Adaptive and Localized Basis Functions for O ( N ) vers. Linear - PowerPoint PPT Presentation

Max Conference on the Materials Design Ecosystem at the Exascale: High-Performance and High-Throughput Computing T RIESTE bigdft.org Cubic vers. Adaptive and Localized Basis Functions for O ( N ) vers. Linear Scaling, Large Systems and Bridging the gap Complex QM Simulations Facilitating processing Outlook Luigi Genovese, Stephan Mohr, L. Ratcliff, D. Caliste, S. Goedecker, T. Deutsch INAC – CEA Grenoble January 30, 2018 Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

BigDFT: A DFT code based on Daubechies wavelets A pseudopotential Kohn-Sham code Daubechies wavelets have unique properties for DFT usage ◮ Systematic, Orthogonal bigdft.org ◮ Localised, Adaptive Cubic vers. ◮ Kohn-Sham operators are analytic O ( N ) vers. Bridging the 1.5 ◮ Efficient Poisson solver, capable gap φ ( x ) Facilitating 1 ψ ( x ) of handling different boundary processing 0.5 conditions – free, wire, surface, Outlook periodic 0 -0.5 ◮ Explicit treatment of charged -1 systems -1.5 -6 -4 -2 0 2 4 6 8 x Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Adaptivity of the mesh Atomic positions (H 2 O) bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Adaptivity of the mesh Fine grid (high resolution) bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Adaptivity of the mesh Coarse grid (low resolution) bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Adaptivity of the mesh No close form: Scaling relations! All functions have compact support, centered on grid points. m ∑ φ ( x ) = h j φ ( 2 x − j ) bigdft.org j = − m Cubic vers. We only use the filters h j : short convolutions (GPU-friendly) O ( N ) vers. 1.5 Bridging the φ ( x ) gap 1 ψ ( x ) Facilitating processing 0.5 Outlook 0 -0.5 -1 -1.5 -6 -4 -2 0 2 4 6 8 x Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

DeltaTest benchmark: ∆ = 1 . 0 Science, 351, 6280 (2016) For the first three rows bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook Screenshot of DeltaTest webpage as of 24/02/16, elements up to Ar, new NLCC - HGH - NC - PSP (S. Saha) Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

http://bigdft.org version 1.8.1 A code both for solid-state and physical chemistry ◮ 3D periodic, surfaces and free BC ( ← Poisson Solver) ◮ Usage of analytic HGH pseudopotentials ◮ Very high precision (analytic Kohn-Sham operators) bigdft.org ◮ All-electron accuracy, benchs in G2-1, (DeltaTest) Cubic vers. O ( N ) vers. Bridging the Present functionalities gap Facilitating Kohn-Sham DFT (metals, van der Walls, Hybrid Functionals), processing Systems embedded in electrostatic environments, Outlook Library of Structural Prediction, O ( N ) calculations Under implementation Non orthorhombic cells, PAW, linear response TD-DFT Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

BigDFT breakdown process (1.8.x) Modularity first Each section of BigDFT is, when appropriate, defined as a module with its own build bigdft.org system and compilation instructions. Cubic vers. O ( N ) vers. At present: Bridging the gap ◮ FUTILE 1.0 (low level) Facilitating ◮ GaIn 1.0 processing Outlook (Gaussian Integral from Fiesta GW code) ◮ CheSS 1.0 (talk Stephan) ◮ PSolver 1.8 Poisson solver + exchange Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

γ =PBE0/PBE Hybrid Functionals for large systems Since 2009, BigDFT ported on GPU. bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook Use for the exchange part ( N 2 Poisson Solver evaluations). Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

