This presentation will cover two topics. First, we will discuss a combination of ideas for a linear solver and preconditioners for unstructured, broad classes of large partial differential equations (PDEs) systems, to accommodate architectural features of 100PF supercomputers (and beyond). In particular, we will discuss the combination of algebraic multigrid (AMG) with a sparse factorization method that exploits numerically low-rank structures such as hierarchically semi-separable (HSS) matrices. Both approaches have optimal ordering complexity for certain PDEs, and a hybrid solution can be adapted to the specifics of the PDEs and the hardware characteristics. We will show preliminary results for this AMG-HSS Linear Solver. Second, we will revisit the singular value decomposition (SVD) of a bidiagonal matrix obtained from eigenpairs of a tridiagonal matrix. We will focus on sequential algorithms implemented in LAPACK and discuss their accuracy and other issues. The goal is to enable the computation of subsets of eigenpairs (singular vectors) in parallel, which could lead to speedups in SVD calculations, in particular in ScaLAPACK.
The first part or the presentation is joint work with Sherry Li, Alexander Druinsky, Brian Austin, Eric Roman (LBNL), Panayot Vassilevski, Andrew Barker and Umberto Villa (LLNL). The second part is joint work with Jim Demmel and Beresford Parlett (UC Berkeley).
 |