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Adaptively Compressed Polarizability Operator For Accelerating Large Scale ab initio Phonon Calculations Ze Xu 1 Lin Lin 2 Lexing Ying 3 1 , 2 UC Berkeley 3 ICME, Stanford University BASCD, December 3rd 2016 Ze Xu (UC Berkeley) ACP BASCD 2016 1

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Adaptively Compressed Polarizability Operator

For Accelerating Large Scale ab initio Phonon Calculations Ze Xu1 Lin Lin2 Lexing Ying3

1,2UC Berkeley 3ICME, Stanford University

BASCD, December 3rd 2016

Ze Xu (UC Berkeley) ACP BASCD 2016 1 / 40

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Table of Contents

1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion

Ze Xu (UC Berkeley) ACP BASCD 2016 2 / 40

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Phonon

Table of Contents

1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion

Ze Xu (UC Berkeley) ACP BASCD 2016 3 / 40

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Phonon

What is Phonon?

“In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter... it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.” —wikipedia

Ze Xu (UC Berkeley) ACP BASCD 2016 4 / 40

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Phonon

Lattice Wave

Figure 1: Lattice wave example. Figure from wikipedia.

Ze Xu (UC Berkeley) ACP BASCD 2016 5 / 40

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Phonon

Phonon Dispersion Relations

Figure 2: Phonon dispersion and densities of states of gallium arsenide (GaAs). Figure from S. Baroni et al., 2001 .

Ze Xu (UC Berkeley) ACP BASCD 2016 6 / 40

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Phonon

Why Care?

Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40

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Phonon

Why Care?

  • Thermal conductivity; specific heat;

Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40

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SLIDE 9

Phonon

Why Care?

  • Thermal conductivity; specific heat;
  • Elastic neutron scattering;

Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40

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SLIDE 10

Phonon

Why Care?

  • Thermal conductivity; specific heat;
  • Elastic neutron scattering;
  • Electrical conductivity;

Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40

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SLIDE 11

Phonon

Why Care?

  • Thermal conductivity; specific heat;
  • Elastic neutron scattering;
  • Electrical conductivity;
  • Electron-phonon interaction related topics;

Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40

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Phonon

Why Care?

  • Thermal conductivity; specific heat;
  • Elastic neutron scattering;
  • Electrical conductivity;
  • Electron-phonon interaction related topics;

Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40

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Phonon

Elastic neutron scattering

  • Thermal conductivity; specific heat;
  • Elastic neutron scattering;
  • Electrical conductivity;
  • Electron-phonon interaction related topics;

Figure 3: In elastic neutron-scattering, the neutron bounces off the bombarded nucleus without exciting or destabilizing it. Figure from The Schlumberger Oilfield Glossary.

Ze Xu (UC Berkeley) ACP BASCD 2016 8 / 40

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Phonon

Electron-phonon interaction

  • Thermal conductivity; specific heat;
  • Elastic neutron scattering;
  • Electrical conductivity;
  • Electron-phonon interaction related topics;
  • major contributor to electrical resistance in most inorganic

metals and semiconductors above zero (very low) temperature;

  • Overheating problem;

Ze Xu (UC Berkeley) ACP BASCD 2016 9 / 40

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Phonon

Dynamical Matrix

DI,J = 1 √MIMJ ∂2Etot({RI}) ∂RI∂RJ ,

  • MI mass of the I-th atom, I = 1, . . . , NA.
  • RI ∈ Rd position of the I-th atom, I = 1, . . . , NA.
  • D ∈ RdNA×dNA

Ze Xu (UC Berkeley) ACP BASCD 2016 10 / 40

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SLIDE 16

DFT

Table of Contents

1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion

Ze Xu (UC Berkeley) ACP BASCD 2016 11 / 40

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DFT

Density Functional Theory

Ab initio calculation – DFT.

  • most widely used;
  • exact description of ground state properties:
  • electron density, energy, and atomic forces.

