that the second order iterative methods exhibit remarkable acceleration properties for stably solving ill-posed problems. The most well-known methods are the Nesterov acceleration scheme (Neubauer (2017)), the ν-method (Engl et al, 1996, § 6.3), and the two-point gradient method (Hubmer & Ramlau (2017)). Recently, the authors in Zhang & Hofmann (2018) have established an initial theory of the second order asymptotical regularization method with fixed damping parameter for solving general linear ill-posed inverse problems.…”