NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast Ewald summation under 2d- and 1d-periodic boundary conditions based on NFFTs
F. Nestler* and D. Potts
Departement of Mathematics
SampTA 2013, Bremen July 5, 2013
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Contents
¶ Nonequispaced fast Fourier transforms
· Coulomb interactions
¸ Fast Ewald Summation under 3d-periodic boundary conditions
¹ A new approach to fast Ewald summation under mixed b.c.
º Conclusion
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
The 3d-NFFT
Notation
• define the torusT:=R/Z'[−1/2,1/2)
• forM ∈2NsetIM :={−M/2, . . . ,M/2−1}3
Fast evaluation of trigonometric sums : NFFT: f(xj) := X
k∈IM
fˆke2πik·xj xj∈T3, j= 1, . . . , N FFT: f(j) := X
k∈IM
fˆke2πik·j/M j∈ IM, N:=|IM|=M3
adjoint NFFT: h(k) :=
N
X
j=1
fje−2πik·xj k∈ IM
Complexity: O(|IM|log|IM|+N)
[Dutt, Rokhlin 1993] [Beylkin 1995] [Potts, Steidl, Tasche 2001] [Greengard, Lee 2004]
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
The 3d-NFFT
Notation
• define the torusT:=R/Z'[−1/2,1/2)
• forM ∈2NsetIM :={−M/2, . . . ,M/2−1}3
Fast evaluation of trigonometric sums : NFFT: f(xj) := X
k∈IM
fˆke2πik·xj xj∈T3, j= 1, . . . , N
FFT: f(j) := X
k∈IM
fˆke2πik·j/M j∈ IM, N:=|IM|=M3
adjoint NFFT: h(k) :=
N
X
j=1
fje−2πik·xj k∈ IM
Complexity: O(|IM|log|IM|+N)
[Dutt, Rokhlin 1993] [Beylkin 1995]
[Potts, Steidl, Tasche 2001] [Greengard, Lee 2004]
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
The 3d-NFFT
Notation
• define the torusT:=R/Z'[−1/2,1/2)
• forM ∈2NsetIM :={−M/2, . . . ,M/2−1}3
Fast evaluation of trigonometric sums : NFFT: f(xj) := X
k∈IM
fˆke2πik·xj xj∈T3, j= 1, . . . , N FFT: f(j) := X
k∈IM
fˆke2πik·j/M j∈ IM, N:=|IM|=M3
adjoint NFFT: h(k) :=
N
X
j=1
fje−2πik·xj k∈ IM
Complexity: O(|IM|log|IM|+N)
[Dutt, Rokhlin 1993] [Beylkin 1995]
[Potts, Steidl, Tasche 2001] [Greengard, Lee 2004]
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
The 3d-NFFT
Notation
• define the torusT:=R/Z'[−1/2,1/2)
• forM ∈2NsetIM :={−M/2, . . . ,M/2−1}3
Fast evaluation of trigonometric sums : NFFT: f(xj) := X
k∈IM
fˆke2πik·xj xj∈T3, j= 1, . . . , N FFT: f(j) := X
k∈IM
fˆke2πik·j/M j∈ IM, N:=|IM|=M3
adjoint NFFT: h(k) :=
N
X
j=1
fje−2πik·xj k∈ IM
Complexity: O(|IM|log|IM|+N)
[Dutt, Rokhlin 1993] [Beylkin 1995]
[Potts, Steidl, Tasche 2001] [Greengard, Lee 2004]
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
The 3d-NFFT
Notation
• define the torusT:=R/Z'[−1/2,1/2)
• forM ∈2NsetIM :={−M/2, . . . ,M/2−1}3
Fast evaluation of trigonometric sums : NFFT: f(xj) := X
k∈IM
fˆke2πik·xj xj∈T3, j= 1, . . . , N FFT: f(j) := X
k∈IM
fˆke2πik·j/M j∈ IM, N:=|IM|=M3
adjoint NFFT: h(k) :=
N
X
j=1
fje−2πik·xj k∈ IM
Complexity: O(|IM|log|IM|+N)
[Dutt, Rokhlin 1993] [Beylkin 1995]
[Potts, Steidl, Tasche 2001] [Greengard, Lee 2004]
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Definition of the Coulomb interaction energy
LetN chargesqj∈Rat positionsxj∈R3 be given.
