Cooperative parallel asynchronous computation of the solution of symmetric linear systems

This paper introduces a new paradigm, called cooperative computation, for the solution of systems of linear equations with symmetric coefficient matrices. The simplest version of the algorithm consists of two agents, each one computing the solution of the whole system, using an iterative method. Infrequent unidirectional communication occurs from one agent to the other, either periodically, or probabilistically, thus characterizing the computation as parallel and asynchronous. Every time one agent communicates its current approximation of the solution to the other, the receiving agent carries out a least squares computation to replace its current value by an affine combination of the current approximations, and the algorithm continues until a stopping criterion is met. Deterministic and probabilistic variants of this algorithm are introduced and shown to be efficient, specifically in relation to the popular Barzilai-Borwein algorithm, particularly for ill-conditioned matrices.