Two Level Domain Decomposition for Multiclusters

We discuss the design of parallel algorithms to solve elliptic problems on multi-clusters computers. Multi-clusters can be seen as two-level parallel architecture machines, since communication between clusters are usually much slower than communication or access to memory within each of the clusters. We introduce special algorithms that use two levels of parallelism and match the multi-cluster architecture. Efficient parallel algorithms that rely on fast uniform communication have been extensively developed in the past: we intend to use them for parallel computation within the clusters. On top of these local parallel algorithms, new robust and parallel algorithms are needed that can work with few clusters linked by a slow communication network. We present a two level domain decomposition algorithm that uses Aitken or Steffensen acceleration procedure combined to Schwarz for the outer loop and standard parallel domain decomposition for the inner loop. We demonstrate finally the interest of our algorithm for metacomputing. We consider the design of parallel algorithms for multi-cluster architecture with few heterogeneous clusters linked by an affordable network of order 10Mb/s bandwidth. Each cluster can be a shared multiprocessors machine or an MIMD computer with a fast internal Network. The elapse time to access memory from a given processor to a given data on such architecture is then strongly dependent on the location of the datas. Fast scalable parallel algorithm for the Laplace problem with domain decomposition and/or multigrid on a uniform MIMD architecture have usually very poor efficiency on multi-cluster machine with slow inter-cluster network. On the contrary a numerically unefficient iterative domain decomposition algorithm such as the classical additive Schwarz procedure for the Laplace problem, is easy to implement, robust and scalable on multi-cluster architecture. So our goal is the design of an acceleration procedure for iterative domain decomposition analogous to additive Schwarz that increases the numerical efficiency of the basic underlined algorithm but stay easy to implement, robust and scalable on multi-clusters. The common procedure to accelerate additive Schwarz method is the introduction of a coarse-grid operator [LSFQ97]. The resulting modified Schwarz algorithms becomes numerically efficient but the coarse grid computation might be a bottle neck for the parallel processing. We adopt here a different point of view and try to extract from a finite sequence of the interfaces generated by the Schwarz iterative procedure or analogous relaxation method, an accurate prediction of the interface’s limit. We will show in simple case as finite difference approximation of Elliptic operator with con-