Subspace correction methods for total variation and $\ell_1-$minimization
Provides a new algorithmic framework for convex optimization in Hilbert spaces, benefiting researchers in inverse problems and imaging by enabling efficient domain decomposition and accelerated sparse recovery.
The paper introduces oblique thresholding for subspace correction methods to minimize total variation and ℓ1-regularized energies, achieving efficient solutions for signal and image processing problems with convergence guarantees.
This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via \emph{oblique thresholding}, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for singular elliptic PDE's arising in total variation minimization and in accelerated sparse recovery algorithms based on $\ell_1$-minimization. We include numerical examples which show efficient solutions to classical problems in signal and image processing.