Pescan benchmarks

Here we show results comparing different eigensolver strategies for five systems, whose properties are indicated in the table below. The calculations were performed on an IBM SP 5 with 8 processors per node.

system number of
atoms
number of
plane waves
E_ref
(eV)
E_cut
(Ryd)
N1xN2xN3 number of
eigestates
number of
processors
20Cd19Se 39 11,331 -4.8 6.88 39x79x39 10 8
83Cd81Se 164 34,143 -4.8 6.88 164x276x164 10 16
232Cd235Se 467 75,645 -4.8 6.88 467x697x467 10 16
534Cd527Se 1071 141,625 -3.8 (CBM)
-4.8 (VBM)
6.88 1061x1467x1061 10 32
dot5 1327 2,717,000 +0.6 (CBM)
-0.4 (VBM)
35.0 - 6 64
Qwire 66,624 2,266,000 -5.1 (CBM)
-5.4 (VBM)
5.0 - 5 64

General information and remarks:
(a) See State-of-the-art Eigensolvers for Electronic Structure Calculations of Large Scale Nano-systems for more information and references therein.
(b) A premature acceptance of not yet fully converged pairs (E,w) may also lead to numerical problems. This is particularly important for PRIMME, since it implements algorithms that are different in nature from PCG and LOBPCG.
Conclusions:
Recommendations (see also guidelines):

Based on the systems we have studied and the number os eigenstates we have computed, we recommend the default PRIMME MIN_MATVECS, with the restart size close to the number of required eigestates, and a basis size equal to 2 or 3 times the restart size. Also, for large systems, it is important to set tol to a smaller value that would be normally set for PCG or LOBPCG.



20Cd19Se
(atomic coordinates, pseudopotential for Cd, pseudopotential for Se)

a) Folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(1) 100 - - 4956 9.4
LOBPCG(1) - - - 4756 19.3
PARPACK(2) - 20 - 14630 27.2
PARPACK(3) - 25 - 9712 18.1
PARPACK(3) - 30 - 7474 14.1
PARPACK(3) - 35 - 5838 11.1
PRIMME MIN_MATVECS(1) - 16 8 1750 3.9
PRIMME MIN_TIME(1) - 16 8 4720 8.0

b) Unfolded spectrum:

algorithm basis size restart size matvecs time (s)
PARPACK(4) 20 - 10020 20.6
PARPACK(5) 25 - 1326 2.9
PARPACK(5) 30 - 1310 2.9
PARPACK(5) 35 - 1293 2.9
PRIMME MIN_MATVECS(1) 16 8 4185 10.0
PRIMME MIN_TIME(1) 16 8 3350 7.0

(1) Converges to the eigenstates -6.19176, -6.19176, -6.34729, -6.38668, -6.43238, -6.43238, -6.60944, -6.60945, -6.71546 and -6.71546.
(2) Converges to only 8 eigenstates if the maximum number of iterations (restarts) is set to 1000: -6.43238, -6.60945, -6.60944, -6.71546, -6.71546, -6.88809, -6.91577 and -6.98363.
(3) Converges to the eigenstates -6.43238, -6.43238, -6.60944, -6.60945, -6.71546, -6.71546, -6.88809, -6.91577, -6.98363 and -7.08253.
(4) Does not converge to any eigenstate with the maximum number of iterations (restarts) set to 1000.
(5) Converges to the eigenstates -6.19176, -6.34729, -6.38668, -6.43238, -6.43238, -6.60944, -6.60945, -6.71546, -6.71546 and -6.88809.



83Cd81Se
(atomic coordinates, pseudopotential for Cd, pseudopotential for Se)

a) Folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(1) 100 - - 17920 65.6
PCG(1) 200 - - 15096 52.7
LOBPCG(1) - - - 10688 69.9
PARPACK(2) - 50 - 24252 86.7
PARPACK(3) - 100 - 15126 60.3
PRIMME MIN_MATVECS(1) - 30 10 3670 12.7
PRIMME MIN_TIME(1) - 30 10 11808 36.7

b) Unfolded spectrum:

algorithm basis size restart size matvecs time (s)
PARPACK(4) 50 - 4073 16.7
PARPACK(3) 100 - 3273 15.7
PRIMME MIN_MATVECS(1) 30 10 5077 23.2
PRIMME MIN_TIME(1) 30 10 5059 19.6

