1 f d 2016 10 2017 2 2 2 High Performance Comp. 3 - - PowerPoint PPT Presentation

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1 f d 2016 10 2017 2 2 2 High Performance Comp. 3 - - PowerPoint PPT Presentation

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1 f d 2016 10 2017 2 2 2 High Performance Comp. 3 JO, Ohzeki, Shinaoka, Yoshimi, arXiv:1702.03056 Shinaoka, JO, Ohzeki, Yoshimi, arXiv:1702.03054 4 INTRODUCTION: Two Problems in Quantum Many-body

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f d –

  • – 2016

10  – 2017 2 2

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High Performance Comp.

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JO, Ohzeki, Shinaoka, Yoshimi, arXiv:1702.03056 Shinaoka, JO, Ohzeki, Yoshimi, arXiv:1702.03054

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INTRODUCTION: Two Problems in Quantum Many-body Computations

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c.f.

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扱いにくい フーリエ 変換 扱いやすい

（ダイアグラム展開 量子モンテカルロ法）

解析接続

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CT-QMC data 8-fold degenerate impurity Anderson model

Vidberg, Serene, 1977

8 lines should coincide The standard method: Pade

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→

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G ρ difficulty: K (ill-conditioned matrix)

Lehmann (NaN) 

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  • A. W. Sandvik, PRB 57, 10287 (1998)
  • S. Fuchs, T. Pruschke, and M. Jarrell, PRE 81, 056701 (2010)
  • K. S. D. Beach, arXiv:cond-mat/0403055
  • A. W. Sandvik, PRE 94, 063308 (2016)
  • K. S. D. Beach, R. J. Gooding, and F. Marsiglio, PRB 61, 5147 (2000)
  • A. Dirks et al., Phys. Rev. E 87, 023305 (2013).
  • F. Bao et al., PRB 94, 125149 (2016)
  • O. Goulko et al., PRB 95, 014102 (2017).
  • G. Bertaina, D. Galli, and E. Vitali, arXiv:1611.08502.

L.-F. Arsenault et al., arXiv:1612.04895.

Stochastic method Growing attempts Maximum entropy method m : “default model” = m

  • M. Jarrell, J. E. Gubernatis, Phys. Rep. 269, 133 (1996)
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PROBLEM I

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More complicated object fermionic bosonic

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URu2Si2 Wiebe et al. 2007

+ + ... + + + ...

テンソル積

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PROBLEM II

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We will show…

  • Two problems are “two sides of the same coin”
  • Solution to

– Problem I (analytical continuation) – Problem II (two-particle objects)

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SOLUTION to problem I Sparse-Modeling (SpM) Analytical Continuation

JO, Ohzeki, Shinaoka, Yoshimi, arXiv:1702.03056

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(SpM)

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劣決定系

スパース性

x N

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スパース性

=

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Q. ρ A.

K (ill-conditioned matrix)

ρ’ G’

∵

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L1

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LASSO (Least Absolute Shrinkage of Selection Operators)

  • R. Tibshirani, J. R. Stat. Soc. B 58, 267 (1996)

L1 スパース性

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ADMM (alternating direction method of multipliers) Boyd et al., Foundations and Trends in Machine Learning 3, 1 (2011)

( )

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QMC (1)

M=4000

(2)

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λ

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For a given λ

automatic!

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✓ SpM

1.

  • 2. L1

–

  • (ill-conditioned

)

–

  • Our code will be available soon on GitHub

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SOLUTION to problem II Intermediate Representation (IR)

Shinaoka, JO, Ohzeki, Yoshimi, arXiv:1702.03054

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M=4000 わずか7成分！

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SVD

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Λ→0で ルジャンドル多項式 に一致！

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c.f. Boehnke et al. 2011

わずか5, 6要素で十分！

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Legendre (Boehnke et al. 2011) (Legendre) SVD

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The solution has been there

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Lehmann (basic equation)

SVD

c.f. SV truncation Creffield et al. 1995 Bryan method in MaxEnt Bryan 1990

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A “new” rep

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Everything in IR basis!

  • QMC measurement
  • Perturbative expansion
  • Bethe-Salpeter equation

…

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  • Two problems in quantum many-body calculations

I. II.

  • Our solution

– SVD – L1

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