Resilient s-ACD for Asynchronous Collaborative Solutions of Systems of Linear Equations

Abstract: Solving systems of linear equations is a critical component of nearly all scientific computing methods. Traditional algorithms that rely on synchronization become prohibitively expensive in computing paradigms where communication is costly, such as heterogeneous hardware, edge computing, and unreliable environments. In this paper, we introduce an s-step Approximate Conjugate Directions (s-ACD) method and develop resiliency measures that can address a variety of different data error scenarios. This method leverages a Conjugate Gradient (CG) approach locally while using Conjugate Directions (CD) globally to achieve asynchronicity. We demonstrate with numerical experiments that s-ACD admits scaling with respect to the condition number that is comparable with CG on the tested 2D Poisson problem. Furthermore, through the addition of resiliency measures, our method is able to cope with data errors, allowing it to be used effectively in unreliable environments.