f dotsuki/pdf/2017-kobe-sparse.pdfspm 1. 2. l1 – • • (ill-conditioned ) – • our code will...
TRANSCRIPT
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– f d
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– 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 Problemsin 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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URu2Si2Wiebe 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
F
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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スパースモデリングで本質を見抜く