A Parallel Implementation of Multilevel Recursive Spectral Besection for Application to Adaptive Unstructured Meshes

The design of a parallel implementation of multilevel recursive spectral bisection is described. The goal is to implement a code _ha_ is fast enough to enable dynamic repartitioning of adaptive meshes. 1 Background The R.ecursive Spectral Bisection (I_SB) _lgori_hm is one of a class of recursive bisection methods for partitioning unstructured problems [4]. RSB is typically used as a preprocessing step prior to running a unstructured-mesh simulation on a massively paraI!el computer. A.pplJcations that change the mesh adaptively throughout the simulation, however, require a fast method to repartition "on the fly." Finding a partition that both balances the work of all processors and minimizes interprocessor commuAication is an NP-hard problem. Therefore, all practical partitioning algorithms are necessarily heuristic appro:dmations. Among these. KSB empirica3.l:." provides the best partitions for a large set of problems, albeit at somewhat more run time than most. other methods. I_SB bisects a graph by first finding the eigenvector (the Fied!er vector) corresponding to the smallest non-trivial eigenvalue of the Laplacian matrix of 3. graph. The graph could represent, for example, the connectivity between elements in a finite-element mesh, or the connectivity between volumes in an unstructured finite-volume mesh. The vertices of the gaph are reordered with the permutation induced by sorting the components of the Fiedler vector, and the _aph is cut in hail, with those vertices in the lower half of the new ordering in one part, and the remaining vertices in the other part. The most straightforward implementation of RSB, which uses the Lanczos algorithm to find the Fiedler vectors, is unacceptably slow for many applications. Multilevel recursive spectral bisection (MR.SB) [1] is a refinement of the algorithm that is much faster, typically by an order of magnitude or more, and has been instrumental in the acceptance of RSB. The basic idea behind MR.SB is to speed up the Fiedler-vector computation by constructing a series of successively smaller contracted gaphs that maintain the global structure of the orig,lnal graph. The Fiedler vector of the smallest gaph is found quickly wi_h _he Lanczos algorithm. That result is interpolated to the next larger graph to form an approx.imate Fiedler vector, which is then refined with Rayleigh Quotient Iteration using the SYMMLQ algorithm. This process is repeated until the Fiedler vector of the original graph is obtained. °Cray Research Inc., NASA Ames Research Center, Moffett Field, CA 94035 tComputer Sciences Corp., NASA Ames R_e_ch Center, Moffett Field, CA 94035 ,Soy) https://ntrs.nasa.gov/search.jsp?R=20020004348 2017-12-05T00:02:07+00:00Z