Adaptive algebraic multigrid methods in lattice computations
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1 Adaptive algebraic multigrid methods in lattice computations Karsten Kahl Bergische Universität Wuppertal January 8, 2009
2 Acknowledgements Matthias Bolten, University of Wuppertal Achi Brandt, Weizmann Institute of Science James Brannick, Penn State University Andreas Frommer, University of Wuppertal Ira Livshits, Ball State University Scott MacLachlan, Tufts University Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 2/35
3 Outline AMG in lattice computations Basic multigrid components Two-grid theory Adaptive coarse-grid generation interpolation Model problem: Gauge Laplacian Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 3/35
4 Motivation Multigrid optimal for certain problems, e.g. elliptic PDEs Linear complexity Convergence independent of system size Algebraic multigrid extended the applicability of multigrid techniques to a broader class of problems, e.g. discontinuous coefficients, complex geometries,... Algebraic multigrid only relies on the system matrix, black-box solver In case AMG fails to yield optimal performance it often serves as a supreme preconditioner for Krylov-Subspace methods, e.g. CG, GMRES,... Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 4/35
5 Challenges for AMG in lattice computations Solver challenges: Systems are nearly singular Non-hermitian and positive real or hermitian and maximally indefinite Low energy modes are unknown: highly oscillatory with oscillations dependent upon fluctuations in background gauge fields (due to heterogeneity of covariant derivatives) Large number of low energy modes What is needed? A method that can Approximate several a priori unknown low energy components within a desired level of accuracy Based solely on the algebraic problem Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 5/35
6 Basic multigrid components Two-grid theory Basic multigrid principles Optimality of multigrid: Resolve problem on different scales Smoother effectively reduces high energy modes Represent low energy modes in coarse space Global convergence Geometric multigrid Fix coarsening, adjust smoother (geometrically smooth error) Algebraic multigrid Fix smoother, adjust coarsening (algebraically smooth error) Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 6/35
7 Basic multigrid components Basic multigrid components Two-grid theory Smooth The Multigrid V cycle Finest Grid Prolongation Restriction Fewer Dofs First Coarse Grid Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 7/35
8 Basic multigrid algorithm Basic multigrid components Two-grid theory MG l (A l, b l ) Initialize u l = 0 for i = 1,..., ν 1 do u l = u l + B 1 l (b l A l u l ) Pre-smooothing end for if l + 1 = L then u l+1 = A 1 l+1 Rl l+1 (b l Au l ) else u l+1 = MG(A l+1, Rl+1 l (b Au l)) end if u l = u l + Pl+1 l u l+1 Coarse-grid correction for i = 1,..., ν 2 do u l = u l + B 1 l (b l A l u l ) Post-smoothing end for Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 8/35
9 Basic multigrid components Two-grid theory Algebraic multigrid Given hermitian system of linear sparse algebraic equations m Au = f or a ij u j = f i, i = 1, 2,..., m j=1 Coarsening in Algebraic multigrid consists of two components: 1. Find appropriate coarse-grid variables u c k = i µ ki u i, k = 1, 2,..., m c 2. Find coarse-grid representation of the linear system A c u c = f c or m c aiju c j c = f c i, i = 1, 2,..., m c j=1 Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 9/35
10 Basic multigrid components Two-grid theory Galerkin-based algebraic multigrid methods: Given the set of coarse-grid variables u c k define interpolation P : C mc C m or u i = j p ij u c j Restriction R = P Coarse-grid operator then given by A c = P AP Variational principle for Galerkin coarse-grid operator: ( ) argmin Ac (I PA 1 c P A)e A = P AP Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 10/35
11 Basic multigrid components Two-grid theory Classical AMG Special case: Coarse-grid variables are subset of all variables. Referred to as C-F splitting. Operators simplify by permutation to [ ] [ ] [ ] W Aff A P = and Au = f fc uf = I Black-box classical AMG: A cf A cc Coarsening relies on strength-of-connection Interpolation based on matrix coefficients only Implicit assumptions on algebraically smooth error! u c [ ff f c ] Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 11/35
12 Basic multigrid components Two-grid theory Convergence of two-level method E tg = (I B 1 A)(I π A )(I B 1 A), with π A := P(P AP) 1 P A. Assuming A is hpd, the convergence factor is obtained via E tg 2 A = 1 1 K(P) ; (I π B K(P) = sup )v 2 B v v 2 A where B := B (B + B A) 1 B and P : C mc C m. Further, if B is hpd, then K(P) <. Analysis of recursive multi level method depends in addition on stability: PR A < η, RP = I nc Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 12/35
