The Maseeh Mathematics and Statistics Colloquium Series*
Numerical Upscaling by Multilevel Methods
by Panayot S. Vassilevski, Ph.D., Center for Applied Scientific Computing, Lawrence Livermore National Laboratory
Wednesday, January 22th, 2014 at 2:00pm
Neuberger Hall room 454, Portland State University
(Refreshments served at 3:00 in presentation room)
This event is free and open to the public.
Multigrid (or MG) is becoming the method of choice to solve large sparse systems of algebraic equations that typically arise from discretized partial differential equations (or PDEs). We give a brief motivation why this is the case; namely, due to the potential for optimality of MG. We describe some necessary and sufficient conditions for an optimal MG iteration method. Next, we focus on the ``algebraic'' versions of MG (or AMG). This refers to the case when the hierarchy of spaces needed to construct a MG is not given, hence it has to be constructed by the user, generally, in some problem dependent way. The construction of operator dependent coarse spaces with some guaranteed approximation properties is also useful for discretization (upscaling) purposes. Based on our theory (necessary conditions), as it turns out, the AMG constructed coarse spaces corresponding to a good AMG solver, have at least some weak approximation properties. In practice, however, stronger approximation properties are needed, in particular, when the spaces are meant as a discretization (upscaling) tool. We present our element-based AMG approach to construct coarse spaces with guaranteed approximation properties targeting general classes of finite elements suitable for deriving coarse (upscaled) discretizations. The performance of the method is illustrated with a number of examples.
Panayot S. Vassilevski is a computational mathematician at CASC. His research interests include numerical linear algebra, discretization and iterative solution methods for partial differential equations (PDEs), and multilevel dimension reduction (upscaling) methods for continuous (PDEs) and discrete mathematics (graph) problems.Dr. Vassilevski joined LLNL in March 1998. He received his Ph.D. in Mathematics from the St. Kliment Ohridski University of Sofia in 1984. Prior to joining LLNL, he held a number of visiting faculty positions at UCLA (1991-1993), Texas A&M University (spring 1996), Bowling Green State University (spring 1997) and UCSD (spring 1998).
* Sponsored by the Maseeh Mathematics and Statistics Colloquium Series Fund and the Fariborz Maseeh Department of Mathematics & Statistics, PSU.