Coulomb interaction energy (xij:=xi−xj):
E:=1 2
N
X
i,j=1 i6=j
qiqj
kxijk
=1 2
N
X
j=1
qjφ(xj) with φ(xj) :=
N
X
i=1 i6=j
qi
kxijk.
3d-periodic boundary conditions chooseS:=Z3,xj∈BT3 and set
φ(xj) :=φS(xj) :=X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Definition of the Coulomb interaction energy
LetN chargesqj∈Rat positionsxj∈R3 be given.
Coulomb interaction energy (xij:=xi−xj):
E:=1 2
N
X
i,j=1 i6=j
qiqj
kxijk=1 2
N
X
j=1
qjφ(xj) with φ(xj) :=
N
X
i=1 i6=j
qi
kxijk.
3d-periodic boundary conditions chooseS:=Z3,xj∈BT3 and set
φ(xj) :=φS(xj) :=X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Definition of the Coulomb interaction energy
LetN chargesqj∈Rat positionsxj∈R3 be given.
Coulomb interaction energy (xij:=xi−xj):
E:=1 2
N
X
i,j=1 i6=j
qiqj
kxijk=1 2
N
X
j=1
qjφ(xj) with φ(xj) :=
N
X
i=1 i6=j
qi
kxijk.
3d-periodic boundary conditions chooseS:=Z3,xj∈BT3 and set
φ(xj) :=φS(xj) :=X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk
B
Franziska Nestler Chemnitz University of Technology, Germany1
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Definition of the Coulomb interaction energy
LetN chargesqj∈Rat positionsxj∈R3 be given.
Coulomb interaction energy (xij:=xi−xj):
E:=1 2
N
X
i,j=1 i6=j
qiqj
kxijk=1 2
N
X
j=1
qjφ(xj) with φ(xj) :=
N
X
i=1 i6=j
qi
kxijk.
3d-periodic boundary conditions chooseS:=Z3,xj∈BT3 and set
φ(xj) :=φS(xj) :=X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk
Franziska Nestler Chemnitz University of Technology, Germany1
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Definition of the Coulomb interaction energy
LetN chargesqj∈Rat positionsxj∈R3 be given.
Coulomb interaction energy (xij:=xi−xj):
E:=1 2
N
X
i,j=1 i6=j
qiqj
kxijk=1 2
N
X
j=1
qjφ(xj) with φ(xj) :=
N
X
i=1 i6=j
qi
kxijk.
3d-periodic boundary conditions chooseS:=Z3,xj∈BT3 and set
φ(xj) :=φS(xj) :=X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk
Franziska Nestler Chemnitz University of Technology, Germany1
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Definition of the Coulomb interaction energy
LetN chargesqj∈Rat positionsxj∈R3 be given.
Coulomb interaction energy (xij:=xi−xj):
E:=1 2
N
X
i,j=1 i6=j
qiqj
kxijk=1 2
N
X
j=1
qjφ(xj) with φ(xj) :=
N
X
i=1 i6=j
qi
kxijk.
3d-periodic boundary conditions chooseS:=Z3,xj∈BT3 and set
φ(xj) :=φS(xj) :=X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk
1
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Coulomb energy s.t. periodic boundary conditions E(S) :=1
2
N
X
j=1
qjφS(xj) with φS(xj) := X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk.
Mixed boundary conditions:
2d-periodic
xj∈BT2×R,S:=Z2× {0}
1
1d-periodic
xj∈BT×R2,S:=Z× {0}2
1
AssumePN
j=1qj= 0⇒conditionalconvergence
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Coulomb energy s.t. periodic boundary conditions E(S) :=1
2
N
X
j=1
qjφS(xj) with φS(xj) := X
n∈S N
X
i=1 i6=jifn=0
qi
kxij+Bnk.
