Eigensolvers for Large Eigensolvers for Large Electronic Structure Calculations Osni Marques ( OAM ( OAMarques@lbl.gov ) @lbl ) Acknowledgments: A. Canning, J. Dongarra, J. Langou, S. Tomov, C. Voemel and L.-W. Wang
Introduction Photo luminescence of semi-conducting materials: 1 1. Electrons in stable initial state El t i t bl i iti l t t 2. Energy ⇒ electron “jumps” to previously unoccupied energy level 3. Electron jumps back ⇒ light Electron jumps back ⇒ light 3 CdSe quantum dot (size) 12/15/2008 2 Eigensolvers for Large Electronic Structure Calculations
Problem and Physical Interpretation Ψ = ε Ψ H Schrödinger Equation i i i = − 1 Δ Δ + + • Complex Hamiltonian • Complex Hamiltonian [ [ ] ] H H V V 2 • Δ is the kinetic energy term • V is the potential energy term p gy • Implicitly defined by matrix-vector product (via FFT) ε • Real eigenvalue i • Discrete energy level • Can be occupied by electron or unoccupied • Clustered multiplicities Clustered, multiplicities Ψ • Complex eigenvector i • Profile gives probability of finding electron at spatial location 12/15/2008 3 Eigensolvers for Large Electronic Structure Calculations
Simulation Code: ESCAN (Energy SCAN) • Solves single particle problem ( density functional theory ) g p p ( y ) y f • Semi-empirical potential • Non-selfconsistent calculations • Plane waves for larger systems • Plane-waves for larger systems • Optical of electronic properties of interest • Interior eigenvalue problem • Folded spectrum method • For more info, contact Lin-Wang Wang ( lwwang@lbl.gov ) , g g ( ) g@ g 12/15/2008 4 Eigensolvers for Large Electronic Structure Calculations
Spectral Transformations • Shift-invert Rayleigh quotient 1 w ρ − − 1 ([ ( ] , ) ) H e I ref • Folded spectrum 2 w ρ − ([ ] , ) H e I ref • • Harmonic Rayleigh quotient Harmonic Rayleigh quotient − * [ ] w H e I w ρ = ( ) ref w * − 2 [ [ ref ] ] w w H H e e I I w w f 12/15/2008 5 Eigensolvers for Large Electronic Structure Calculations
ESCAN: Folded Spectrum Approach 1 1 − ∇ ∇ + ψ = ε ψ 2 2 [ [ ( ( )] )] ( ( ) ) ( ( ) ) V V r r r i i i 2 ψ = ε ψ − ε ψ = ε − ε ψ 2 2 ( ) ( ) H H ref I i i i i i ref i Lowest 200 eigenstates of a CdSe system ( n=4241 ), ε ref =-5.0 M 47 -6.85525E+00 48 -6.85525E+00 49 49 -6.70916E+00 6 70916 00 50 -6.56302E+00 51 -6.38244E+00 52 -6.38244E+00 53 -2.11151E+00 54 -1.28873E+00 55 55 9.98501E 01 -9 98501E-01 56 -8.93434E-01 57 -8.93433E-01 58 -7.61729E-01 M 12/15/2008 6 Eigensolvers for Large Electronic Structure Calculations
Eigensolvers of Choice Algorithm Details Parameters Banded PCG Conjugate-Gradient (CG)-based nline Rayleigh-Quotient Minimization; i implemented by Wang and Zunger. l d b W d Z PARPACK Implicit restarted Arnoldi (IRA); ncv implemented by Lehoucq, Maschoff, Sorensen and Yang Sorensen and Yang. LOBPCG Locally Optimal Block-Preconditioned - CG; based on A. Knyazev. PRIMME PRIMME Jacobi-Davidson, Preconditioned Jacobi Davidson Preconditioned max basis size max basis size Iterative Multimethod Eigensolver; min restart size max block size implemented by A. Stathopoulos and J. max prev retain Combs. max inner iterations i it ti • • • → → − − ε ε 2 − − ε ε − − ε ε 2 ψ ψ PRIMME PRIMME on on [( [( ) ) ( ( ) ) ] ] tol tol H H ref I I i ref i 12/15/2008 7 Eigensolvers for Large Electronic Structure Calculations
Banded PCG / LOBPCG * Hx x x Hx = ( ) f x * x x * x Hx ∇ ∇ = − = ( ( ) ) ( ( ) ) ( ( ) ) f f x x Hx Hx x x r r x x * x x nline iterations of nonlinear CG it ti f li CG li CdSe system ( n=4241 ), ε ref =-4.25, 30 line minimizations per iteration iteration iteration line minimizations line minimizations 1 30-30-30-30-30-30-30-30 2 30-30-30-30-30-30-30-30 10 26-23-1-30-30-30-30-30 15 1-1-1-1-30-30-30-30 20 1-1-1-1-1-30-30-30 25 1-1-1-1-1-1-1-30 30 30 1-1-1-1-1-1-1-30 1 1 1 1 1 1 1 30 12/15/2008 8 Eigensolvers for Large Electronic Structure Calculations
