FFT in quantum physicsMaciej E. Marchwiany, Maciej Szpindler

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Table of contents

●Fourier tansform●FFT●Poisson equation●Solving Poisson equation●Poisson equation in DFT●P3DFFT●P3DFFT: initialization●P3DFFT: Array Decomposition●P3DFFT: Forward transform●P3DFFT: Backward transform

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Fourier transform

FT :ℝN→ℂ

N

The Fourier transform is a mathematical operation with many applications in physics and engineering that expresses a mathematical function of time as a function of frequency, known as its frequency spectrum; Fourier's theorem guarantees that this can always be done.

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Applications

● engineering● signal processing● electronics● cosmology ● NMR spectroscopy● quantum chemistry● computational material science● ...

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Fourier transform

f ∈L1 ℝN

f k =∫ℝ

N f x e−2 i x k d x

f k = 1

2π N ∫ℝ

N f x e−2 i xk d x

f k = 1

2π N /2∫ℝ

N f x e−2 i xk d x

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Inverse Fourier transform

f x =1

2πN∫ℝ

N f k e2 i k xd k

f k =∫ℝ

N f x e2 i k xd k

f k = 1

2π N /2∫ℝ

N f x e2 i k x d k

f k =∫ℝ

N f x e−2 i x k d x

f k = 1

2π N ∫ℝ

N f x e−2 i xk d x

f k = 1

2π N /2∫ℝ

N f x e−2 i xk d x

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Interpretation

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Discrete Fourier transform

xk=∑ xne−2i π

kN

nxn=

1N

∑ xk e2i π

kN

n

∫dx →∑n

k=k0,k1,…, kN−1

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FFT

A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory.

By far the most common FFT is the Cooley–Tukey algorithm. This is a divide and conquer algorithm that recursively breaks down a DFT of any composite size N = N1N2 into many smaller DFTs of sizes N1 and N2, along with O(N) multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966).

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FFT algorithms

● Cooley–Tukey algorithm● Prime-factor FFT algorithm● Bruun's FFT algorithm● Rader's FFT algorithm● Bluestein's FFT algorithm● Odlyzko–Schönhage algorithm

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Fourier transform on operators

B x B k

∂

∂ x ik x

∇2 x,y,z −k 2 k 2=∣k∣2

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Poisson equation

∇2V H r =−4 πρ r

V H r

ρ r

V H r =∫ρ R

∣r−R∣d R Coulomb potential

−k 2V Hk =−4πρ k

Electrons density

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Solving Poisson equation

∇2V H r =−4 πρ r

V H r

Poisson equation solver

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Solving Poisson equation

∇2V H r =−4 πρ r −k 2V H

k =−4πρ k

V H k =4π

k 2ρ k V H r

FFT

iFFT

1/k2

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k=0

k=0

ρ k=0 =∫ ρ x d x=⟨ ρ⟩

mean value of x⟨ x ⟩

f k=0=∫ℝN f x e−2 i x 0d x=∫ℝN f x d x

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k=0

⟨ ρ⟩=0 ρ k=0 =0

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k=0

⟨ ρ⟩=0 ρ k=0 =0

⟨ ρ⟩≠0 ρ k=0 ≠0

and - easy to compute= − : ⟨ ⟩=0

V H V H − =V H −V H

V H

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Electron densinity

ρ r =∑∣Ψ n r ∣2

Schrödinger equation

∇ 2+V Ψ=EΨ

H Ψ=EΨ

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Wave function in 1D

Ψ r =Ae ikr+Be−ikr

Ψ r =a sin kr +b cos kr

Ψ r =a sin kr

∫ d r∣Ψ r ∣2=1

V r :Ψ 0 =0∧Ψ L =0

∇ 2+V r Ψ r =EΨ r

∂2

∂r 2+V r Ψ r =EΨ r

k=nπL

n∈ℤ

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Wave function in 3D

Ψ r =Ax eik x x+B x e

−ik x x Ay eik y y+B y e

−ik y y A z eik z z+B z e

−ik z z Ψ r =a x sin k x x +bx cos k x r x a y sin k y y +b y cos k y r y az sin k z z +b zcos k z r z

