IML FISTA: A Multilevel Framework for Inexact and Inertial Forward-Backward. Application to Image Restoration
arxiv(2023)
摘要
This paper presents a multilevel framework for inertial and inexact proximal
algorithms, that encompasses multilevel versions of classical algorithms such
as forward-backward and FISTA. The methods are supported by strong theoretical
guarantees: we prove both the rate of convergence and the convergence of the
iterates to a minimum in the convex case, an important result for ill-posed
problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA,
based on the use of the Moreau envelope to build efficient and useful coarse
corrections, fully adapted to solve problems in image restoration. Such a
construction is derived for a broad class of composite optimization problems
with proximable functions. We evaluate our approach on several image
reconstruction problems and we show that it considerably accelerates the
convergence of the corresponding one-level (i.e. standard) version of the
methods, for large-scale images.
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