f dotsuki/pdf/2017-kobe-sparse.pdfspm 1. 2. l1 – • • (ill-conditioned ) – • our code will...

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– f d

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– 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

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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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

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’

∵

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!

✓ SpM

1.

2. L1

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• (ill-conditioned )

–

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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

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

A “new” rep

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Everything in IR basis!

- QMC measurement

- Perturbative expansion

- Bethe-Salpeter equation

…

• Two problems in quantum many-body calculations

I.

II.

• Our solution

– SVD

– L1

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