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Setting up diagonalization engine

ELPA Library

For static calculations, it is recommended to use ELPA Library, which has better performance than ScaLapack. In particular, ELPA allows for the utilization of GPUs which provide a significant boost for calculations. In order to activate ELPA lib in predefines.h set:

// select diagonalization routine
#define DIAGONALIZATION_ROUTINE ELPA

Moreover, you need to inspect carefully part:

// ---------------------- ELPA SETTINGS ---------------------------
// Fill this part only if ELPA library is used for diagonnalization

// uncomment it if you want to activate GPU for diagonalizations 
#define ELPA_USE_GPU

// Select ELPA kernels
#define ELPS_USE_SOLVER ELPA_SOLVER_1STAGE
#define ELPA_USE_COMPLEX_KERNEL ELPA_2STAGE_COMPLEX_DEFAULT
#define ELPA_USE_REAL_KERNEL ELPA_2STAGE_REAL_DEFAULT

// Fraction of eigenvectors to be extracted in each cycle.
// 1.0 corresponds to extraction if all eigenvectors (USE IT IF YOU YOU ARE NOT SURE)
// NOTE: value of this parameter should assure that all eigenstates below requested Ec are extracted.  
// NOTE: For 3D case this value typically can be set to 0.78
#define ELPA_NEV_FRACTION 1.0

Documentation

  1. Eigenvalue SoLvers for Petaflop-Applications (ELPA)
  2. Wiki: Eigenvalue SoLvers for Petaflop-Applications (ELPA)
  3. ELPA installation guide

Publications about ELPA performance

  1. GPU-Acceleration of the ELPA2 Distributed Eigensolver for Dense Symmetric and Hermitian Eigenproblems

ScaLapack library

If the target system does not provide ELPA library user can use (standard) diagonalization library: ScaLAPACK. W-SLDA Toolkit can utilize the following ScaLapack diagonalization engines:

#define DIAGONALIZATION_ROUTINE PZHEEVR

or

#define DIAGONALIZATION_ROUTINE PZHEEVD

It is recommended to use PZHEEVR. This engine takes advantage from the fact that typically we extract only a fraction of eigenstates. However, we find that in some rare cases (system dependent) this routine does not work correctly. In such a case, PZHEEVD should be used.

Benchmarks & Scalings

All tests correspond to the extraction of all eigenvectors.

Table

matrix size p q mb nb prec. routine system time [sec] cost
32,768 = 2x128^2 6 8 16 16 real ELPA (2-GPU) Cygnus 93 0.052 nh
45,000 = 2x150^2 6 8 16 16 real ELPA (2-GPU) Cygnus 217 0.12 nh
45,000 = 2x150^2 6 8 16 16 complex ELPA (2-GPU) Cygnus 860 0.478 nh
65,536 = 2x32^3 24 28 32 32 complex ELPA (2-GPU) Summit 115 0.52 nh
65,536 = 2x32^3 24 28 8 8 complex ELPA (1-GPU) Summit 118 0.52 nh
128,000 = 2x40^3 24 28 8 8 complex ELPA (1-GPU) Summit 435 1.93 nh
128,000 24 28 32 32 complex ELPA (2-GPU) Summit 511 2.27 nh
128,000 = 2x40^3 20 20 32 32 complex ELPA (1-GPU) Daint 220 24.4 nh
128,000 54 64 32 32 complex ELPA (2-CPU) Daint 677 54.1 nh
128,000 54 64 32 32 complex PZHEEVR Daint 945 75.6 nh
147,456 = 4x64x24^2 24 25 32 32 complex ELPA (1-GPU) Daint 375 62.5 nh
147,456 = 2x768x96 18 18 16 16 double ELPA (1-GPU) Daint 395 35.6 nh
221,184 = 2x48^3 46 84 32 32 complex ELPA (2-GPU) Summit 603 15.4 nh
221,184 46 84 16 16 complex ELPA (1-GPU) Summit 736 18.8 nh
221,184 46 84 16 16 complex ELPA (2-GPU) Summit 3098 79.2 nh
221,184 46 84 16 16 complex PZHEEVD Summit 5995 153.2 nh
500,000 = 2x50^2x100 96 112 16 16 complex ELPA (1-GPU) Summit 2,109 150.0 nh
524,288 = 2x64^3 96 112 16 16 complex ELPA (1-GPU) Summit 2,217 157.7 nh
746,496 = 2x72^3 112 192 16 16 complex ELPA (1-GPU) Summit 3,436 488.7 nh
1,769,472 = 2x96^3 300 560 32 32 complex ELPA (1-GPU) Summit 52,024 57,804 nh

(1-GPU): ELPA_SOLVER_1STAGE, ELPA_2STAGE_COMPLEX_GPU or ELPA_2STAGE_REAL_GPU
(1-CPU): ELPA_SOLVER_1STAGE, ELPA_2STAGE_COMPLEX_DEFAULT or ELPA_2STAGE_REAL_DEFAULT
(2-GPU): ELPA_SOLVER_2STAGE, ELPA_2STAGE_COMPLEX_GPU or ELPA_2STAGE_REAL_GPU
(2-CPU): ELPA_SOLVER_2STAGE, ELPA_2STAGE_COMPLEX_DEFAULT or ELPA_2STAGE_REAL_DEFAULT

Plots

These scalings are derived empirically: points correspond to real measurement on target system, while line shows a fit of ideal scaling for level-3 rutines (\sim N^3)

Summit

The scaling was derived within ALCC grant Quantum Turbulence in Fermi Superfluids.