To solve such problems, where matrix sizes hover above a million by a million and often reach billion(!), we have to use iterative methods. Therefore, the issues that I have to deal with on daily basis are those of speed of convergence of the method, the amount of memory it uses, how efficient it can be implemented, etc.
You may be asking who needs to solve these ridiculously large problems. Apparently everyone! Numerical linear algebra is the substrate of any computation that is going on out there: finite element methods, optimization, graphics, finance, quantum physics, structural engineering, and the list goes on.
I have been lucky enough to be involved in projects from materials science and recently from lattice quantum chromodynamics. The two projects below deal with numerical linear algebra problems that stem from these applications. The third project is also exciting and deals with the algorithm Google uses to provide the ranking of its several billions of webpages.
I invite you to take a look at the abstracts, perhaps browse my list of publications , and talk to me in person if you are further interested.
For more information email me at: My_First_Name @cs.wm.edu