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CS 267: Applications of Parallel Computers Lecture 23: Solving the Poisson Equation Kathy Yelick /cs267 *1CS267, Yelick Lecture Schedule 11/19: Solving the Poisson Equation 11/21: Solving the Poisson Equation 11/26: Tree-based computation (Poisson Again) 11/28: Visit to NERSC Visualization group need to pick up “pass” for the bus 12/3: TBD 12/5: The Future of Parallel Computing 12/12: CS267 Poster Session (1-3pm, Woz) 12/14: Final Papers due Date2CS267, Yelick Outline Review Poisson equation Overview of Methods for Poisson Equation Jacobis method Red-Black SOR method Conjugate Gradients FFT Multigrid Comparison of methods Particle methods (next week) 2D Poissons equation Consider the continuous 2D Poisson equation, again d2u/dx2 + d2u/dy2 = b The discrete version is: T * x = b 4 -1 -1 -1 -1 4 -1 -1 -1 4 -1 -1 -1 4 -1 -1 4 -1 -1 -1 -1 4 -1 -1 -1 -1 4 -1 -1 4 -1 -1 -1 4 -1 -1 -1 4 T = Graph and “stencil” Date4CS267, Yelick Details of Discretization Approximate d2u/dx2 by differences u(x,y) = u(x+1/2, y) u(x-1/2, y) = u(x+1,y)-u(x,y) (u(x,y) u(x-1,y) = -2u(x,y) + u(x-1,y) + u(x+1,y) Similarly for d2u/dy2 So discrete Poisson for 2D mesh is: 4u(x,y) u(x-1,y) u(x+1,y) u(x,y-1) u(x,y+1) (with sign change) Date5CS267, Yelick Algorithms for 2D Poisson with N Unknowns AlgorithmSerialPRAMMemory#Procs Dense LUN3NN2N2 Band LUN2NN3/2N JacobiN2 NNN Explicit Inv.N log NNN Conj.Grad.N 3/2N 1/2 *log NNN RB SORN 3/2N 1/2NN Sparse LUN 3/2N 1/2N*log NN FFTN*log Nlog NNN MultigridNlog2 NNN Lower bound Nlog NN PRAM is an idealized parallel model with zero cost communication 222 Date6CS267, Yelick Multigrid Motivation Recall that Jacobi, SOR, CG, or any other sparse-matrix -vector-multiply-based algorithm can only move information one grid call at a time Can show that decreasing error by fixed factor c= k, then cost = O(4j-k ) . Flops, proportional to the number of grid points/processor + O( 1 ) a . Send a constant # messages to neighbors + O( 2j-k) b . Number of words sent If level j b Date32CS267, Yelick Practicalities In practice, we dont go all the way to P(1) In sequential code, the coarsest grids cost is negligible, but on a parallel machine they are not. Consider 1000 points per processor In 2D, the surface to communicate is 4xsqrt(1000) = 128, or 13% In 3D, the surface is 1000-83 = 500, or 50% See Tuminaro and Womble, SIAM J. Sci. Comp., v14, n5, 1993 for analysis of MG on 1024 nCUBE2 on 64x64 grid of unknowns, only 4 per processor efficiency of 1 V-cycle was .02, and on FMG .008 on 1024x1024 grid efficiencies were .7 (MG Vcycle) and .42 (FMG) although worse parallel efficiency, FMG is 2.6 times faster that V- cycles alone nCUBE had fast communication, slow processors Date33CS267, Yelick Multigrid on an Adaptive Mesh For problems with very large dynamic range, another level of refinement is needed Build adaptive mesh and solve multigrid (typically) at each level Cant afford to use finest mesh everywhere Date34CS267, Yelick Multiblock Applications Solve system of equations on a union of rectangles subproblem of AMR E.g., Date35CS267, Yelick Adaptive Mesh Refinement Data structures in AMR Usual parallelism