[petsc-users] Direct Schur complement domain decomposition

[Barry Smith]

  My off the cuff response is that "computing the exact Schur complements for the subdomains is sooooo expensive that it swamps out any savings in reducing the amount of communication" plus it requires soooo much memory. Thus solvers like these may make sense only when the problem is "non-standard" enough that iterative methods simply don't work (perhaps due to extreme ill-conditioning), such problems do exist but for most "PDE" problems with enough time and effort one can cook up the right combination of "block-splittings" and multilevel (multigrid) methods to get a much more efficient solver that gives you the accuracy you need long before the Schur complements have been computed.

   Barry

On Dec 29, 2012, at 12:59 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:

> Sorry for the slow reply. What you are describing _is_ multifrontal factorization, or alternatively, (non-iterative) substructuring. It is a direct solve and boils down to a few large dense direct solves. Incomplete factorization is one way of preventing the Schur complements from getting too dense, but it's not very reliable.
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> There are many other ways of retaining structure in the supernodes (i.e., avoid unstructured dense matrices), at the expense of some error. These methods "compress" the Schur complement using low-rank representations for long-range interaction. These are typically combined with an iterative method.
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> Multigrid and multilevel DD methods can be thought of as an alternate way to compress (approximately) the long-range interaction coming from inexact elimination (dimensional reduction of interfaces).
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> On Wed, Dec 19, 2012 at 10:25 AM, Stefan Kurzbach <stefan.kurzbach at tuhh.de> wrote:
> Hello everybody,
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> 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?
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> 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.
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> Now my question:
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> 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?).
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> 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?
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> Best regards
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> Stefan
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