Arnoldi with Implicit Restarts = j − 1 1 Krylov Subspace : ( , , ) ( , , ) K j K A q j span q Aq A q 1 1 1 1 Arnoldi factorization at step k + p . QR iteration on H with “special” shifts to promote convergence to • the k eigenvalues with largest real part, or • the k eigenvalues with largest magnitude, or • the k eigenvalues with smallest real part, or • the k eigenvalues with smallest magnitude becomes non zero After discarding the last p columns, the final set represents a length k Arnoldi factorization. 12/15/2008 9 Eigensolvers for Large Electronic Structure Calculations
Davidson / Jacobi-Davidson = = (re)starting vector [ ], 1 V v v (2) … ] 1 1 1 (1) v 1 block strategy: V 1 =[v 1 = 1 , 2 , for L j p = a) T projection into subspace W V AV j j j ˆ = θ ˆ ˆ b) solve W y y j ˆ ˆ θ ˆ c) ( , ) choose min or max eigenvalue/eigenvector y j j = ˆ ˆ d) x V y Ritz vector j j j ˆ = θ θ − ˆ ˆ e) ) ( ( ) ) r I I A A x residual vector j j j ≤ • preconditioning of an auxiliary problem f) , if tol stop r j • depends on diagonal dominancy ˆ = θ θ − − 1 1 g) ) ( ( ( ( )) )) = θ θ − ( 1 ( ( ( )) )) • may be ill conditioned b ill diti d t t I I diag di A A r P P I I diag diag A A j j j • Jacobi-Davidson solves approximately ⇒ = h) [ ] , T V t V V V I ˆ − − θ − = − ˆ ˆ T ˆ ˆ T + + + ( )( )( ) 1 1 1 I x x A I I x x t r j j j j j j j j j j j j e d end for o ( y Q (by QMR for example) p ) 12/15/2008 10 Eigensolvers for Large Electronic Structure Calculations
Test Cases IBM SP3, 16 , processors System atoms n time matvec (s) Cd20Se19 39 11,331 0.005 (1.0) Cd83Se81 164 34,143 0.014 (2.8) Cd232Se235 467 75,645 0.043 (8.6) Cd534S 527 Cd534Se527 1071 1071 141 625 141,625 0 105 0.105 (21.) • neig = 10 • tol =10 -6 ε ref = − 4.8eV • = + − ∇ + − ε 1 2 2 • diagonal preconditioner: ( ( ) / ) P I V E 2 avg ref k • IBM SP5 (8 to 32 processors) 12/15/2008 11 Eigensolvers for Large Electronic Structure Calculations
Cd20Se19 ( n=11331, 8 procs ) F ld d S Folded Spectrum ALGORITHM nline basis size rest. size prev. ret. inner iter. matvecs time (s) PCG 100 - - - - 4956 9.4 LOBPCG LOBPCG - - - - - 4756 4756 19 3 19.3 PARPACK - 20 - - - 14630 27.2 PARPACK - 25 - - - 9712 18.1 PARPACK - 30 - - - 7474 14.1 PARPACK PARPACK - 35 35 - - - 5838 5838 11 1 11.1 PRIMME JDQMR - 16 8 1 - 8546 14.6 PRIMME MIN_MATVECS - 16 8 2 0 1750 3.9 PRIMME MIN_TIME - 16 8 1 -1 4720 8.0 eigenvalues eigenvalues PARPACK PARPACK PRIMME JDQMR: -6.19176 -6.43238 • adaptive stopping criterion for inner QMR -6.19176 -6.43238 PRIMME MIN_MATVECS: currently GD_Olsen_plusk -6.34729 -6.60944 • GD+k GD+k -6.38668 -6.60945 -6.43238 -6.71546 • preconditioner applied to ( r + ε x ) -6.43238 -6.71546 PRIMME MIN_TIME: currently JDQMR_Etol -6.60944 -6.88809 • JDQMR -6.60945 -6.91577 • stops after resid reduces by a 0.1 factor f id d b 0 1 f -6.71546 6 71546 -6.98363 6 98363 -6.71546 -7.08253 12/15/2008 12 Eigensolvers for Large Electronic Structure Calculations
Cd20Se19 ( n=11331, 8 procs ) Folded Spectrum 16000 30 14000 25 12000 20 10000 matvecs 8000 15 time 6000 10 4000 5 2000 0 0 G G R K K K K E S C C C C C C M M C P P A A A A Q E I T B P P P P V D _ O R R R R T J N A A A A L A I E P P P P M M M _ E M M N N M M I I I I R R M M P I E R M P M I R P 12/15/2008 13 Eigensolvers for Large Electronic Structure Calculations
Cd20Se19 ( n=11331, 8 procs ) Unfolded Spectrum ALGORITHM nline basis size rest. size prev. ret. inner iter. matvecs time (s) PARPACK - 20 - - - **** **** PARPACK - 25 - - - 1326 2.9 PARPACK - 30 - - - 1310 2.9 1293 2.9 PARPACK - 35 - - - PRIMME MIN_MATVECS - 16 8 2 0 4185 10.0 3350 3350 7 0 7.0 PRIMME MIN_TIME - 16 8 1 0 eigenvalues PARPACK Implicit restarted Lanczos is (surprisingly) fast but (as in the -6.19176 -6.19176 folded spectrum) misses some eigenvalues. p ) g -6.19176 6.19176 -6.34729 6.34729 -6.34729 -6.38668 -6.38668 -6.43238 -6.43238 -6.43238 -6.43238 -6.60944 -6.60945 -6.60945 -6.60945 -6.71546 -6.71546 -6.71546 -6.71546 -6.88809 12/15/2008 14 Eigensolvers for Large Electronic Structure Calculations
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