New Horizons

As you probably may (or may not!) know, molecular biology often study biological functions from interaction network between molecules rather than studying each component one-by-one. It's the opposite of the universal divide-and-conquer strategy, I would call it the all-inclusive strategy. Those interactions networks involves myriads (10.000) of molecules that interacts by various chemical ways, which is generally represented as an oriented graph between each molecular compound. The transcriptional networks describe the relationship between genes and proteins, the protein-protein network s defines the cascades of interactions between some, ingenuously lumped, proteins, the metabolic networks attempt to mimic the flush of metabolic reactions inside living organisms.  So the idea is to understand how the  main 'thing' works from all those interactions linked together. Of course, other kind of networks are used in many different domain to study more-or-less linked ...

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 r...