Mixed boundary conditions:
2d-periodic
xj∈BT2×R,S:=Z2× {0}
1
1d-periodic
xj∈BT×R2,S:=Z× {0}2
1
AssumePN
j=1qj= 0⇒conditionalconvergence
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast algorithms
Non Fourier methods
• Multigrid,O(N) [Brandt, Hackbusch, Trottenberg 1977]
• Fast Multipole method,O(N) [Greengard, Rokhlin 1987]
Open boundary conditions - finite sums
Fast summation methods based on NFFTs [Potts, Steidl 2003] Periodic boundary conditions - infinite sums
Basis: Ewald summation formulas [Ewald 1921]
• 3d-periodic: particle mesh approaches (P3M), P2NFFT,O(NlogN) [Hockney, Eastwood 1988] [Deserno, Holm 1998] [Pippig, Potts 2011]
• 2d-periodic: particle mesh like methods,O(NlogN) [Kawata, Mikami 2001] [Lindbo, Tornberg 2011]
• 1d-periodic: ?
Aim: propose efficient NFFT based methods for 2d- and 1d-periodic b.c.
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast algorithms
Non Fourier methods
• Multigrid,O(N) [Brandt, Hackbusch, Trottenberg 1977]
• Fast Multipole method,O(N) [Greengard, Rokhlin 1987]
Open boundary conditions - finite sums
Fast summation methods based on NFFTs [Potts, Steidl 2003]
Periodic boundary conditions - infinite sums Basis: Ewald summation formulas [Ewald 1921]
• 3d-periodic: particle mesh approaches (P3M), P2NFFT,O(NlogN) [Hockney, Eastwood 1988] [Deserno, Holm 1998] [Pippig, Potts 2011]
• 2d-periodic: particle mesh like methods,O(NlogN) [Kawata, Mikami 2001] [Lindbo, Tornberg 2011]
• 1d-periodic: ?
Aim: propose efficient NFFT based methods for 2d- and 1d-periodic b.c.
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast algorithms
Non Fourier methods
• Multigrid,O(N) [Brandt, Hackbusch, Trottenberg 1977]
• Fast Multipole method,O(N) [Greengard, Rokhlin 1987]
Open boundary conditions - finite sums
Fast summation methods based on NFFTs [Potts, Steidl 2003]
Periodic boundary conditions - infinite sums Basis: Ewald summation formulas [Ewald 1921]
• 3d-periodic: particle mesh approaches (P3M), P2NFFT,O(NlogN) [Hockney, Eastwood 1988] [Deserno, Holm 1998] [Pippig, Potts 2011]
• 2d-periodic: particle mesh like methods,O(NlogN) [Kawata, Mikami 2001] [Lindbo, Tornberg 2011]
• 1d-periodic: ?
Aim: propose efficient NFFT based methods for 2d- and 1d-periodic b.c.
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast algorithms
Non Fourier methods
• Multigrid,O(N) [Brandt, Hackbusch, Trottenberg 1977]
• Fast Multipole method,O(N) [Greengard, Rokhlin 1987]
Open boundary conditions - finite sums
Fast summation methods based on NFFTs [Potts, Steidl 2003]
Periodic boundary conditions - infinite sums Basis: Ewald summation formulas [Ewald 1921]
• 3d-periodic: particle mesh approaches (P3M), P2NFFT,O(NlogN) [Hockney, Eastwood 1988] [Deserno, Holm 1998] [Pippig, Potts 2011]
• 2d-periodic: particle mesh like methods,O(NlogN) [Kawata, Mikami 2001] [Lindbo, Tornberg 2011]
• 1d-periodic: ?
Aim: propose efficient NFFT based methods for 2d- and 1d-periodic b.c.
