Optimal-order convergence of Nesterov acceleration for linear ill-posed problems

TL;DR

This paper proves that Nesterov acceleration achieves optimal convergence rates as an iterative regularization method for linear ill-posed problems when parameters are chosen based on solution smoothness, using Gegenbauer polynomials.

Contribution

It establishes the optimal-order convergence of Nesterov acceleration for linear ill-posed problems with parameter choice strategies, a novel theoretical result.

Findings

Nesterov acceleration is optimal-order for ill-posed problems.

Parameter choice based on smoothness ensures optimal convergence.

Representation via Gegenbauer polynomials is key to the proof.

Abstract

We show that Nesterov acceleration is an optimal-order iterative regularization method for linear ill-posed problems provided that a parameter is chosen accordingly to the smoothness of the solution. This result is proven both for an a priori stopping rule and for the discrepancy principle. The essential tool to obtain this result is a representation of the residual polynomials via Gegenbauer polynomials.