(1) Converges to the eigenstates -5.72654, -5.72654, -5.78686, -5.83003, -5.83003, -5.85207, -5.98438, -6.01278, -6.01278 and -6.02422.
(2) Converges to the eigenstates -5.83003, -5.83003, -5.85207, -5.98438, -6.01278, -6.01278, -6.02422, -6.02751, -6.02751 and -6.11332.
(3) Converges to the eigenstates -5.83003, -5.83003, -5.85207, -5.98438, -6.01278, -6.01278, -6.02422, -6.02422, -6.02751 and -6.02751.
(4) Converges to the eigenstates -5.83003, -5.98438, -6.01278, -6.01278, -6.02422, -6.02422, -6.02751, -6.02751, -6.07613 and -6.11332.



232Cd235Se (atomic coordinates, pseudopotential for Cd, pseudopotential for Se)

a) Folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(1) 200 - - 15754 106.4
LOBPCG(1) - - - 11864 121.4
PARPACK(2) - 30 - 20060 133.0
PRIMME MIN_MATVECS(1) - 16 8 3742 25.0
PRIMME MIN_TIME(1) - 16 8 11708 73.4

b) Unfolded spectrum:

algorithm basis size restart size matvecs time (s)
PARPACK(1) 30 - 6205 47.6
PRIMME MIN_MATVECS(1) 16 8 11661 94.0
PRIMME MIN_TIME(1) 16 8 8736 61.6

(1) Converges to the eigenstates -5.51570, -5.51570, -5.53926, -5.58286, -5.58286, -5.60869, -5.67889, -5.69688, -5.69688 and -5.71672.
(2) Does not converge to any eigenstate with the maximum number of iterations (restarts) set to 500.


534Cd527Se (atomic coordinates, pseudopotential for Cd, pseudopotential for Se)

a) CBM, folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(1) 200 - - 7066 62.5
LOBPCG(1) - - - 6880 92.4
PARPACK(2) - 30 - 9362 86.1
PRIMME MIN_MATVECS(1) - 16 8 2176 21.3
PRIMME MIN_TIME(1) - 16 8 6212 52.3

b) CBM, unfolded spectrum:

algorithm basis size restart size matvecs time (s)
PARPACK(1) 30 - 2434 24.4
PRIMME MIN_MATVECS(1) 16 8 12813 143.7
PRIMME MIN_TIME(1) 16 8 6383 59.2

c) VBM, folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(3) 200 - - 23810 228.0
LOBPCG(3) - - - 16862 254.7
PARPACK(4) - 30 - 20060 190.9
PRIMME MIN_MATVECS(3) - 16 8 4762 46.0
PRIMME MIN_TIME(3) - 16 8 11259 109.1

d) VBM, unfolded spectrum:

algorithm basis size restart size matvecs time (s)
PARPACK(5) 30 - 7450 73.2
PRIMME MIN_MATVECS(3) 16 8 15449 172.7
PRIMME MIN_TIME(3) 16 8 11259 98.8

(1) Converges to the eigenstates -3.10118, -2.83770, -2.81099, -2.81099, -2.56043, -2.52280, -2.52280, -2.51159, -2.51159 and -2.37371.
(2) Converges to the eigenstates -2.81099, -2.81099, -2.56043, -2.52280, -2.52280, -2.51159, -2.51159, -2.37371, -2.28992 and -2.23882.
(3) Converges to the eigenstates -5.39076, -5.39076, -5.40313, -5.44361, -5.44361, -5.48316, -5.49335, -5.51804, -5.51804 and -5.52054.
(4) Does not converge to any eigenstate with the maximum number of iterations set to 500.
(5) Converges to the eigenstates -5.39076, -5.39076, -5.44361, -5.44361, -5.48316, -5.49335, -5.51804, -5.51804, -5.52054 and -5.52054.