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast algorithms
Non Fourier methods
• Multigrid,O(N) [Brandt, Hackbusch, Trottenberg 1977]
• Fast Multipole method,O(N) [Greengard, Rokhlin 1987]
Open boundary conditions - finite sums
Fast summation methods based on NFFTs [Potts, Steidl 2003]
Periodic boundary conditions - infinite sums Basis: Ewald summation formulas [Ewald 1921]
• 3d-periodic: particle mesh approaches (P3M), P2NFFT,O(NlogN) [Hockney, Eastwood 1988] [Deserno, Holm 1998] [Pippig, Potts 2011]
• 2d-periodic: particle mesh like methods,O(NlogN) [Kawata, Mikami 2001] [Lindbo, Tornberg 2011]
• 1d-periodic: ?
Aim: propose efficient NFFT based methods for 2d- and 1d-periodic b.c.
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast algorithms
Non Fourier methods
• Multigrid,O(N) [Brandt, Hackbusch, Trottenberg 1977]
• Fast Multipole method,O(N) [Greengard, Rokhlin 1987]
Open boundary conditions - finite sums
Fast summation methods based on NFFTs [Potts, Steidl 2003]
Periodic boundary conditions - infinite sums Basis: Ewald summation formulas [Ewald 1921]
• 3d-periodic: particle mesh approaches (P3M), P2NFFT,O(NlogN) [Hockney, Eastwood 1988] [Deserno, Holm 1998] [Pippig, Potts 2011]
• 2d-periodic: particle mesh like methods,O(NlogN) [Kawata, Mikami 2001] [Lindbo, Tornberg 2011]
• 1d-periodic: ?
Aim: propose efficient NFFT based methods for 2d- and 1d-periodic b.c.
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast algorithms
Non Fourier methods
• Multigrid,O(N) [Brandt, Hackbusch, Trottenberg 1977]
• Fast Multipole method,O(N) [Greengard, Rokhlin 1987]
Open boundary conditions - finite sums
Fast summation methods based on NFFTs [Potts, Steidl 2003]
Periodic boundary conditions - infinite sums Basis: Ewald summation formulas [Ewald 1921]
• 3d-periodic: particle mesh approaches (P3M), P2NFFT,O(NlogN) [Hockney, Eastwood 1988] [Deserno, Holm 1998] [Pippig, Potts 2011]
• 2d-periodic: particle mesh like methods,O(NlogN) [Kawata, Mikami 2001] [Lindbo, Tornberg 2011]
• 1d-periodic: ?
Aim: propose efficient NFFT based methods for 2d- and 1d-periodic b.c.
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Ewald’s idea
Idea of the Ewald summation[Ewald 1921]:
Ewald splitting 1
r = erf(αr) r
| {z } long ranged, continuous
+ erfc(αr) r
| {z } singular in 0, short ranged
0 0.2 0.4 0.6 0.8 1
0 2 4 6 8 10
r
• erf(x) := √2πRx
0 e−t2dt(error function)
• erfc(x) := 1−erf(x)(complementary error function)
• α >0(scaling parameter)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
3d Ewald Summation
We apply 1r = erfc(αr)r +erf(αr)r withr:=kxij+Bnkand obtain
X
n∈Z3 N
X
i,j=1 i6=jifn=0
qiqj
kxij+Bnk
= X
n∈Z3 N
X
i,j=1 i6=jifn=0
qiqj
erfc(αkxij+Bnk) kxij+Bnk +
X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxij+Bnk) kxij+Bnk − 2α
√π
N
X
j=1
q2j
• short range partcan be obtained via direct evaluation after truncation
• limr→0erf(αr)
r =√2απ ⇒substractself energy
• write thelong range partas a sum in Fourier space
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
3d Ewald Summation
We apply 1r = erfc(αr)r +erf(αr)r withr:=kxij+Bnkand obtain
X
n∈Z3 N
X
i,j=1 i6=jifn=0
qiqj
kxij+Bnk = X
n∈Z3 N
X
i,j=1 i6=jifn=0
qiqj
erfc(αkxij+Bnk) kxij+Bnk +
X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxij+Bnk) kxij+Bnk − 2α
