[petsc-users] Direct Schur complement domain decomposition

[Stefan Kurzbach]

Hello everybody,

 

in my recent research on parallelization of a 2D unstructured flow model
code I came upon a question on domain decomposition techniques in "grids".
Maybe someone knows of any previous results on this?

 

Typically, when doing large simulations with many unknowns, the problem is
distributed to many computer nodes and solved in parallel by some iterative
method. Many of these iterative methods boil down to a large number of
distributed matrix-vector multiplications (in the order of the number of
iterations). This means there are many synchronization points in the
algorithms, which makes them tightly coupled. This has been found to work
well on clusters with fast networks.

 

Now my question:

What if there is a small number of very powerful nodes (say less than 10),
which are connected by a slow network, e.g. several computer clusters
connected over the internet (some people call this "grid computing"). I
expect that the traditional iterative methods will not be as efficient here
(any references?).

 

My guess is that a solution method with fewer synchronization points will
work better, even though that method may be computationally more expensive
than traditional methods. An example would be a domain composition approach
with direct solution of the Schur complement on the interface. This requires
that the interface size has to be small compared to the subdomain size. As
this algorithm basically works in three decoupled phases (solve the
subdomains for several right hand sides, assemble and solve the Schur
complement system, correct the subdomain results) it should be suited well,
but I have no idea how to test or otherwise prove it. Has anybody made any
thoughts on this before, possibly dating back to the 80ies and 90ies, where
slow networks were more common?

 

Best regards

Stefan

 

 

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