dot5 (atomic coordinates, pseudopotential for Cd, pseudopotential for Se)

a) CBM, folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(1) 100 - - 220212 36726.8
PCG(2) 200 - - 335966 53670.2
LOBPCG(3) - - - 106518 20996.1
LOBPCG(4) - - - 148486 28379.9
PRIMME MIN_MATVECS(5) - 16 8 4788 823.1
PRIMME MIN_MATVECS(6) - 16 8 62334 10211.3
PRIMME MIN_TIME(7) - 16 8 271492 43242.0

b) VBM, folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(8) 100 - - 197712 31399.3
PCG(8) 200 - - 101904 15670.7
LOBPCG(9) - - - 120030 20779.0
LOBPCG(9) - - - 240030 41399.9
PRIMME MIN_MATVECS(10) - 16 8 11644 1926.7
PRIMME MIN_MATVECS(11) - 16 8 42910 7013.3
PRIMME MIN_MATVECS(12) - 16 8 54362 8757.7
PRIMME MIN_TIME(12) - 16 8 254810 39112.0

(1) Converges to 1.35724, 1.64617, 1.64617, 1.64617, 1.91451 and 1.92275 with the maximum number of allowed iterations set to 500. For two states r(E,w)~1.0E-3.
(2) Converges to 1.35724, 1.64617, 1.64617, 1.64617, 1.92353 and 1.92354 with the maximum number of allowed iterations set to 500. For two states r(E,w)~1.0E-3.
(3) Similar to (2) with the maximum number of allowed iterations set to 10000. For two states r(E,w)~1.0E-2.
(4) Similar to (2) with the maximum number of allowed iterations set to 20000. For two states r(E,w)~1.0E-2.
(5) Converges to 1.13394, 1.23783, 1.35674, 1.64388, 1.79981 and 1.85452. For two states r(E,w)~1.0E-2.
(6) Converges to 1.35724, 1.64617, 1.64617, 1.64617, 1.92364 and 1.92364 with tol=1.0E-10.
(7) Converges to 1.35724, 1.64617, 1.64617, 1.64617, 1.91777 and 1.92363 with tol=1.0E-10 but for two states r(E,w)~1.0E-3.
(8) Converges to -0.72398, -0.72398, -0.72398, -0.72946, -0.72946 and -0.72946.
(9) Similar to (8) with the maximum number of allowed iterations set to 10000. For six states r(E,w)~1.0E-3.
(10) Converges to -0.72400, -0.72948, -0.77577, -0.78304, -0.81558 and -0.84278. For all states r(E,w)~1.0E-4.
(11) Converges to -0.72398, -0.72398, -0.72398, -0.72946, -0.72946 and -0.72946 with tol=1.0E-9.
(12) Similar to (11); tol=1.0E-10.



Qwire

a) CBM, folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(1) 100 - - 21931 1072.0
LOBPCG(1) - - - 20337 1376.8
PRIMME MIN_MATVECS(2) - 16 8 5438 292.0
PRIMME MIN_MATVECS(3) - 16 8 8504 418.0
PRIMME MIN_TIME(2) - 16 8 16490 757.1
PRIMME MIN_TIME(3) - 16 8 28076 1392.0

b) VBM, folded spectrum:

algorithm nline basis size restart size matvecs time (s)
PCG(4) 100 - - 307270 15546.8
PCG(4) 200 - - 149726 7278.1
LOBPCG(5) - - - 56207 3690.1
PRIMME MIN_MATVECS(6) - 16 8 12002 606.1
PRIMME MIN_MATVECS(6) - 20 10 11596 1502.6
PRIMME MIN_MATVECS(6) - 30 10 12670 2571.6
PRIMME MIN_MATVECS(7) - 20 10 24068 1240.6
PRIMME MIN_MATVECS(7) - 30 10 26326 1424.2
PRIMME MIN_TIME(6) - 16 8 36310 1682.6

(1) Converges to -4.89017, -4.71187, -4.68034, -4.68034 and -4.55008.
(2) Similar to (1) but for all states r(E,w)~1.0E-5.
(3) Similar to (1); tol=1.0E-8.
(4) Converges to -5.73241, -5.73241, -5.73423, -5.74245 and -5.74360 with the maximum number of allowed iterations set to 500. For one state r(E,w)~1.0E-5.
(5) Similar to (4) but for all states r(E,w) < tol.
(6) Converges to -5.73241, -5.73423, -5.74249, -5.74365 and -5.75513 but for all states r(E,w)~1.0E-5.
(7) Similar to (4); tol=1.0E-8.