Analysis of the Block Coordinate Descent Method for Non-linear Ill-Posed Problems

Simon Rabanser, Lukas Neumann, Markus Haltmeier

Published 2019-02-13Version 1

Block coordinate descent (BCD) methods approach optimization problems by performing gradient steps along alternating subgroups of coordinates. This is in contrast to full gradient descent, where a gradient step updates all coordinates simultaneously. BCD has been demonstrated to accelerate the gradient method in many practical large-scale applications. Despite its success no convergence analysis for inverse problems is known so far. In this paper, we investigate the BCD method for solving linear and non-linear ill-posed inverse problems. As main theoretical result, we show that for operators having a particular tensor product form, the BCD method combined with an appropriate stopping criterion yields a convergent regularization method. To illustrate the theory, we perform numerical experiments comparing the BCD and the full gradient descent method for the non-linear inverse problem of one-step inversion in multi-spectral X-ray tomography.