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 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 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.
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