Ψ r =a sin k x x sin k y y sin k z z

∫ d r∣Ψ r ∣2=1

V r :Ψ 0 =0∧Ψ L =0

∇ 2+V r Ψ r =EΨ r

∂2

∂ x 2

∂2

∂ y2

∂2

∂ z 2+V r Ψ r =EΨ r

k i=ni π

Li

ni∈ℤ

i=x,y,z

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Electron densinity

ρ r =∑∣Ψ n r ∣2

ρ r =a2sin2 k x x sin2 k y y sin2 k z z

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Time ewolution

Ψ r ,t =e−iωtΨ r

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Time ewolution in 1D

Ψ 0,0 =0

Ψ r,t =a sin kr−ωt

Ψ r,t =e−iωtΨ r

Ψ r,t =Ae ikr−iωt+Be−ikr−iωt

+c sin kr+ωt +d cos kr+ωt Ψ r =a sin kr−ωt +b cos kr−ωt

Ψ 1kω

,1=0

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Time ewolution in 3D

ρ r ,t =∑∣Ψ n r ,t ∣2

Ψ r ,t =asin k x x−ωt sin k y y sin k z z

ρ r ,t =∣a∣2sin2 k x x−ωt sin2 k y y sin2 k z z

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DFT library

FFTW P3DFFT GSL ACML ESSL FFTPACK FFTEASY JTransforms GPUFFTW ...

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Deklaracja biblioteki

#include "p3dfft.h"

#include <mpi.h>

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P3DFFT: initialization

p3dfft_setup(proc_dims,nx,ny,nz,overwrite,memsize)

proc_dims - An array of two integers, specifying how the processor grid should be decomposed. Either 1D or 2D decomposition can be specified.

nx,ny,nz - Dimensions of the 3D transform

overwrite - When set to .true. (or 1 in C) this argument indicates that itis safe to overwrite the input of the btran (backward transform)routine.

memsize – An array of tree integers, specifying memoru used

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P3DFFT: Array Decomposition

p3dfft_get_dims(start,end,size,ip)

start - An array containing 3 integers, defining the beginning indices of the local array for the given task within the global grid

end - An array containing 3 integers, defining the ending indices of the local array within the global grid

size - An array containing 3 integers, defining the local array’s dimensions

ip - An integer argument specifying one of the two choices for array types (1 for forward, 2 for backward, 3 for in-place transform)

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P3DFFT: Array Decomposition

P3DFFT uses 2D block decomposition to assign local arrays for each task. Let P

1 and P

2 be processor grid

dimensions defined in the call to p3dfft_setup.

For ip=1 Array is distributed among P1 tasks in Y

dimension and P2 tasks in Z dimension

For ip=2 Array is distributed among P1 tasks in X

dimension and P2 tasks in Y dimension

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Array Decomposition

1D (FFTW) 2D (P3DFFT)

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Array Decomposition

x

y

z

x

y

z

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Array Decomposition

X = X =X 0 N x−1

0 N y−10 N z−1

X =∑ X p 0 size [0 ]0 size [1 ]0 size [2 ]

X =∑ X p start [0 ] end [0 ]start [1 ] end [1 ]start [2 ] end [2 ]

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P3DFFT: Forward transform

p3dfft_ftran_r2c(IN,OUT,op)

IN - an array of real numbers with dimensions defined by array type with ip=1

OUT - an array of complex numbers with dimensions defined by array type with ip=2

op - 3-letter character string indicating the type of transform desired in X, Y, Z direction

t Fourier transform

c Cosine transform

s Sine transform

n or 0 Empty transform (no operation, output is identical to input)

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P3DFFT: Forward transform

IN - an array of double numbers in order:

x

y

3

1

2

z

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P3DFFT: Forward transform

IN - an array of double numbers in order:

I N =[IN 0,0,0 IN 1,0,0

IN N x−1,0,0 IN 0,1,0 IN 1,1,0

IN N x−1,N y−1,0 IN 0,0,1

IN N x−1,N y−1,N z−1

]

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P3DFFT: Forward transform

OUT - an array of complex numbers in order:

O U T= [ℜ OUT 1ℑ OUT 1ℜ OUT 2ℑ OUT 2

ℜ OUT N−1ℑ OUT N−1

]

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What is compute

x k=∑ xn e−2i π

kN

n

x n=1

2π∑ x k e

2i πkN

n

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What is compute

x k=∑ xn e−2i π

kN

n

x n=1

2π∑ xk e

2i πkN

n

x nFFT xk

iFFT Nxn

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P3DFFT: Backward transform

p3dfft_ftran_c2r(IN,OUT,op)

IN - an array of real numbers with dimensions defined by array type with ip=2

OUT - an array of complex numbers with dimensions defined by array type with ip=1

op - 3-letter character string indicating the type of transform desired in X, Y, Z direction

t Fourier transform

c Cosine transform

s Sine transform

n or 0 Empty transform (no operation, output is identical to input)

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Terminates P3DFFT

p3dfft_clean() - Terminates P3DFFT