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 under Hölder source conditions. Furthermore, some converse results and logarithmic rates are verified. The essential tool to obtain these results is a representation of the residual polynomials via Gegenbauer polynomials.
Highlights
One option to calculate a regularized solution to a linear ill-posed problem Ax = y, with A : X → Y linear and bounded and X, Y being Hilbert spaces, when only noisy data yδ with y − yδ = δ are available is to employ iterative regularization schemes
We show that Nesterov acceleration is an optimal-order iterative regularization method for linear ill-posed problems provided that a parameter is chosen to the smoothness of the solution
The essential tool to obtain these results is a representation of the residual polynomials via Gegenbauer polynomials
Summary
We remark that other alternatives for the sequence αk are possible as well, but for the main analysis of this paper we only consider (2) This iteration (in a general nonlinear context) was suggested by Yurii Nesterov for general convex optimization problems [14]. The background and main motivation of the present article is the recent interesting work of Neubauer [15] for ill-posed problems in the linear case He showed that (1) is an iterative regularization scheme and, more important, proved convergence rates, which are of optimal order only for a priori parameter choices and in case of low smoothness of the solution while being suboptimal otherwise. The index δ of yδ indicates noisy data, and analogous, xδk denotes the iterates of (1) with noisy data yδ, while the lack of δ indicates exact data y and correspondingly the iteration xk with exact data y in place of yδ in (1)
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