Ze Xu (UC Berkeley) ACP BASCD 2016 12 / 40

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DFT

Kohn-Sham Density Functional Theory

EKS({ψi}; {RI}) =1 2

Ne

  • i=1
  • |∇ψi(r)|2 dr +
  • Vion(r; {RI})ρ(r) dr

+ 1 2

  • vc(r, r′)ρ(r)ρ(r′) dr dr′ + Exc[ρ]

+ EII({RI}). (1)

Ze Xu (UC Berkeley) ACP BASCD 2016 13 / 40

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DFT

Derivative

FI = −

  • ∂VI

∂RI (r − RI)ρ(r) dr − ∂EII({RI}) ∂RI . ∂2Etot({RI}) ∂RI∂RJ =

  • ∂VI

∂RI (r − RI)δρ(r) δRJ dr + δI,J

  • ρ(r)∂2VI

∂R2

I

(r − RI) dr + ∂2EII({RI}) ∂RI∂RJ .

Ze Xu (UC Berkeley) ACP BASCD 2016 14 / 40

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SLIDE 20

DFT

Derivative

FI = −

  • ∂VI

∂RI (r − RI)ρ(r) dr − ∂EII({RI}) ∂RI . (2) ∂2Etot({RI}) ∂RI∂RJ =

  • ∂VI

∂RI (r − RI)δρ(r) δRJ dr+δI,J

  • ρ(r)∂2VI

∂R2

I

(r − RI) dr +∂2EII({RI}) ∂RI∂RJ . (3)

Ze Xu (UC Berkeley) ACP BASCD 2016 15 / 40

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DFT

Polarizability Operator

δρ(r) δRJ =

  • δρ(r)

δVion(r′) ∂VJ ∂RJ (r′ − RJ) dr′. Fr´ echet derivative χ(r, r′) := δρ(r) δVion(r′) is the reducible polarizability

  • perator.

Ze Xu (UC Berkeley) ACP BASCD 2016 16 / 40

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DFT

KSDFT Cont.

Kohn-Sham equations (W. Kohn and L. Sham 1965) H[ρ]ψi =

  • −1

2∆ + V[ρ]

  • ψi = εiψi,

(4)

  • ψi(r)ψj(r) dr = δij,

ρ(r) =

Ne

  • i=1

|ψi(r)|2 . (5)

  • εi energy level; ψi orbitals;
  • index i = 1, . . . , Ne called occupied states; i = Ne + 1, . . . unoccupied

states;

  • εg = εNe+1 − εNe band gap;
  • V[ρ](r) = Vion(r; {RI}) +
  • vc(r, r′)ρ(r′) dr′ + Vxc[ρ](r)
  • Vion =

I VI,

gJ = ∂VJ

∂RJ .

Ze Xu (UC Berkeley) ACP BASCD 2016 17 / 40

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DFT

Sternheimer Equations

Q(εi − H)Qζij = Q(ψi ⊙ gj). i = 1, . . . , Ne, j = 1, . . . , dNa (6)

  • Q projection onto unoccupied states;
  • ζij defined as the solution to the equation.

This is the core part of density-functional perturbation theory (DFPT) (S. Baroni et al. 2001).

Ze Xu (UC Berkeley) ACP BASCD 2016 18 / 40

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DFT

How to solve that?

Q(εi − H)Qζij = Q(ψi ⊙ gj). i = 1, . . . , Ne, j = 1, . . . , dNa

  • SVD?
  • Frozen phonon approach? (Or finite difference in math)
  • Compression?

Ze Xu (UC Berkeley) ACP BASCD 2016 19 / 40

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DFT

How to solve that? cont.

Q(εi − H)Qζij = Q(ψi ⊙ gj). i = 1, . . . , Ne, j = 1, . . . , dNa

  • SVD?
  • Frozen phonon approach? (Or finite difference in math)
  • Compression!

ACP formulation reduces the computational complexity of phonon calculations from O(N4

e ) to O(N3 e ) for the first time (X.et al. 2016).