13 Adaptive coarse-grid generation interpolation Get rid of implicit assumptions on algebraically smooth error. Classical smoothers (Jacobi, Gauss-Seidel) reduce eigenmodes to large eigenvalues (high energy) of hpd operators well Algebraically smooth error consists of eigenmodes to small eigenvalues (low energy) Guided by two-grid theory: weak approximation property K(P) K c (P) = A sup v (I PR)v 2 v 2 A Approximate eigenmodes proportionally accurate to the inverse of their eigenvalue Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 13/35
14 techniques Adaptive coarse-grid generation interpolation Recall: Coarsening in AMG consists of two components 1. Find appropriate coarse-spaces 2. Define interpolation capable of representing algebraically smooth error Approaches for adaptivity in algebraic multigrid context 1. Coarse-grid variables Compatible relaxation 2. Interpolation Adaptive reduction based AMG (αamgr) Adaptive smoothed aggregation (αsa) Bootstrap AMG (BAMG) Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 14/35
15 Adaptive coarse-grid generation interpolation Introduction to Compatible relaxation (CR) Compatible relaxation leaves coarse-grid variables unchanged. Allows measuring interplay of smoother and coarse-grid correction Finds splitting of variables Ω = F C Tool to measure the quality of C In C-F-splitting mind-set CR is F-smoothing If CR converges fast Algebraically smooth error can be represented using C. Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 15/35
16 Adaptive coarse-grid generation interpolation Compatible relaxation for Coarse-grid generation Initialize C = C 0 or C = Initialize U = Ω C while U do Perform ν Compatible Relaxation sweeps U = {i : u ν i u ν 1 i > θ} C = C {independent set of U} end while Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 16/35
17 Compatible relaxation theory Adaptive coarse-grid generation interpolation Recall that K(P) = (I π B(P))v 2 B (I PR)v 2 B for all v Now, given R (i.e. the coarse variables) let K m (P) = inf sup (I PR)v 2 B P v v 2 A The minimum and minimizer are then given by [ A 1 1 K m = λ min ( B 1 ff A and P = ff A fc ff ) I Further, the asymptotic convergence factor of CR provides an upper bound for the above minimum: K m c 1 ρ f, ρ f = E f 2 A ff, E f = (I B 1 ff A ff ) Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 17/35 ]
18 Adaptive coarse-grid generation interpolation Compatible relaxation for general interpolation So far: CR reduces to F-smoothing, thus assuming [ ] W P = I Coarse variables are subset of all variables In general interpolation is given by P = [ Pf Lemma The multigrid error propagator remains unchanged if we replace P by PX for any nonsingular matrix X. P c ] Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 18/35
19 Adaptive coarse-grid generation interpolation Using previous result, for the reordered general interpolation [ ] Pf P =, with P c some invertible matrix, we can consider instead the interpolation operator [ ] P = PPc 1 W =. I P c Appropriate form for the coarse variables is given Coarsening based on CR generalized in this way Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 19/35
20 Adaptive coarse-grid generation interpolation Basic principles of adaptive AMG interpolation Given a set of coarse variables u c Find interpolation P such that 1. P is sparse 2. e approximated well in range(p) for algebraically smooth e (I B 1 A)e e Motivated by weak approximation property Approaches: 1. Reduction based AMG 2. Aggregation based AMG 3. Bootstrap AMG Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 20/35
21 Adaptive coarse-grid generation interpolation Reduction based adaptive AMG After relaxation [ ] [ ] Aff A fc ef A cf A cc e c Assume this holds exactly, then [ ] 0 0 A ff e f + A fc e c = 0 e f + A 1 ff A fce c = 0 Ideal interpolation [ef ] [ A 1 = ff I Multiplying by P A e c A fc ] e c = Pe c P APe c = r c with P AP = A cc A cf A 1 ff A fc := S A (Schur complement) Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 21/35
22 Adaptive coarse-grid generation interpolation Reduction-based AMG Spectral equivalence to A ff Let [ ] [ Aff A A = fc D 1, P = P I A cf A cc A fc with σ = 2 Λ+λ. Assume A is hpd, and further that 1. λd S A ff ΛD S, with λ, Λ R +, and 2. γs A P AP ΓS A, with γ, Γ R +, Then with E TG = (I π A )S, ] ( [ ] ) D 1, S = I σ S 0 A, 0 0 E TG 2 A m S A 1 + m SA (1 + κ2 m SA ) = 1 1 κ2 1 + m SA < 1, where κ = Λ λ Λ+λ, and m S A = max( γ 1, Γ 1 ). Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 22/35
23 Adaptive coarse-grid generation interpolation Smoothed Aggregation (A)MG: defining P Partition of variables into aggregates {A i } nc i=1 Define a tentative prolongation ˆP by a partition of unity based on {A i } nc i=1 Smoothed interpolation given by P = (I ωa)ˆp with corresponding coarse-grid system A c = P A P Smoothed aggregation still implies geometrical smoothness 1. 1 ˆP = }. } A 1 A nc Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 23/35
