A knowledge-based approach to high-performance computing in ab initio simulations.

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1 Mitglied der Helmholtz-Gemeinschaft A knowledge-based approach to high-performance computing in ab initio simulations. AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli

2 Academic background Laurea in Physics Università di Roma I La Sapienza. Ph.D. in Physics University of Texas at Austin. Postdoctoral Research Associate University of North Carolina at Chapel Hill. Head of the Simulation Laboratory ab initio Jülich Supercomputing Centre. Junior Research Group Leader Aachen Institute for Advanced Study in Computational Engineering Science. AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 2

3 High-performance simulations Two Observations Often software libraries are used as black boxes. Very little information coming from the physics of the specific application is exploited by scientific computing codes. AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 3

4 High-performance simulations Two Observations Often software libraries are used as black boxes. Very little information coming from the physics of the specific application is exploited by scientific computing codes. One Objective Exploiting physical information extracted from the simulations in order to: increase the performance of large legacy codes; improve the computational paradigm on which the codes are based. enabling access to more physics AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 3

5 Main active projects HPC and scalable eigensolvers tailored to Density Functional Theory (DFT) Chebyshev Filtered Subspace Iteration Development of a block iterative eigensolver tailored to sequences of dense eigenvalue problems arising in DFT methods based on the LAPW basis set. Collaboration with M. Berljafa (University of Manchester) Spectrum-slicing methods Development of a integral-based iterative eigensolver for sparse generalized hermitian eigenvalue problems appearing in real-space DFT methods. Collaboration with Y. Saad (U. of Minnesota) and E. Polizzi (U. of Massachussets). HPC Tensor algebra Development of taxonomy rules and tailored kernels for high-performance multicontraction operations between high dimensional tensors for quantum chemistry. Collaboration with P. Bientinesi (AICES) and J. Hammond (Intel) AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 4

6 Density Functional Theory simulations General framework Initial guess for charge density n start (r) Compute discretized Kohn-Sham equations Solve a set of eigenproblems P (l) k 1...P (l) k N No OUTPUT Electronic structure,... Yes Converged? n (l) n (l 1) < η Compute new charge density n (l) (r) 1 every P (l) k : A(l) k x = B(l) k λx is a generalized eigenvalue problem; 2 A and B are hermitian (B is positive definite); 3 required: lower 2 10 % of eigenpairs; 4 k-vector index: k = 1 : ; 5 iteration cycle index: l = 1 : AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 5

7 Chebyshev Filtered Subspace Iteration Alternative solving strategy P (l) k 1 ITERATION (l) iterative solver (X (l) k 1,Λ (l) k 1 ) P (l+1) k 1 ITERATION (l + 1) iterative solver (X (l+1) k 1,Λ (l+1) k 1 ) P (l) k 2 iterative solver (X (l) k 2,Λ (l) k 2 ) P (l+1) k 2 iterative solver (X (l+1) k 2,Λ (l+1) k 2 ) Next cycle P (l) k N iterative solver (X (l) k N,Λ (l) k N ) P (l+1) k N iterative solver (X (l+1) k N,Λ (l+1) k N ) X {x 1,...,x n } Λ diag(λ 1,...,λ n ) AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 6

8 Chebyshev Filtered Subspace Iteration Chebyshev filter A generic vector v = n i=1 s ix i is very quickly aligned in the direction of the eigenvector corresponding to the extremal eigenvalue λ 1 v m n n = p m (A)v = s i p m (A)x i = s i p m (λ i )x i i=1 i=1 n C m ( = s 1 x 1 + λ i c e ) s i i=2 C m ( λ 1 c e ) x i s 1 x 1 AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 7

9 Chebyshev Filtered Subspace Iteration Speed-up Speed-up = CPU time (input random vectors) CPU time (input approximate eigenvectors) 4 Au 98 Ag 10 - n =13, cores. 3 Speed-up Iteration Index AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 8

10 Chebyshev Filtered Subspace Iteration Strong scalability Size n of the eigenproblems are kept fixed while the number of cores is progressively increased Three systems of size n = 13,379 12,455 9,273. time [s] 10 3 Au 98 Ag TiO Na 15 Cl 14 Li number of cores AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 9

11 Chebyshev Filtered Subspace Iteration Optimized ChFSI versus direct solvers For the size of eigenproblems here tested the ScaLAPACK implementation of BXINV or MRRR is on par or worse than EleMRRR. For this reason a direct comparison with ScaLAPACK is not included time [s] EleMRRR EleChFSI, 1e-10 EleChFSI OPT, 1e cores 128 cores eigensystem index l AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 10

12 Spectrum-slicing methods Integral based projector λ min few eigenvalues a b cluster λ max P = 1 (A zb) 1 B dz 2iπ u i u T i B Γ λ i [a b] Approximation n c P χ nc (A,B) = w j (A z j B) 1 B j=1 AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 11

13 Spectrum-slicing methods Integral based projector λ min few eigenvalues a b cluster λ max Integral-based Subspace Iteration While{CONV < NEV} Q i = PQ i 1 χ nc (A,B) = w j (A z j B) 1 BQ i 1 j=1 n c Core problems 1 Relies on good estimates of number µ [a b] of eigenvalues in [a b] 2 Solve for multiple right-hand side linear systems per integration node 3 Accuracy depends on the quadrature method used AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 11

14 Spectrum-slicing methods Under investigation Stochastic estimator Trace(P) n n c n v n v γ j v T k (A σ ji) 1 v k j=1 k=1 Eigenvalue Count Case: na5; n = 25; n = 300 c v RQ samples (n = 5) c RQ samples (n c = 25) Running mean (n c = 5) Running mean (n c = 25) Exact Sample vectors Sparse linear systems Solving with Generalized Minimal Residual method (GMRES) Exploiting rational Krylov methods AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 12

15 Spectrum-slicing methods Under investigation λ min few eigenvalues a b cluster λ max Exact Number CMQP GQP Number of integration nodes AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 12

16 Other active projects Matrix structure exploitation Analysis of structure variations in Hamiltonians and Overlap matrices across sequences of DFT eigenproblems. Re-use of data stored in Householder projectors in combination with tailored Jacobi rotations. Collaboration with P. Bientinesi (AICES) Data generation in LAPW-based methods Analisys of matrix entries generation in the FLEUR code with the aim of setting alternative strategies enabling modularity, flexibility and scalability. Collaboration with P. Bientinesi (AICES) and D. Wortmann (FZJ) Ab initio nanoscale interfacial heat transfer modeling Development of a method incorporating ab initio calculations in classical nonequilibrium molecular dynamics (NEMD) modeling. Aim at providing a detailed picture of phonon transport across interfaces from quantum physics simulations. Collaboration with M. Hu (AICES) AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 13

17 Thank you! For more information AICES Advisory Board Meeting. July 14th 2014 Edoardo Di Napoli Folie 14