  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.