24 Adaptive coarse-grid generation interpolation Multiple vector preserving interpolation For set of multiple vectors X := [x (1),..., x (r) ], x (i) C n, ˆP = X 1... X nc } A 1 Q 1 R 1 } A nc. Q nc R nc ˆP ˆP = Q Q = I ˆPX c = ˆPR = X Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 24/35
25 Bootstrap AMG AMG in lattice computations Adaptive coarse-grid generation interpolation Assume given C-F splitting Generate random test-vectors u (1),..., u (k) Slight smoothing yields rich local representation of smoothness For each F variable i compute algebraic distance to C points j in graph neighborhood d α ij = n=1 k n=1 ( u (n) i ) δ ij u (n) 2 j Set P i to the algebraically closest coarse neighbors of i Compute interpolatory weights p ij that minimize k u (n) i 2 p ij u (n) j j P i Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 25/35
26 Adaptive coarse-grid generation interpolation Start with initial random function Smooth functions Define tentative restriction Define interpolation by Least Squares fit for each fine grid point Figure: 1D Poisson problem u = 0; 4 test-vectors; caliber 2 interpolation Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 26/35
27 Model problem: Gauge Laplacian Model problem Systems arising in lattice computations possess properties challenging for AMG: Complex and random coefficients Localized low energy modes Couplings by matrix blocks Non-hermitian system matrices To get better understanding of each new property, we first investigate a model problem with reduced complexity. For this model problem numerical results for several approaches were obtained. Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 27/35
28 Model problem: Gauge Laplacian Gauge Laplacian Diagonal blocks of the spin-permuted linear Wilson-Dirac system (Non-physical stabilization term) Discretized onto a 2-dimensional regular grid Unknown fermionic field f (x) C for all grid points x Gauge variables u(x, m) = e iφ U(1) for m = 1, 2 φ distributed in [0, 2π) according to probability density function of the gauge theory, can appear random from one link to next Define co-variant derivatives for both spins m = 1, 2 b δ + f (x, ) = u(x, m)f (x + e m, ) f (x, ), δ f (x, ) = f (x, ) u (x e m, m)f (x e m, ) Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 28/35
29 Gauge Laplacian AMG in lattice computations Model problem: Gauge Laplacian Gauge Laplacian given by ( (Hf )(x) = ρ + 2 m=1 1 ( δ + + δ )) f (x) 2 Stencil representation [ u(x, 2) ] u (x e 1, 1) 2(2 + ρ) u(x, 1) u (x e 2, 2) Periodic boundary conditions Hermitian operator H = H Positive definite for ρ > ρ cr Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 29/35
30 AMG in lattice computations Model problem: Gauge Laplacian Spectral properties of the Gauge Laplacian I Real eigenvalues 0 < λ1 λ2... λm I ρ ρcr λ1 0 I Eigenmodes to small eigenvalues particularly random and local Modulus Modulus Phase Phase Real Imaginary Real Imaginary Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl /35
31 Model problem: Gauge Laplacian Adaptive Reduction based AMG β/n (11).48(11).48(10) 5.6(10).6(12).58(12) 10.52(10).64(12).65(13) Table: 2-level GS V(2,2)-cycle, asymptotic convergence (PCG iterations) 2-level V(2, 2)-cycles with Gauss-Seidel smoothing yield size independent convergence factors Good preconditioning properties Rather expensive method, operator complexities between 2.6 and 2.8 due to odd-even reduction Multi-level method unstable Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 31/35
32 Model problem: Gauge Laplacian Adaptive Smoothed Aggregation β/n (8).72(9).49(8) 5.69(9).78(9).9(9) 10.58(9).73(9).91(11) Table: 2-level GS V(2,2)-cycle, asymptotic convergence (PCG iterations) Geometric, aggregation-based coarsening (2 2 1 DoF) Promising method, robust preconditioner, cheaper setup, smaller operator complexity Preliminary 2-level results Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 32/35
33 Model problem: Gauge Laplacian Bootstrap AMG β/n (10).67 (10).78 (9) 5.39 (10).67 (10).79 (9) (9).55 (10).69 (9) Table: Multi-level GS V(2,2)-cycle, asymptotic convergence (PCG iterations) Caliber 4 interpolation; 8 to 12 test-vectors CR coarsening with full coarsening on finest grid Fully adaptive setup Operator complexities between 1.6 and 1.9 for 4-level method Very good multi-level results for the model problem; Highly customizable to control performance and complexity Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 33/35
34 Conclusions Lattice computations pose challenging problems to state-of-the-art AMG methods Adaptivity can overcome many of these challenges We investigated different adaptive strategies for the Gauge Laplacian: Adaptive reduction-based AMG Adaptive smoothed aggregation Bootstrap AMG Promising results for the model problem, especially for BAMG Theory for Bootstrap AMG approach in preparation Extension to real physical problems of Lattice QCD Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 34/35
35 Thank you for your attention! Adaptive algebraic multigrid methods in lattice computations, Karsten Kahl 35/35
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