Dear Jed Thanks. Non-iterative or direct substructuring is my understanding as well. You said this is the same as multifrontal factorization as well. Could you point me to some source where I can see the parallels? Maybe this is obvious for people who have grown up with solving sparse systems, but not for me :)
I will have to spend some more time to find out about the other hints you gave. Best regards Stefan Am 29.12.2012 19:59, schrieb Jed Brown: > 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. > > 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. > > 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). > > On Wed, Dec 19, 2012 at 10:25 AM, Stefan Kurzbach > <stefan.kurzbach at tuhh.de <mailto:stefan.kurzbach at tuhh.de>> wrote: > > 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 > > -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20130101/13edff18/attachment-0001.html>
