Convergence control of relaxation processes with fine-grain locally-connected two-scale automata networks

Relaxation algorithms are discrete-time dynamical systems tamed for problem-solving purposes. We are interested in systems with a 2D-structured state-space and neighborhood dependencies, such as can be formally represented by locally-coupled automata networks. These models are applicable in fields ranging from image processing to computational physics (continuum model). They have been touted as ideally suited to parallel implementation, especially on isomorphic algorithmically-specialized architectures. Parallelism applies to each iteration of the state-update which can be performed independently at each site using purely local operations. Testing the convergence of these algorithms normally involves, by contrast, a global operation, at variance with a locality that is strongly valued for architectural reasons. We propose to monitor convergence with a locally-coupled network of purely logic automata, superimposed on the computation network. A coherent convergence test can result in this way from the aggregation of purely local operations. This is akin to the general concept of a two-scale automata network, where control states propagate iteratively on a fast time-scale, nested within iterations of the computation network