Problems and Techniques
Authors: Joe Flaherty
Venue: SIAM Review (1999)

Our second installment of Problems and Techniques features articles on the controllability of the Navier--Stokes equations and the parallel partitioning of irregular graphs. Fernández-Cara surveys theoretical results on controlling the incompressible Navier--Stokes equations. Needless to say, flow control has important applications in several areas, including reducing turbulence and enhancing the performance of flight vehicles or turbo-machinery, or increasing the efficiency of flow-dominated manufacturing processes. Motivated by a conjecture of J. L. Lions, Fernández-Cara examines approximate and null controllability. The former refers tofinding a control function such that the fluid's velocity at a given time is (arbitrarily) close to a prescribed velocity. The latter implies the existence of a control such that the velocity vanishes at a given time. Lions conjectured the approximate controllability of the Navier--Stokes equations, but this and null controllability remain unanswered in general situations. Fernández-Cara reviews known results and also examines the controllability of finite-dimensional approximations of the Navier-Stokes equations through Galerkin's method. Nice analyses of some very difficult problems are presented. In our second article, Karypis and Kumar present a multilevel strategy for partitioning irregular graphs. Thus, they seek to divide a graph into a number of subgraphs having the same number of vertices such that the number of edges connecting vertices in the different subgraphs is minimal. This problem has numerous applications in parallel scientific computation including the solution of sparse linear algebraic systems and the load balancing of unstructured-grid finite element computations. With this latter application, which is near and dear to my heart, elements may comprise graph vertices while connections of elements through common faces correspond to edges. The graph partitioning problem is NP-complete, so various approximation techniques have been developed, including graph-theoretic strategies and partitionings based on spectral and geometric decompositions. From a software perspective, Hendrickson and Leland set the stage with their CHACO system. Many approaches involve a multilevel strategy that coarsens the original graph until it can easily be partitioned and then projects these partitionings onto finer graphs. This is akin to the popular multigrid strategies for iterative solutions of linear algebraic systems. Multilevel strategies are central to the popular and very successful METIS system that was developed by our authors. So what's the problem? Most partitioning algorithms execute in serial mode. This is often acceptable; however, serial partitioning may not be possible for very large problems and it certainly would not be possible for algorithms such as adaptive finite element computation, which typically require a dynamic re-partitioning with each mesh or method alteration. Hence there is a need for parallel partitioning procedures, which are the subject of the article. Coarsening, partitioning, and prolongation procedures are described and applied to a suite of graphs obtained from finite element computation. These procedures have good speed-ups and efficiencies close to the best serial procedures. The Karypis and Kumar article should be of interest to both discrete mathematicians and computational scientists. Please join me in thanking our authors, Enrique Fernández-Cara, George Karypis, and Vipin Kumar, for providing two excellent articles on subjects of widespread interest. We're always eager to receive exciting articles on recent mathematical developments, so please send them our way. We would also be happy to have input on subject matter that you would like to see in the section.