γ =PBE0/PBE Hybrid Functionals for large systems UO 2 systems: Atoms Orbitals 12 200 96 1432 bigdft.org 324 5400 768 12800 Cubic vers. 1029 17150 O ( N ) vers. Bridging the gap Facilitating processing Outlook Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Scaling of “Traditional” BigDFT ( linear scaling ) “Traditional” BigDFT code We can reach systems containing up to a few thousand electrons thanks to wavelet properties and efficient parallelization: (MPI + OpenMP + GPU) bigdft.org 60 DFT operations scale differently: Time / iteration (s) Cubic vers. 50 ◮ O ( N log N ) : Poisson solver O ( N ) vers. 40 Bridging the ◮ O ( N 2 ) : convolutions 30 gap ◮ O ( N 3 ) : linear algebra 20 Facilitating processing 10 and have different prefactors: Outlook 0 c O ( N 3 ) ≪ c O ( N 2 ) ≪ c O ( N log N ) 0 1000 2000 Number of atoms For bigger systems the O ( N 3 ) will dominate ☛ Motivation for a new approach Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Localized optimized minimal basis set ( linear scaling ) Kohn-Sham orbitals Density Matrix Defined via the kernel K αβ in Linear combinations of support functions φ α ( r ) : the φ α ( r ) basis: Ψ i ( r ) = ∑ c α i φ α ( r ) ρ ( r , r ′ ) = ∑ f i Ψ i ( r ) Ψ i ( r ′ ) α ◮ localized around atoms i bigdft.org = ∑ φ α ( r ) K αβ φ β ( r ′ ) ◮ expanded in wavelets Cubic vers. α , β O ( N ) vers. ◮ optimized in-situ Bridging the gap Facilitating processing Outlook Localization → Sparse matrices ( H αβ , K αβ ) → O ( N ) Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Comparison with the cubic version ( linear scaling ) 5000 total runtime linear total runtime cubic 4500 linear extrapolation, reference 3000 atoms 4000 3500 ◮ 20 min for 18 000 time (seconds) 3000 atoms 2500 2000 ◮ CPU Time and bigdft.org 1500 memory ∝ number Cubic vers. 1000 of atoms O ( N ) vers. 500 0 Bridging the 2000 4000 6000 8000 10000 12000 14000 gap number of atoms Facilitating processing High flexibility, like the cubic code Outlook ◮ Different levels of precision via the cutoff radii: Without fine-tuning converges to absolute energy differences of the order of 10 meV/atom, and almost exact forces. ◮ System sizes: 100 - 200K atoms � 1M basis functions Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Features of the localized optimized minimal basis set Ideal properties to work at the many thousand atoms scale ◮ Accurate results with good localization ◮ Low No. of degrees of freedom ☛ Low condition number (quasi-orthogonal) ☛ Small Spectral Width (thanks to pseudo-potential) bigdft.org S H Cubic vers. O ( N ) vers. Bridging the gap system (#atoms) sparsity κ sparsity SW (eV) Facilitating processing Outlook DNA (15613) 99.57% 2.29 98.46% 49.25 bulk pentacene (6876) 98.96% 2.26 97.11% 42.30 perovskite (768) 90.34% 2.15 76.47% 47.25 Si nanowire (706) 93.24% 2.16 81.61% 41.54 H 2 O droplet (1800) 96.71% 1.57 90.06% 38.26 Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Our record: 250,000 atoms (Stephan Mohr) Algorithm is robust and reliable on a variety of systems Accurate and efficient linear scaling DFT calculations with universal applicability S. Mohr, L. E. Ratcliff, L. Genovese, D. Caliste, P. Boulanger, S. Goedecker and T. Deutsch Phys. Chem. Chem. Phys. , 2015, 17 , 47, 31360-31370. bigdft.org DOI: 10.1039/c5cp00437c Cubic vers. O ( N ) vers. Bridging the Included in the Real-space numerical gap grid methods in quantum chemistry Facilitating themed issue of PCCP processing Guest-edited by Luca Frediani (The Arctic University of Norway) and Outlook Dage Sundholm (University of Helsinki) Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Why do we need so Large scale DFT? bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Review of O ( N ) DFT calculations bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook New calculation paradigms are emerging! Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Bridging the gap between different methods! bigdft.org Cubic vers. O ( N ) vers. Bridging the gap Facilitating processing Outlook Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese

Testing different approaches and models ◮ Fragments : Constrained DFT, Charge transfer, Excitations bigdft.org ◮ Atomic charge analysis Cubic vers. O ( N ) vers. ◮ Statistics of atomic configurations Bridging the gap (snapshots from MD with force fields) Facilitating processing Outlook ◮ Impact of the (electrostatic) environment ◮ Comparison between Full QM, QM/QM, and MM calculations Laboratoire de Simulation Atomistique http://inac.cea.fr/L_Sim T. Deutsch, L. Genovese