Adaptive Multigrid for QCD. Lattice University of Regensburg

Transcription

1 Lattice 2007 University of Regensburg Michael Clark Boston University with J. Brannick, R. Brower, J. Osborn and C. Rebbi -1- Lattice 2007, University of Regensburg

2 Talk Outline Introduction to Multigrid (Geometric) Adaptive Smooth Aggregation MG Results Summary and Outlook -2- Lattice 2007, University of Regensburg

3 Introduction Dominant cost in Lattice QCD originates in evaluating D(U)x = b e.g., Dynamical fermions in HMC (determinant): O(1) Point to All Propagators: O(12) All to All Propagators, Disconnected diagrams: O(100+) Krylov solvers typical for this problem (CG, BiCGstab, etc.) These scale as O(N 2 ) in cost Cost explodes as m 0: Critical slowing down Multigrid approaches remove this, use for QCD? -3- Lattice 2007, University of Regensburg

4 Intro to Multigrid: Failure of stationary solvers Stationary iterative solvers (relaxation) effective on high frequency error Minimal effect on low frequency error Example: Free Laplace operator in 2D Ax = 0, x 0 random Gauss-Seidel relaxation Plot error = x -4- Lattice 2007, University of Regensburg

5 Intro to Multigrid: Failure of stationary solvers Stationary iterative solvers (relaxation) effective on high frequency error Minimal effect on low frequency error Example: Free Laplace operator in 2D Gauss-Seidel relaxation Ax = 0, x 0 random Plot error = x -5- Lattice 2007, University of Regensburg

6 Intro to Multigrid: Failure of stationary solvers Stationary iterative solvers (relaxation) effective on high frequency error Minimal effect on low frequency error Example: Free Laplace operator in 2D Gauss-Seidel relaxation Ax = 0, x 0 random Plot error = x -6- Lattice 2007, University of Regensburg

7 Intro to Multigrid: Failure of stationary solvers Stationary iterative solvers (relaxation) effective on high frequency error Minimal effect on low frequency error Example: Free Laplace operator in 2D Gauss-Seidel relaxation Ax = 0, x 0 random Plot error = x -7- Lattice 2007, University of Regensburg

8 Intro to Multigrid: Coarse Grid Representation I Low frequency error components slow down inversion Low frequency modes are smooth represent on coarse grid Low frequency on fine grid = High frequency on coarse grid Relax system on coarse grid Prolongate coarse grid correction back to fine grid 2 Grid Correction Scheme much better than relaxation -8- Lattice 2007, University of Regensburg

9 Intro to Multigrid: Coarse Grid Representation II Define restriction operator R Define prolongation operator P Notice R = P Define coarse grid operator A c = P AP (Galerkin prescription) (Almost) Equivalent to rediscretization in free theory -9- Lattice 2007, University of Regensburg

10 Intro to Multigrid: V-Cycle (Brandt) In general 2 level algorithm not enough to remove error Iterate coarsening process until exact solve is possible Remove any prolongation errors with additional relaxation This is the essence of the multigrid V-Cycle -10- Lattice 2007, University of Regensburg

11 Geometric Multigrid (Brandt) Classical geometric multigrid O(N) work Use a preconditioner for Krylov solver Removes critical slowing down Why is nobody using it then? -11- Lattice 2007, University of Regensburg

12 Lattice QCD Multigrid Gauge field U is not geometrically smooth Low frequency components of Dirac operator locally oscillatory Geometric multigrid completely fails -12- Lattice 2007, University of Regensburg

13 Lattice QCD Multigrid Lattice QCD and MG have long and painful history, e.g., PTMG (Lauwers et al) Projective MG (Brower et al) RG approaches (de Forcrand et al) Previous MG methods: Work for smooth gauge fields (µ 1 < l σ ) Fail as m 0 (µ 1 > l σ ) Failed because not considering why MG works in the first place! -13- Lattice 2007, University of Regensburg

14 Why Multigrid Works (Free Field Theory) Zero mode = constant, Exactly preserved by restriction operator: ψ 0 = P P ψ 0 Also near zero modes well preserved ψ k P P ψ k Need to preserve low modes on coarse grid Adaptivity required for gauge fields -14- Lattice 2007, University of Regensburg

15 Adaptive Smooth Aggregation MG (αsa) Adaptively use (s)low modes to define required V-cycle (Brezina et al) Initial Algorithm setup: 1. Relax on homogenous problem Ax = 0, random x 0 resulting error vector is a representation of slow modes 2. Cut vector x into subsets to be aggregated (blocked) 3. This defines the prolongator such that x = P P x 4. Additional smoothing applied to P (Richardson iteration) P = (1 3λ 4 A)P Better conditioned coarse operator 5. Define coarse grid operator A c = P AP 6. Relax on coarse grid A c x c = 0, x c 0 = P x, goto 2. This defines out initial V-Cycle -15- Lattice 2007, University of Regensburg

16 Adaptive Smooth Aggregation MG αsa (Brezina et al) -16- Lattice 2007, University of Regensburg

17 Adaptive Smooth Aggregation MG (Brezina et al) -17- Lattice 2007, University of Regensburg

18 Adaptive Smooth Aggregation MG αsa In general require more a single vector (Brezina et al) Now repeat setup process replacing relaxation with V-cycle Previously found errors components quickly reduced Error vector rich in new error components Augment V-Cycle to preserve additional vector space Cut vectors x 1, x 2 into blocks On each block use QR decomposition to define augmented prolongator Each vector corresponds to an extra dof per coarse site Add more vectors until satisfactory solver found -18- Lattice 2007, University of Regensburg

19 Geometric Adaptive Smoothed Aggregated QCD-MG αsa designed for problems without an underlying geometry Uses algebraic strength of connection to block system QCD has regular geometric lattice with unitary connections use regular geometric blocking strategy (i.e., 4 d N s N c ) Maintains simple geometry for coarse lattice Current algorithm requires HPD A = D D -19- Lattice 2007, University of Regensburg

20 Results Test algorithm for 2D Wilson fermions with U(1) background lattice, β = 6, 10, Q = 0, 4, ˆm = MG algorithm: 4 4( 2) blocking, 3 levels, 8 vectors Use under-relaxed MinRes for smoothing Exact solve on coarse grid Apply MG V-cycle as a preconditioner for CG Compare total number of D, D applications with plain CG -20- Lattice 2007, University of Regensburg

21 Results (β = 6) -21- Lattice 2007, University of Regensburg

22 Results (β = 10) -22- Lattice 2007, University of Regensburg

23 Results and Summary Critical slowing down virtually gone No slow down as µ 1 > l σ No dependence on β Bad news: Setup costs O(4) inversions Bad for HMC Ok for Propagators Great for all to all / disconnected diagrams No large volume 4d SU(3) results but confirmed for small volumes (8 4 ) -23- Lattice 2007, University of Regensburg

24 Outlook Need to tackle D directly, not D D Better sparsity on coarser grids Less vectors required? Less setup cost? P R Full parallel implementation for 4d Wilson operator Long term goal is solver for domain wall fermions For those more interested: arxiv: Lattice 2007, University of Regensburg