"FISTA" in Banach spaces with adaptive discretisations

TL;DR

This paper extends the FISTA algorithm to Banach spaces with adaptive discretisations, allowing for convergence guarantees in more general settings like L1-penalised reconstructions, with theoretical and numerical validation.

Contribution

It introduces a modified FISTA algorithm with adaptive subsets in Banach spaces, providing convergence conditions and analysis for L1-penalised problems.

Findings

Convergence guaranteed under new conditions in Banach spaces.

Reduced convergence rate depending on problem conditioning.

Numerical results demonstrate practical effectiveness.

Abstract

FISTA is a popular convex optimisation algorithm which is known to converge at an optimal rate whenever a minimiser is contained in a suitable Hilbert space. We propose a modified algorithm where each iteration is performed in a subset which is allowed to change at every iteration. Sufficient conditions are provided for guaranteed convergence, although at a reduced rate depending on the conditioning of the specific problem. These conditions have a natural interpretation when a minimiser exists in an underlying Banach space. Typical examples are L1-penalised reconstructions where we provide detailed theoretical and numerical analysis.