Ze Xu (UC Berkeley) ACP BASCD 2016 20 / 40

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DFT

The Dyson Equation

Irreducible polarizability operator χ0 = δρ

δV

χ = χ0 + χ0vhxcχ

  • r

U = χ0G + χ0vhxcU, U = χG. (7) So compression of χ0 can only be done adaptively.

Ze Xu (UC Berkeley) ACP BASCD 2016 21 / 40

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ACP

Table of Contents

1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion

Ze Xu (UC Berkeley) ACP BASCD 2016 22 / 40

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ACP

Adaptively Compressed Polarizability Operator

Q(εi − H)Qζij = Q(ψi ⊙ gj). i = 1, . . . , Ne, j = 1, . . . , dNa

  • Chebyshev interpolation – disentangle the energy dependence on the

left

  • Interpolative separable density fitting method – compress the rhs

vectors.

  • Adaptively compressed – cost stays the same across iterations.

Ze Xu (UC Berkeley) ACP BASCD 2016 23 / 40

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ACP

Chebyshev Interpolation

Pick {εc} ∈ I ≡ [ε1, εNe]. Lagrange interpolation ζ =

Nc

  • c=1
  • ζc
  • c′=c

ε − εc′

  • εc −

εc′ .

Ze Xu (UC Berkeley) ACP BASCD 2016 24 / 40

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ACP

Interpolative Decomposition

Mij = ψi ⊙ gj,

  • r

Mij(r) = ψi(r)gj(r). Mij(r) ≈

Nµ

  • µ=1

ξµ(r)Mij(rµ) ≡

Nµ

  • µ=1

ξµ(r)ψi(rµ)gj(rµ). (8) (H. Cheng, Z. Gimbutas, P. G. Martinsson, and V. Rokhlin, 2005)

Ze Xu (UC Berkeley) ACP BASCD 2016 25 / 40

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ACP

Interpolative Separable Density Fitting

Two-step procedure: subsampled random Fourier Transform and QR decomposition. (J. Lu and L. Ying, 2015)

Ze Xu (UC Berkeley) ACP BASCD 2016 26 / 40

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ACP

Reconstruction

To avoid O(N4

e ) complexity, construct

Wµ = 2

Ne

  • i=1

ψi ⊙  

Nc

  • c=1
  • ζcµ
  • c′=c

εi − εc′

  • εc −

εc′   ψi(rµ). (9) χ0gj ≈

Nµ

  • µ=1

Wµgj(rµ), (10)

  • r formally, we have χ0 ≈

χ0 := WΠT .

Ze Xu (UC Berkeley) ACP BASCD 2016 27 / 40

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ACP

Iterative scheme

Introduce the following change of variable U = U − B, B = v−1

hxcG,

(11)

  • U =

χ0[ U]vhxc U + B. (12) Iterative scheme: (a)Construct χk

0[

U k] = W k(Πk)T (b) U k+1 =

  • I − W k(Πk)T vhxc

−1 B = B + W k I − (Πk)T vhxcW k−1 (Πk)T vhxcB. (13)

Ze Xu (UC Berkeley) ACP BASCD 2016 28 / 40

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Numerical Examples

Table of Contents

1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion

Ze Xu (UC Berkeley) ACP BASCD 2016 29 / 40

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Numerical Examples

One-dimensional Model

  • 0.2

0.2 0.4 0.6 0.8 1

phonon frequency

0.5 1 1.5 2 2.5 3 3.5 4

ϱD

DFPT ACP FD

  • 0.2

0.2 0.4 0.6

phonon frequency

1 2 3 4 5 6 7 8 9

ϱD

DFPT ACP FD

Figure 4: Phonon spectrum for the 1D systems computed using ACP, DFPT, and FD, for both (a) insulating and (b) semiconducting systems.

Ze Xu (UC Berkeley) ACP BASCD 2016 30 / 40

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Numerical Examples

One-dimensional Model cont.

40 60 80 100 120 140

System size: # of Atom

102 103 104

Time (s)

DFPT ACP FD

40 60 80 100 120 140

System size: # of Atom

102 103

Time (s)

DFPT ACP FD

Figure 5: Computational time of 1D examples. Comparison among DFPT, ACP, and FD for (a) insulating, and (b) semiconducting systems, respectively.