√π
N
X
j=1
q2j
• short range partcan be obtained via direct evaluation after truncation
• limr→0erf(αr)
r =√2απ ⇒substractself energy
• write thelong range partas a sum in Fourier space
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
3d Ewald Summation
We apply 1r = erfc(αr)r +erf(αr)r withr:=kxij+Bnkand obtain
X
n∈Z3 N
X
i,j=1 i6=jifn=0
qiqj
kxij+Bnk = X
n∈Z3 N
X
i,j=1 i6=jifn=0
qiqj
erfc(αkxij+Bnk) kxij+Bnk +
X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxij+Bnk) kxij+Bnk − 2α
√π
N
X
j=1
q2j
• short range partcan be obtained via direct evaluation after truncation
• limr→0erf(αr)
r = √2απ ⇒substractself energy
• write thelong range partas a sum in Fourier space
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
An efficient algorithm
• calculate the self energy
• compute short range part: direct evaluation
• long range part: conditional convergence
assume: charge neutrality + spherical order of summation 1
2 X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxi−xj+Bnk) kxi−xj+Bnk
= 1
2πB X
k∈Z3\{0}
e−π2kkk2/(α2B2)
kkk2 |S(k)|2+ 2π 3B2
N
X
j=1
qjxj
2
,
whereS(k) :=PN
j=1qje2πik·xj/B
• truncate the infinite sum (Z3→ IM)
• S(k),k∈ IM, by adjoint 3d-NFFT
• resulting method: O(NlogN)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
An efficient algorithm
• calculate the self energy
• compute short range part: direct evaluation
• long range part: conditional convergence
assume: charge neutrality + spherical order of summation 1
2 X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxi−xj+Bnk) kxi−xj+Bnk
= 1
2πB X
k∈Z3\{0}
e−π2kkk2/(α2B2)
kkk2 |S(k)|2+ 2π 3B2
N
X
j=1
qjxj
2
,
whereS(k) :=PN
j=1qje2πik·xj/B
• truncate the infinite sum (Z3→ IM)
• S(k),k∈ IM, by adjoint 3d-NFFT
• resulting method: O(NlogN)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
An efficient algorithm
• calculate the self energy
• compute short range part: direct evaluation
• long range part: conditional convergence
assume: charge neutrality + spherical order of summation 1
2 X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxi−xj+Bnk) kxi−xj+Bnk
= 1
2πB X
k∈Z3\{0}
e−π2kkk2/(α2B2)
kkk2 |S(k)|2+ 2π 3B2
N
X
j=1
qjxj
2
,
whereS(k) :=PN
j=1qje2πik·xj/B
• truncate the infinite sum (Z3→ IM)
• S(k),k∈ IM, by adjoint 3d-NFFT
• resulting method: O(NlogN)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
An efficient algorithm
• calculate the self energy
• compute short range part: direct evaluation
• long range part: conditional convergence
assume: charge neutrality + spherical order of summation 1
2 X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxi−xj+Bnk) kxi−xj+Bnk
= 1
2πB X
k∈Z3\{0}
e−π2kkk2/(α2B2)
kkk2 |S(k)|2+ 2π 3B2
N
X
j=1
qjxj
2
,
whereS(k) :=PN
j=1qje2πik·xj/B
• truncate the infinite sum (Z3→ IM)
• S(k),k∈ IM, by adjoint 3d-NFFT
• resulting method: O(NlogN)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
An efficient algorithm
• calculate the self energy
• compute short range part: direct evaluation
• long range part: conditional convergence
assume: charge neutrality + spherical order of summation 1
2 X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxi−xj+Bnk) kxi−xj+Bnk
= 1
2πB X
k∈Z3\{0}
e−π2kkk2/(α2B2)
kkk2 |S(k)|2+ 2π 3B2
N
X
j=1
qjxj
2
,
whereS(k) :=PN
j=1qje2πik·xj/B
