http://www.elses.jp

A01 星班

新しい数理アルゴリズムを中核とした電子状態計算と計算機科学の融合（鳥取大学）MD 星健夫

1, Physics (result)2, Algorithm
3, Issues beyond
the present work

研究協力者 (A01 張班)：
曽我部知広（愛知県立大）張紹良、宮田考史（名古屋大）
山本有作（神戸大）
関連講演：P27 秋山（Python を用いた原子構造可視化ツール）http://www.elses.jp‘ELSES’ : Our code for ‘order-N’ electronic structure calculations( ELSES = Extra Large-Scale Electronic Structure calculation )Elapse time (sec / MD step)

Benchmark for amourphous-like conjugated polymer; poly-(9,9 dioctyl-fluorene)
Hoshi et al., JPCM in press; Preprint (arXiv:1202.0098)

105(a) ‘Order-N’ scaling
upto 107 atoms(b) Parrallel effiency
upto 1,024 coreswith 107 atoms

104

MPII/OpenMP103
102
105

106

107

Number of atoms

108 101102

103

104

Number of CPU coreshttp://www.elses.jp‘ELSES’ : Our code for ‘order-N’ electronic structure calculations( ELSES = Extra Large-Scale Electronic Structure calculation )Benchmark for amourphous-like conjugated polymer; poly-(9,9 dioctyl-fluorene)
Hoshi et al., JPCM in press; Preprint (arXiv:1202.0098)

Elapse time (sec / MD step)

See the web page for papers

105(a) ‘Order-N’ scaling
upto 107 atoms(b) Parrallel effiency
upto 1,024 cores・Oringinal linear�algebraic order�N algorithms
with generalized eigen�valueequation
with 107 atoms

104
・Applicable to both metals and insulators
・Modelled �TB�type� Hamiltonians
based on ab initio calculaiton

103
102
105

106

107

Number of atoms

108 101102103

104

Number of CPU coreshttp://www.elses.jp

ELSESによるナノ物質研究→産学連携・新物質(日本発)・基準物質

アモルファス状π共役高分子系（Hoshi et al. ISTCP-VII, Waseda, 2011;
JPCM in press; arXiv:1202.0098 )螺旋型金ナノワイヤ
L=12nmナノ多結晶ダイアモンド形成 (sp2-sp3 ナノドメイン変換)
17 nm

シリコン脆性MD

イオン溶液/リチウム電解質（西野他, SSI18, Warsaw (2011);電池討論会(2012)）

Krylov subspace : a linear-albegraic concept
A: matrix, b: ʻseedʼ vector,: subspace dimension (iteration number)→ foundation of many iterative algorithm
such as conjugate-gradient (CG) algorithm
ExamplesProblem

Solution at the ν-th step: initial guess

Methods for the large-scale electronic state calculation
Krylov subspace algorithms is used
for the generalized shifted linear equations(1)(: the j-th unit vector )instead of the generalized eigen-value equation(a)H, SNG!

eigen states y(1)( physical quantity )Methods for the large-scale electronic state calculation
Krylov subspace algorithms is used
for the generalized shifted linear equations(1)(: the j-th unit vector )insteadof theCalculaiton
ofgeneralized
the Greenʼs eigen-valuefunction andequation
the density matrix
(a)
(2)
(3)
(4)

Novel Krylov subspace algorithms for generalized shifted equation
Teng et al. PRB83, 165103 (2011); Sogabe et al., talk (2009); paper submitted
Yamashita et al.,Trans of JSIAM 21, 241 (2011);
Hoshi et al.., JPCM in press (Preprint:arXiv:1202.0098).
(1) How to construct a sub(Hilbert) spaceby matrix-vector multiplications with H and S(2) How to determine the solution with a given subspace(2-a) Gerlerkin Principle(2-b) Collinear ResidualS≠I

S=I *gLanczos,gArnoldi,mArnodi, Arnoldi(M;W;G)

subs. diag.

gsCOCG,
gsQMR

sCOCG,sQMR*Takayama et al., JPSJ73, 1519 (2004); Takayama et al. PRB 73, 165108(2006);
Sogabe et al. ETNA31, 126 (2008)

Novel Krylov subspace algorithms for generalized shifted equation
Teng et al. PRB83, 165103 (2011); Sogabe et al., talk (2009); paper submitted
Yamashita et al.,Trans of JSIAM 21, 241 (2011);
Hoshi et al.., JPCM in press (Preprint:arXiv:1202.0098).

Six algorithms(1) How to construct a sub(Hilbert) spaceby matrix-vector multiplications with H and S(2) How to determine the solution with a given subspace→ Which is the bestfor electronic structure calculation ?
S≠I
S=I
*
→ It may depend on material and purpose(physicalgLanczos,gArnoldi,
quantity to be calculated)(2-a) Gerlerkin Principle(2-b) Collinear Residual

mArnodi, Arnoldi(M;W;G)
gsCOCG,gsQMR

subs. diag.
sCOCG,sQMR*Takayama et al., JPSJ73, 1519 (2004); Takayama et al. PRB 73, 165108(2006);
Sogabe et al. ETNA31, 126 (2008)

Physical quantities calculated from the Krylov subpace method
ex. πconjugated polymer: poly-(9,9 dioctil ﬂuorene)
Ref. Hoshi, et al. ISTCP-VII, Waseda (2011);
JPCM in press; arXiv:1202.0098(1) Energy and force(for molecular-dynamics simulation)(2) Electronic spectrum(density of states, and so on)(3) Speciﬁc eigen states(Fig. HOMO and LUMO)

Summary of the present work
(1) Large-scale eletrnic state calculation calculation
Order-N property with upto 10M atoms
MPI/OpenMP parallelism upto 1024 CPU cores
MD study (upto 10nm-scale material ; upto 10ps ~ ns)(2) Algorithm design for physical quantities
Six algorithms
based on the generalized shifted linear eqns (zS-H)x=b)
Three types of physical quantities

H, SNG!

eigen states y( physical quantity )Issues beyond the present works(1) Building numerial libraries with
the novel direct (O(N3)) solver suitable to parallellism →P02山本→ Level-3 BLAS type (Y. Yamamoto (Kobe U), 2005)

Two-stage tridiagonalization ( dense→band→tridiag. )→ Choice or hybrid between
the above solver and the Krylov subspace solvers(2) Auto-optimization (auto-choice, auto-tuning)in the Application-Algorithm-Architecture collaboration→ operation cost vs communication cost vs memory cost

ex. memory-saving scheme for 105 atom of NPD( memory size : 12GB→1GB )研究の俯瞰図（将来展望まで含めた)

新しい数理アルゴリズムを中核とした電子状態計算と計算機科学の融合
electrnic state calculation
Application : Large-scale(nano material with 103-107 atoms)
Algorithm design for application(physical quantitiy)

Algorithm :Novel linear-algebraic solvers( Krylov solvers and/or Direct solvers)(this work)(future work)

Algorithm design for archtecture(with many-core CPU) ~Architecture : PC
(post-petascale) supercomputer

Auto-optimization(Auto-choice)(Auto-tuning)(future work)