Ze Xu (UC Berkeley) ACP BASCD 2016 31 / 40

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Numerical Examples

Two-dimensional Model

20 40 60 80 100 120 10 20 30 40 50 60 70 10 20 30 40 50 60 70

index(i)

1 2 3 4 5

εi

  • ccupied

unoccupied

Figure 6: The electron density ρ of the 2D system with defects (a), and the

  • ccupied and unoccupied eigenvalues (b).

Ze Xu (UC Berkeley) ACP BASCD 2016 32 / 40

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Numerical Examples

Two-dimensional Model cont.

  • 5

5 10 15 20 25

phonon frequency

0.02 0.04 0.06 0.08 0.1 0.12 0.14

ϱD

DFPT ACP

Figure 7: Phonon spectrum for the 2D system with defects. NA = 69.

Ze Xu (UC Berkeley) ACP BASCD 2016 33 / 40

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What’s More?

Table of Contents

1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion

Ze Xu (UC Berkeley) ACP BASCD 2016 34 / 40

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SLIDE 40

What’s More?

Split Representation of ACP

χ0(r, r′) = 2

Ne

  • i=1

∞

  • j=Ne+1

ψi(r)ψj(r)ψi(r′)ψj(r′) εi − εj = 2  

Ne

  • i=1

Nt

  • j=Ne+1

ψi(r)ψj(r)ψi(r′)ψj(r′) εi − εj +

Ne

  • i=1

∞

  • j=Nt+1

ψi(r)ψj(r)ψi(r′)ψj(r′) εi − εj   := χ(1) + χ(2)

0 .

Ze Xu (UC Berkeley) ACP BASCD 2016 35 / 40

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What’s More?

Silicon with 8 Atoms

External Perturbation (avg along z-axis) 5 10 15 20 25 30 5 10 15 20 25 30

  • 1.5
  • 1
  • 0.5

0.5 1 1.5

ACP Response (avg along z-axis) 5 10 15 20 25 30 5 10 15 20 25 30

  • 0.08
  • 0.06
  • 0.04
  • 0.02

0.02 0.04 0.06

Figure 8: External perturbation and response.

Ze Xu (UC Berkeley) ACP BASCD 2016 36 / 40

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What’s More?

Silicon with 8 Atoms cont.

Error of ACP Response (avg along z-axis) 5 10 15 20 25 30 5 10 15 20 25 30

  • 4
  • 2

2 4 6 #10-8

Error of FD Response (avg along z-axis) 5 10 15 20 25 30 5 10 15 20 25 30

  • 6
  • 4
  • 2

2 4 6 8 10 12 #10-6

Figure 9: Error on response.

Ze Xu (UC Berkeley) ACP BASCD 2016 37 / 40

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Conclusion

Table of Contents

1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion

Ze Xu (UC Berkeley) ACP BASCD 2016 38 / 40

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SLIDE 44

Conclusion

Conclusion

  • ACP formulation reduces the computational complexity of phonon

calculations from O(N4

e ) to O(N3 e ) for the first time.

  • Recently we extend the formulation to cope with real materials.
  • L. Lin, Z. Xu and L. Ying, Adaptively compressed polarizability
  • perator for accelerating large scale ab initio phonon calculations,

SIAM Multiscale Model. Simul. accepted, 2016

Ze Xu (UC Berkeley) ACP BASCD 2016 39 / 40

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SLIDE 45

Conclusion

Conclusion

  • ACP formulation reduces the computational complexity of phonon

calculations from O(N4

e ) to O(N3 e ) for the first time.

  • Recently we extend the formulation to cope with real materials.
  • L. Lin, Z. Xu and L. Ying, Adaptively compressed polarizability
  • perator for accelerating large scale ab initio phonon calculations,

SIAM Multiscale Model. Simul. accepted, 2016 Thanks for your attention.

Ze Xu (UC Berkeley) ACP BASCD 2016 40 / 40