# Analysis of the Block Coordinate Descent Method for Linear Ill-Posed Problems

## TL;DR

This paper analyzes the convergence of block coordinate descent (BCD) methods for linear inverse problems, demonstrating that under certain conditions, BCD with proper stopping criteria acts as a regularization method, supported by numerical experiments.

## Contribution

The paper provides the first convergence analysis of BCD for inverse problems and shows it can serve as a regularization method under specific tensor product operator conditions.

## Key findings

- BCD with stopping criteria converges for tensor product operators
- Numerical experiments compare BCD and full gradient descent
- Tests include linear and non-linear inverse problems

## Abstract

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 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 a system of integral equations. We also present numerical tests for a non-linear inverse problem not covered by our theory, namely one-step inversion in multi-spectral X-ray tomography.