is to deal grids on each level to processors Load balancing is a problem Date36CS267, Yelick Support for AMR Domains in Titanium designed for this problem Kelp, Chombo, and AMR+ are libraries for this Primitives to help with boundary value updates, etc. Date37CS267, Yelick Multigrid on an Unstructured Mesh Another approach to variable activity is to use an unstructured mesh that is more refined in areas of interest Controversy over adaptive rectangular vs. unstructured Numerics easier on rectangular Supposedly easier to implement (arrays without indirection) but boundary cases tend to dominate Date38CS267, Yelick Multigrid on an Unstructured Mesh Need to partition graph for parallelism What does it mean to do Multigrid anyway? Need to be able to coarsen grid (hard problem) Cant just pick “every other grid point” anymore Use “maximal independent sets” again How to make coarse graph approximate fine one Need to define R() and In() How do we convert from coarse to fine mesh and back? Need to define S() How do we define coarse matrix (no longer formula, like Poisson) Dealing with coarse meshes efficiently Should we switch to using fewer processors on coarse meshes? Should we switch to another solver on coarse meshes? See, for example, the Prometheus system by Mark Adams Solved up to 39M unknowns on 960 processors with 50% efficiency Date39CS267, Yelick Irregular mesh: Tapered Tube (Multigrid) Date40CS267, Yelick Coarsening Unstructured Meshes The Prometheus system uses maximal independent sets for coarsening Same idea used for multilevel spectral partitioning A simple greedy algorithm computes a maximal independent set Prometheus uses several heuristics for FEM problems: Order vertices for MIS algorithm Corner, edge, surface, interior Modify graph by deleting edges within a class Move nodes to cover “lost” vertices Date41CS267, Yelick Interpolation and Restriction Operators Prometheus functions (high level) Date42CS267, Yelick a:73N)J%G#CWySuOqhnd9;52=M(I$EYAVxRtjpflb 8.40-K*H!DXzTvPsioeka:63N)J%FZCWySuOqgmd9;51+M(I$EYAUwRtjpflb8.40-K63N)J%G#CWySuOrhnd9;52=M(I$EYBVxRtjpflc 8.40-K*H!DXzTvQsioeka:63N)J%F#CWySuOqgnd9;51+M(I$EYAUxRtjpflb8.40-K53N)J%F#CWySuOqhnd9;51=M(I$EYAUxRtjpflb 8.40-K63N)J53N)J%F#CWySuOqgnd9;51=M(I$EYAUxRtjpflb 8.40-K53N)J%G#CWySuOqhnd9;51=M(I$EYAVxRtjpflb 8.40-K*H!DXzTvPsioeka:640-K53N)J%G#CWySuOqhnd9;51=M(I$EYAVxRtjpflb 8.40-K*H!DXzTvPsioeka:640-K51=M(I$EYAUxRtjpflb8.40-K53N)J%F#CWySuOqhnd9;51=M(I$EYAVxRtjpflb 8.40-K51=M(I$EYAUxRtjpflb8.40-K53N)J%F#CWySuOqhnd9;51=M(I$EYAVxRtjpflb 8.40-K52=M(I$EYBVxRtjpflc 8.40-L*H!DXzTvQsioeka:63N)J%F#CWySuOqgnd9;51=M(I$EYAUxRtjpflb8.40-K530-K63N)J%G#CWySuOrhnd9;52=M(I$EYBVxRtjpflc 8.40-K*H!DXzTvQsioeka:63N)J%F#CWySuOqgnd9;51+M(I$EYAUxRtjpflb8.40-K52=M(I$EYAVxRtjpflb 8.40-K*H!DXzTvPsioeka:63N)J%FZCWySuOqgmd9;51+M(I$EYAUwRtjpflb8.40-K63N)J%G#CWySuOrhnd9;52=M(I$EYBVxRtjpflc 8.40-K*H!DXzTvQsioeka:63N)J%F#CWySuOqgnd9;51*H!DYAUwQsioflb73N-K53N)J%G#CWySuOqhnd9;51=M(I$EYAVxRtjpflb 8.40-K*H!DXzTvPsioeka:640-K530-K63N)J52=M(I$EYBVxRtjpflc 8.40-L*H!DXzTvQsioeka:63N)J%F#CWySuOqgnd9;51=M(I$EYAUxRtjpflb8.40-KG#CWyTvPrhnd9:63N)K63N)J%G#CWySuOqhnd9;52=M(I$EYAVxRtjpflc 8.40-K*H!DXzTvPsioeka:63N)J%FZCWySu