Recent developments of Discontinuous Galerkin Density functional theory
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1 Lin Lin Discontinuous Galerkin DFT 1 Recent developments of Discontinuous Galerkin Density functional theory Lin Lin Department of Mathematics, UC Berkeley; Computational Research Division, LBNL Numerical Methods for Quantum Chemistry Tromsø, January, 2015
2 2 Brief description of DGDFT Kohn-Sham DFT (LDA/GGA) Discontinuous basis functions Continuous density? Finite kinetic energy? Relatively simple control of accuracy Relatively small number of basis per atom
3 3 Feature of DGDFT Real space formulation (Discontinuous Galerkin) Exchange-correlation used by libxc HGH norm conserving pseudopotential. Geometry optimization Molecular dynamics (NVE/NVT) Massively parallel (>50000 procs) Compatible with the Pole Expansion and Selected Inversion method (PEXSI) for overcoming O(N^3) wall [LL-Lu-Ying-E, J. Comput. Phys. 231, 2140 (2012)]
4 4 Why another basis set Example: electron density of Si (HGH pseudopotential)
5 5 Our goal: Automatic dimension reduction Construct the local basis functions adaptively by solving a small part of the system, and obtain discontinuous basis functions. Discontinuous Galerkin (DG) framework to patch basis functions together. Basis (discontinuous) Density (nearly continuous)
6 6 Adaptive local basis functions Element: EE kk, Extended element: QQ kk. Solving the linear eigenvalue problem (NO pseudohydrogen saturation etc.) inside the extended element region with periodic boundary conditions. Here QQ 2 = EE 1 EE 2 EE 3
7 7 Adaptive local basis functions Restrict φφ kk,jj from extended element to element φφ kk,jj Orthogonalize φφ kk,jj (via singular value decomposition): adaptive local basis functions.
8 8 Discontinuous Galerkin method Finite element method with discontinuous basis functions. Inter-element continuity (consistency): e.g. Penalty on the inter-element jump [Arnold, 1982]
9 9 Kohn-Sham energy functional
10 10 Kohn-Sham energy functional
11 11 Discontinuous Galerkin method Kohn-Sham New terms [LL-Lu-Ying-E, J. Comput. Phys. 231, 2140 (2012)] Interior penalty method [Arnold, 1982]
12 12 Flowchart of adaptive local basis functions
13 13 Different dimensional systems numbasis/atom numprocessor
14 14 Two level parallelization in DGDFT 1. Intra-element parallelization numbasis: 10 ~ 100 procs 2. Inter-element parallelization numelement: 4 ~ 1,000 procs Scale to 100,000 procs or more
15 15 DG matrix as a block finite difference stencil Sparsity pattern of a DG matrix for a graphene system with 128 atoms (40 basis per atom)
16 16 Pole EXpansion and Selected Inversion Pole expansion, QQ OO log ββδee Selected Inversion QQ ρρ(xx) ii=1 ωω ii HH zz ii II ; QQ = 40~80 in practice (xx, xx) Scale at most as OO(NN 2 ) even for metals (zero gap). Rely on sparsity of H rather than near-sightedness [LL-Lu-Ying-E, Chin. Ann. Math. 30B 729, 2009] [LL-Lu-Ying-Car-E, Commun. Math. Sci. 7, 755, 2009] [LL-Yang-Meza-Lu-Ying-E, ACM Trans. Math. Software 37, 40, 2011] [LL-Chen-Yang-He, J. Phys. Condens. Matter 25, , 2013] [LL-Garcia-Huhs-Yang, J. Phys. Condens. Matter 26, , 2014]
17 17 Work with sparse H, S matrices Integrated with SIESTA as SIESTA-PEXSI [LL-Garcia-Huhs-Yang, J. Phys. Condens. Matter 26, , 2014] Scale to >10,000 processors for over atoms solved in DZP basis set.
18 18 DGDFT for the same system but with different partitioning Three kinds of partition: 1D, 2D and 3D Partition numbasis/element numprocessors 1D 6*1*1 600 (12 basis/atom) 6*600 = D 6*6*1 300 (34 basis/atom) 36*300 = D 6*6*6 100 (68 basis/atom) 216*100 = 21600
19 19 1D, 2D and 3D Partition Energy can be as accurate as au/atom. Compared to converged planewave calculation using ABINIT
20 20 PhosphoreneNanoribbons ACPNR4_3: 54 atoms ACPNR4_60: 1080 atoms ACPNR4_120: 2160 atoms ACPNR4_240: 4320 atoms
21 21 All units are atomic unit. Accuracy compared to converged ABINIT results. Accuracy
22 22 Efficiency: Construction of basis (a) and Hamiltonian (b)
23 23 Efficiency: Diagonalization (c) and Total wall clock time (d) Empty: DIAG Filled: PEXSI
24 24 Molecular dynamics Single level Nose-Hoover thermostat. Temperature: 300 K Time step: 2 fs Conserved quantity MM ppii 2 HH = + VV + pp ηη 2MM II 2QQ + 3NNkk BBTTTT II=1 2 HH tt HH 0 DDDDDDDDDD = HH 0 or au/ps per atom
25 25 Molecular dynamics
26 26 Edge reconstruction of ACPNR
27 27 Conclusion Adaptive local basis functions (ALB). Automatic dimension reduction Atomic and environmental effect Accurate and small number of basis per atom. Local Hamiltonian matrix. Parallel computing. Combined with the pole expansion and selected inversion algorithm for low order scaling computation (at most OO(NN 2 ) scaling) of KSDFT.
28 28 Acknowledgment Joint work with Mathematicians: Weinan E (Princeton) Jianfeng Lu (Duke) Chao Yang (LBNL) Lexing Ying (Stanford) Physicists / Chemists: Wei Hu (LBNL) John Pask (LLNL) Gaigong Zhang (LBNL) Thank you for your attention! Support from DOE Alvarez fellowship, SciDAC, CAMERA funding
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