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Spanning Trees and Direct Solvers


This blog entry will be entirely dedicated to a subject that absorbed me for more than one year: support graph preconditioners. When one have to solve say a laplacian equation on a mesh the numerical discretization  the problem to solve is generally of the form Ax=b, where A is a sparse matrix with many zero entries. More importantly the non-zero entries have a pattern direclty inherited from the mesh connectivity. For linear elements, (i,j) entry is non void if and only if there is an edge between node i and node j (what's cool with the laplacian is that a node is a degree of freedom).

So, as long as the mesh has a moderate connectivity pattern, the matrix inherits  that pattern and is therefore relatively sparse. If you factorize your matrix with a direct solver then the work your computer has to do is directly linked to the connectivity of the underlying mesh. For instance if the underlying mesh is a tree, then the complexity of the direct solver will be linear with respect to the number of nodes in the tree. Not bad.

If your graph is planar (like in the 4-color theorem), then complexity is proportional to the number of node at power 3/2. Not linear, but almost.
 So trees are provably the easiest graph (or mesh, who care) to factorize because they have very low (linear) complexity. The idea of support graph preconditioners is to exploit that fact and construct a approximation of the original matrix which underlying graph resembles as much as possible to a tree.
For a laplacian matrix, it is possible to construct such approximation by considering a minimum spanning tree of the mesh. Of course, it is not as easy as that! To get an efficient preconditioner one have to do something on the approximation, generally adding few additional edges in the spanning tree to reduce the condition number of the preconditioned system (see [3] for instance).

I think it's funny to see that my current work to make solvers faster is very similar to the child game of reconnecting a set of points without lifting the pen and without passing two times on the same edge.


A regular mesh...



... and one of its spanning trees !


However we can talk about the (undirected) graph of a matrix only for particular matrices, like laplacian matrices which are symmetric positive definite matrices with positives entries on the diagonal and only negative entries on the extra-diagonal. In that particular case, we have a isomorphism between the weighted undirected graph and its matrix, but that is a rather technical footnote.

References:
[1] Dan Spielman website (here) have a wonderful talk by him (here)
[2] Bruce Hendrickson webpage (here)
[3] Doron Chen and Sivan Toledo: Vaidya's preconditioners: Implementation and experimental study

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