• truncate theinfinite sum(Z3→ IM)
• S(k),k∈ IM, by adjoint 3d-NFFT
• resulting method: O(NlogN)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
An efficient algorithm
• calculate the self energy
• compute short range part: direct evaluation
• long range part: conditional convergence
assume: charge neutrality + spherical order of summation 1
2 X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxi−xj+Bnk) kxi−xj+Bnk
= 1
2πB X
k∈Z3\{0}
e−π2kkk2/(α2B2)
kkk2 |S(k)|2+ 2π 3B2
N
X
j=1
qjxj
2
,
whereS(k) :=PN
j=1qje2πik·xj/B
• truncate the infinite sum (Z3→ IM)
• S(k),k∈ IM, by adjoint 3d-NFFT
• resulting method: O(NlogN)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
An efficient algorithm
• calculate the self energy
• compute short range part: direct evaluation
• long range part: conditional convergence
assume: charge neutrality + spherical order of summation 1
2 X
n∈Z3 N
X
i,j=1
qiqj
erf(αkxi−xj+Bnk) kxi−xj+Bnk
= 1
2πB X
k∈Z3\{0}
e−π2kkk2/(α2B2)
kkk2 |S(k)|2+ 2π 3B2
N
X
j=1
qjxj
2
,
whereS(k) :=PN
j=1qje2πik·xj/B
• truncate the infinite sum (Z3→ IM)
• S(k),k∈ IM, by adjoint 3d-NFFT
• resulting method: O(NlogN)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Computing the potentials
If we are interested in the potentials:
Write the Fourier part of the potentials in the form 1
πB X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xi/B
!
e−2πik·xj/B
. . . and use adjoint NFFT + NFFT
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Computing the potentials
If we are interested in the potentials:
Write the Fourier part of the potentials in the form 1
πB X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xi/B
!
| {z }
adjoint NFFT
e−2πik·xj/B
| {z }
NFFT
. . . and use adjoint NFFT + NFFT
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Ewald summation s.t. 2d periodic boundary conditions
Long range part under 2d-periodic b.c. [Grzybowski, Gwóźdź, Bródka 2000]
We definexij:=xi−xj= (˜xij, xij,3)and obtain 1
4B X
k∈Z2\{0}
N
X
i,j=1
qiqje2πik·˜xij/BΘp2(kkk, xij,3)−
√π B2
N
X
i,j=1
qiqjΘp20 (xij,3)
2d Fourier series k=0part
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5
Θp2(k, r) k= 1 k=√
2 k=√
3 . . .
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
e2πkr/Berfc(αBπk+αr)+e−2πkr/Berfc(αBπk−αr)
k
Θp20 (r) =α1e−α2r2+r√ πerf(αr)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Ewald summation s.t. 2d periodic boundary conditions
Long range part under 2d-periodic b.c. [Grzybowski, Gwóźdź, Bródka 2000]
We definexij:=xi−xj= (˜xij, xij,3)and obtain 1
4B X
k∈Z2\{0}
N
X
i,j=1
qiqje2πik·˜xij/BΘp2(kkk, xij,3)−
√π B2
N
X
i,j=1
qiqjΘp20 (xij,3)
2d Fourier series k=0part
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5
Θp2(k, r) k= 1 k=√
2 k=√
3 . . .
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
e2πkr/Berfc(αBπk+αr)+e−2πkr/Berfc(αBπk−αr)
k
Θp20 (r) =α1e−α2r2+r√ πerf(αr)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Ewald summation s.t. 2d periodic boundary conditions
Long range part under 2d-periodic b.c. [Grzybowski, Gwóźdź, Bródka 2000]
We definexij:=xi−xj= (˜xij, xij,3)and obtain 1
4B X
k∈Z2\{0}
N
X
i,j=1
qiqje2πik·˜xij/BΘp2(kkk, xij,3)−
√π B2
N
X
i,j=1
qiqjΘp20 (xij,3)
2d Fourier series k=0part
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5
Θp2(k, r) k= 1 k=√
2 k=√
3 . . .
