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Adaptive Multigrid for Lattice QCD James Brannick October 22, 2007 Penn State

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Presentation plan Intro to Lattice Quantum Chromodynamics Review of Multigrid Basics (Adaptive) MG for QCD Numerical Experiments

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Participants in the MG-QCD project A. Bessen (Columbia University) A. Brandt (UCLA, WIS) J. Brannick (PSU) M. Brezina (CU Boulder) R. Brower (BU) M. Clark (BU) R. Falgout (CASC-LLNL) A. Frommer (Wuppertal) K. Kahl (Wuppertal) C. Ketelson (CU Boulder) D. Keyes (Columbia University) O. Livne (Univ. of Utah) I. Livshits (Ball State) S. MacLachlan (TU Delft) T. Manteuffel (CU Boulder) S. McCormick (CU Boulder) V. Nistor (PSU) K. Osterlee (TU Delft) J. Osborne (ANL) C. Rebbi (BU) J. Ruge (CU Boulder) P. Varanas (LLNL)

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Forces in Standard Model: SU(N) Atoms: Maxwell N=1(charge) Nuclei: Weak N=2 (isospin) Sub nuclear: Strong N=3 (color) Standard Model: U(1) £ SU(2) £ SU(3)

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QCD path integral Anti-quark quark Gauge Dirac Operator Generalized Curl (Maxwell)

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The Dirac PDE (for Quarks) x = (x 1,x 2,x 3,x 4 ) (space,time) 4 x 4 sparse spin matrices: 4 non-zero entries 1,-1, i, -i 3x3 color gauge matrices Wilson (1974): discretization on a hypercubic lattice U(x,x+ ) = exp[i hg A (x)]

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Discrete Dirac operator on hypercubic lattice x x+ Dimension: 1,2,…,d Color a,b = 1,2,3 Spin i,j = 1,2,3,4 x 1 axis x 2 axis Spin projection Operator

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Wilson fermion matrix: M Typical lattice size : 32 3 х 24 Typical lattice size : 32 3 х 24 Dimension of then Dimension of M then 16 3 х 24 х 2 х 3 х 4 ≈ 10 6 х 10 6 !! 16 3 х 24 х 2 х 3 х 4 ≈ 10 6 х 10 6 !! Need Need M -1 (U), Tr[ M -1 (U) ], Det[ M(U) ] Together account for dominant cost more than 80% of the overall flops! Different types of algorithms used for different fermionic actions: Krylov methods typical, as M* = M M(U) Goal of our NSF PetaApps project is inversion on lattice - overall Goal of our NSF PetaApps project is inversion on lattice - overall simulation at this resolution requires sustained Petaflop years! simulation at this resolution requires sustained Petaflop years! /

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2-d “toy” problem: Schwinger Model Space time is 2-d Gauge links: U(x,x+ ) = exp[i e A (x)], U(1) theory Dirac fields have 2 spins (not 4) Operator is quaternionic (Pauli) matrix involving 1, 2

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Spectrum of Schwinger matrix & “critical slowing down’’ mass gap =.1 mass gap =.01 mass gap =.001

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Stationary Linear Iterations Consider solving the linear system using a SLI: Let e k =u - u k be the error, and note that r k =Ae k. The error propagation operator is then, with r k =f - Au k

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Multigrid V-cycle Multigrid solvers are optimal ( O(N) operations), and hence have good scaling potential MG uses a sequence of coarse-grid problems to accelerate the solution of the original problem smoothing Fine Grid Smaller Coarse Grid restriction prolongation (interpolation)

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Why does standard MG work? Low mode is constant -- key to MG success is smooth error, e.g., Laplacian Constant exactly preserved on coarse level All near zero modes also preserved!

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Lattice QCD MG Gauge field and hence low modes not geometrically smooth (locally oscillatory) Geometric MG completely fails, preserving low modes for gauge fields requires adaptivity Real Part Imaginary Part Lowest mode of M, ¯ = 6

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Lattice QCD & (A)MG Lattice QCD and MG have a long and painful history (15+ yrs.) * PTMG (Lauwers et al, 93) * Renormilazation MG (de Forcrand et al, 91) * Projective MG (Brower et al, 91) * Many others.. All failed for non-smooth fields in the m ! m cr limit, failed because did not consider why MG works in first place! Diamond: Jacobi Circle: CG Square: V-cycle Star: W-cycle

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Adaptive Smooth Aggregation AMG Adaptive Smoothed Aggregation ( SA) Multigrid, Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge, SIAM Review, Adaptive Smoothed Aggregation Multigrid for Lattice QCD, Brannick, Brezina, Keyes, Livne, Livshits, MacLachlan, Manteuffel, McCormick, and Ruge, Zikatanov,J. Comp. Physics, Adaptively use (s)low modes to define the evolving V-cycle Initial Algorithm Setup 1. Relax on Ax = 0, with random initial guess 1. Relax on Ax = 0, with random initial guess (resulting error is repesentative of (s)low modes). (resulting error is repesentative of (s)low modes). 2. Cut vector into pieces over aggregates (blocks). 2. Cut vector into pieces over aggregates (blocks). 3. Define the prolongator so that x = Px. 3. Define the prolongator so that x = Px. 4. Smooth the vector using simple Richardson 4. Smooth the vector using simple Richardson iteration: P = (I - ¸ A)P, ¸ choosen to minimize iteration: P = (I - ¸ A)P, ¸ choosen to minimize condition number of coarse scale operator: condition number of coarse scale operator: A = ((I-¸ A)P)A(I-¸ A)P. A = ((I-¸ A)P)A(I-¸ A)P. * c c General setup adaptive process repeated with current General setup adaptive process repeated with current solver to find additional vectors as needed solver to find additional vectors as needed

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Adaptive SA designed for problem without underlying geometry Uses algebraic defintion of “strength of connection” to define aggregates QCD defined on regular lattice with unitary connections and so use regular geometric blocking strategy (i.e., 4 x n x n ) Maintains simple regular geometry on coarse scales, allowing for perfect load balancing and minimal comm. Original algorithm requires HPD operator and thus we solve M M Adaptive Smooth Aggregation MGdsc *

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Results for M M Schwinger model, 2-d with U(1) background on128 x 128 lattice with ¯ = 6,10, Q = 0,4, mass gap = Use 4 x 4 ( x 2 ) blocking and 3 levels with 8 vectors Under-relaxed MinRes smoother (Bank and Douglas) Compare MG-PCG with CG *

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Results… ¯ = 6¯ = 10 Critical slowing down eliminated! Critical slowing down eliminated! No dependence on No dependence on ¯ Dramatic improvement over CG Dramatic improvement over CG Adaptive Multigrid for QCD, Brannick, Brower, Clark, Osborne, and Rebbi, Phys. Rev. Letters, sub Adaptive Multigrid for Wilson Fermions, Brannick, Brower, Clark, Osborne, and Rebbi, Proc. of Science: Lattice 07, sub

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Results: MG for M ¯ = 6, N = 128 Huge reduction of flops Huge reduction of flops Fewer vectors needed Fewer vectors needed Develop. of code for full 4-d QCD underway Develop. of code for full 4-d QCD underway Adaptive Multigrid for the non-hermitian Wilson operator, Brannick, Brower, Clark, Osborne, and Rebbi, in preparation.

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Possible future collaborations Current projects (with J. Xu & L. Zikatanov) Radiation transport Radiation transport Electromagnetics Electromagnetics Oil resevoir simulations Oil resevoir simulations Fuel cell dynamics Fuel cell dynamics Stochastic matrices Stochastic matrices Lattice field theories Lattice field theories

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