Accelerated Landweber methods based on co-dilated orthogonal polynomials

TL;DR

This paper introduces accelerated Landweber methods that leverage co-dilated orthogonal polynomials to improve convergence in solving linear ill-posed problems, especially when spectra are clustered at zero.

Contribution

It proposes a novel modification of u-methods using co-dilated ultraspherical polynomials, enhancing decay of residuals and convergence without increasing order.

Findings

Enhanced decay of residual polynomials at the origin.

Maintained convergence order of original u-methods.

Numerical tests confirm improved performance with adaptive dilation.

Abstract

In this article, we introduce and study accelerated Landweber methods for linear ill-posed problems obtained by an alteration of the coefficients in the three-term recurrence relation of the \nu-methods. The residual polynomials of the semi-iterative methods under consideration are linked to a family of co-dilated ultraspherical polynomials. This connection makes it possible to increase the decay of the residual polynomials at the origin by means of a dilation parameter. This increased decay has advantages when solving linear ill-posed equations in which the spectrum of the involved operators is clustered at the origin. The convergence order of the new semi-iterative methods turns out to be the same as the convergence order of the original \nu-methods. The new algorithms are tested numerically and a simple adaptive scheme is developed in which an optimal dilation parameter is computed.