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
e2πkr/Berfc(αBπk+αr)+e−2πkr/Berfc(αBπk−αr)
k
Θp20 (r) =α1e−α2r2+r√ πerf(αr)
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm
Long range part 1
4B X
k6=0 N
X
i,j=1
qiqje2πik·x˜ij/BΘp2(kkk, xij,3) −
√π B2
N
X
i,j=1
qiqjΘp20 (xij,3)
• truncate theinfinite sum
, assume that|xij,3|< hfor alli, j
• chooseε >0,∀remainingk=kkk 6= 0construct a smooth function
Rk(r) :=
Θp2(k, r) :|r| ≤h Bk(r) :h <|r| ≤h+ε
Rk(r)≈
m/2−1
X
l=−m/2
bk,leπilr/(h+ε)
0
−(h+ε) h+ε
−h h
Θp2(k,·)
1
• m∈2Nsufficiently large
• replaceΘp2(kkk,·)by the Fourier sums
• by adjoint NFFT:S(k, l) :=
N
X
j=1
qje2πik·x˜j/B+πilxj,3/(h+ε) (k6=0)
• analog for thek=0part
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm
Long range part 1
4B X
k6=0 N
X
i,j=1
qiqje2πik·x˜ij/BΘp2(kkk,xij,3) −
√π B2
N
X
i,j=1
qiqjΘp20 (xij,3)
• truncate the infinite sum, assume that|xij,3|< hfor alli, j
• chooseε >0,∀remainingk=kkk 6= 0construct a smooth function
Rk(r) :=
Θp2(k, r) :|r| ≤h Bk(r) :h <|r| ≤h+ε
Rk(r)≈
m/2−1
X
l=−m/2
bk,leπilr/(h+ε)
0
−(h+ε) h+ε
−h h
Θp2(k,·)
1
• m∈2Nsufficiently large
• replaceΘp2(kkk,·)by the Fourier sums
• by adjoint NFFT:S(k, l) :=
N
X
j=1
qje2πik·x˜j/B+πilxj,3/(h+ε) (k6=0)
• analog for thek=0part
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm
Long range part 1
4B X
k6=0 N
X
i,j=1
qiqje2πik·x˜ij/BΘp2(kkk, xij,3) −
√π B2
N
X
i,j=1
qiqjΘp20 (xij,3)
• truncate the infinite sum, assume that|xij,3|< hfor alli, j
• chooseε >0,∀remainingk=kkk 6= 0construct a smooth function
Rk(r) :=
Θp2(k, r) :|r| ≤h Bk(r) :h <|r| ≤h+ε
Rk(r)≈
m/2−1
X
l=−m/2
bk,leπilr/(h+ε)
0
−(h+ε) h+ε
−h h
Θp2(k,·)
1
• m∈2Nsufficiently large
• replaceΘp2(kkk,·)by the Fourier sums
• by adjoint NFFT:S(k, l) :=
N
X
j=1
qje2πik·x˜j/B+πilxj,3/(h+ε) (k6=0)
• analog for thek=0part
Franziska Nestler Chemnitz University of Technology, Germany
NFFT Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm
Long range part 1
4B X
k6=0 N
X
i,j=1
qiqje2πik·x˜ij/BΘp2(kkk, xij,3) −
√π B2
N
X
i,j=1
qiqjΘp20 (xij,3)
• truncate the infinite sum, assume that|xij,3|< hfor alli, j
• chooseε >0,∀remainingk=kkk 6= 0construct a smooth function
Rk(r) :=
Θp2(k, r) :|r| ≤h Bk(r) :h <|r| ≤h+ε
Rk(r)≈
m/2−1
X
l=−m/2
bk,leπilr/(h+ε)
0
−(h+ε) h+ε
−h h
Θp2(k,·)
1
• m∈2Nsufficiently large
• replaceΘp2(kkk,·)by the Fourier sums
• by adjoint NFFT:S(k, l) :=
N
X
j=1
qje2πik·x˜j/B+πilxj,3/(h+ε) (k6=0)
• analog for thek=0part
Franziska Nestler Chemnitz University of Technology, Germany
