Convergence of the conjugate gradient method with unbounded operators

TL;DR

This paper proves the convergence of the conjugate gradient method for inverse linear problems in infinite-dimensional Hilbert spaces with unbounded operators, extending previous results for bounded operators.

Contribution

It establishes convergence under minimal assumptions for unbounded, self-adjoint, non-negative operators, broadening the applicability of the conjugate gradient method.

Findings

Convergence holds in the Hilbert space norm.

Convergence also occurs at various regularity levels depending on iterates.

Numerical tests illustrate differences from bounded operator cases.

Abstract

In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm. The convergence is proved to always hold in the Hilbert space norm (error convergence), as well as at other levels of regularity (energy norm, residual, etc.) depending on the regularity of the iterates. We also discuss, both analytically and through a selection of numerical tests, the main features and differences of our convergence result as compared to the case, already available in the literature, where the operator is bounded.