On Nesterov acceleration for Landweber iteration of linear ill-posed problems

Abstract

Abstract In this paper we deal with Nesterov acceleration and show that it speeds up Landweber iteration when applied to linear ill-posed problems. It is proven that, if the exact solution x † ∈ ℛ ⁢ ( ( T * ⁢ T ) μ ) {x^{\dagger}\in{\cal R}((T^{*}T)^{\mu})} , then optimal convergence rates are obtained if μ ≤ 1 2 {\mu\leq\frac{1}{2}} and if the iteration is terminated according to an a priori stopping rule. If μ > 1 2 {\mu>\frac{1}{2}} or if the iteration is terminated according to the discrepancy principle, only suboptimal convergence rates can be guaranteed. Nevertheless, the number of iterations for Nesterov acceleration is always much smaller if the dimension of the problem is large. Numerical results verify the theoretical ones.