Adaptive SketchandProject Methods for Solving Linear Systems
Abstract
We present new adaptive sampling rules for the sketchandproject method for solving linear systems. To deduce our new sampling rules, we first show how the progress of one step of the sketchandproject method depends directly on a sketched residual. Based on this insight, we derive a 1) maxdistance sampling rule, by sampling the sketch with the largest sketched residual 2) a proportional sampling rule, by sampling proportional to the sketched residual, and finally 3) a capped sampling rule. The capped sampling rule is a generalization of the recently introduced adaptive sampling rules for the Kaczmarz method. We provide a global linear convergence theorem for each sampling rule and show that the maxdistance rule enjoys the fastest convergence. This finding is also verified in extensive numerical experiments that lead us to conclude that the maxdistance sampling rule is superior both experimentally and theoretically to the capped sampling rule. We also provide numerical insights into implementing the adaptive strategies so that the per iteration cost is of the same order as using a fixed sampling strategy when the number of sketches times the sketch size is not significantly larger than the number of columns.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.03604
 Bibcode:
 2019arXiv190903604G
 Keywords:

 Mathematics  Numerical Analysis;
 15A06;
 15B52;
 65F10;
 68W20;
 65N75;
 65Y20;
 68Q25;
 68W40;
 90C20