HPGMG

Abstract

HPGMG is a scalable reimplementation of miniGMG (see previous software disclosure) and is a compact geometric multigrid (MG benchmark designed to proxy the performance characteristics of the solves found in adaptive mesh refinement multigrid (AMR MG) applications. It solves the equation a*alpha*u - b div beta grad u = f on a large cubical domain using cell-centered values and either homogenous dirichlet or periodic boundaries. a and b are scalar constants, alpha and beta are space varying constants, f is the right hand side, and u is the solution. The cubical domain is divided into cubical subdomains which are distributed across the supercomputer. Whereas miniGMG enforced a static spatial decomposition across all levels in the MG V-cycle, HPGMG is free to redistribute data at each level including the possibility of moving all data to a single process (memory permitting). Thus, where miniGMG's U-cycle demanded a coarse grid solve across the entire supercomputer (a performance bottleneck), HPGMG can avoid this by combining distributed data and continuing the V-cycle. Technically, HPGMG can still implement a U-cycle or a Krylov bottom solve, but this solve can now be done on a global problem of only 2^3-8^3 cells. Additionally, HPGMG supports the MG F-cycle which avoids the iterative process of the v-cycle with a direct approach that can reach the discretization error in one pass. This provides a substantial performance boosts and opens avenues for exploiting systems that include hierarchical memories or heterogeneous processors. Like miniGMG, the righthand side f in HPGMG is generated from a continuous function for u. Thus, one can solve.