Subspace Recycling-based Regularization Methods

TL;DR

This paper extends subspace recycling techniques to ill-posed problems in Hilbert spaces, demonstrating their regularization properties and developing an augmented gradient descent method effective in image reconstruction tasks.

Contribution

It introduces a framework showing subspace recycling as a regularization for ill-posed problems and develops an augmented gradient descent method within this framework.

Findings

Recycling methods satisfy regularization conditions under certain assumptions.

The augmented gradient descent method improves convergence in ill-posed problems.

Effective in applications like Gaussian blur models and astronomical image reconstruction.

Abstract

Subspace recycling techniques have been used quite successfully for the acceleration of iterative methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace generated iteratively by a known algorithm with a fixed subspace of vectors which are ``useful'' for solving the problem. Often, this has the effect of inducing a projected version of the original linear system to which the known iterative method is then applied, and this projection can act as a deflation preconditioner, accelerating convergence. Most often, these methods have been applied for the solution of well-posed problems. However, they have also begun to be considered for the solution of ill-posed problems. In this paper, we consider subspace augmentation-type iterative schemes applied to linear ill-posed problems in a continuous Hilbert space setting, based on a recently developed framework describing these methods. We show that under suitable assumptions, a recycling method satisfies the formal definition of a regularization, as long as the underlying scheme is itself a regularization. We then develop an augmented subspace version of the gradient descent method and demonstrate its effectiveness, both on an academic Gaussian blur model and on problems arising from the adaptive optics community for the resolution of large sky images by ground-based extremely large telescopes.