Section: New Results

Algebraic preconditioners

Our work focused on the design of robust algebraic preconditioners and domain decomposition methods to accelerate the convergence of iterative methods.

In [5] we present a communication avoiding ILU0 preconditioner for solving large linear systems of equations by using iterative Krylov subspace methods. Recent research has focused on communication avoiding Krylov subspace methods based on so called s-step methods. However there is no communication avoiding preconditioner yet, and this represents a serious limitation of these methods. Our preconditioner allows to perform $s$ iterations of the iterative method with no communication, through ghosting some of the input data and performing redundant computation. It thus reduces data movement by a factor of $3s$ between different levels of the memory hierarchy in a serial computation and between different processors in a parallel computation. To avoid communication, an alternating reordering algorithm is introduced for structured and unstructured matrices, that requires the input matrix to be ordered by using a graph partitioning technique such as kway or nested dissection. We show that the reordering does not affect the convergence rate of the ILU0 preconditioned system as compared to kway or nested dissection ordering, while it reduces data movement and should improve the expected time needed for convergence. In addition to communication avoiding Krylov subspace methods, our preconditioner can be used with classical methods such as GMRES or s-step methods to reduce communication.