Randomized Block Adaptive Linear System Solvers

Abstract

. Randomized linear solvers leverage randomization to structure-blindly compress and solve a linear system to produce an inexpensive solution. While such a property is highly desirable, randomized linear solvers often suffer when it comes to performance as either (1) problem structure is not being exploited, and (2) hardware is inefficiently used. Thus, randomized adaptive solvers are starting to appear that use the benefits of randomness while attempting to still exploit problem structure and reduce hardware inefficiencies. Unfortunately, such randomized adaptive solvers are likely to be without a theoretical foundation to show that they will work (i.e., find a solution). Accordingly, here, we distill three general criteria for randomized block adaptive solvers, which, as we show, will guarantee convergence of the randomized adaptive solver and supply a worst-case rate of convergence. We will demonstrate that these results apply to existing randomized block adaptive solvers, and to several that we devise for demonstrative purposes.