Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
NFFT basierte schnelle Ewald-Summation für gemischt periodische Randbedingungen
Franziska Nestler
Fakultät für Mathematik
24. Rhein-Ruhr-Workshop Bestwig, 31. Januar 2014
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Contents
¶ Coulomb interactions
· Fast Ewald Summation under 3d-periodic boundary conditions
¸ A new approach to fast Ewald summation under mixed b.c.
¹ Conclusion
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
T:=R/Z'[−1/2,1/2)
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
T:=R/Z'[−1/2,1/2)
B
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
T:=R/Z'[−1/2,1/2)
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
T:=R/Z'[−1/2,1/2)
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
• fully periodic: xj∈BT3,S:=Z3
1
• mixed boundary conditions:
2d-periodic
xj∈BT2×R,S :=Z2× {0}
B
B
1d-periodic
xj∈BT×R2,S:=Z× {0}2
B
AssumePN
j=1qj= 0⇒conditional convergence
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
• fully periodic: xj∈BT3,S:=Z3
1
• mixed boundary conditions:
2d-periodic
xj∈BT2×R,S :=Z2× {0}
B
B
1d-periodic
xj∈BT×R2,S:=Z× {0}2
B
AssumePN
j=1qj= 0⇒conditional convergence
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)
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=1 i6=jifn=0
qi
kxij+Bnk
= X
n∈Z3 N
X
i=1 i6=jifn=0
qi
erfc(αkxij+Bnk) kxij+Bnk +
X
n∈Z3 N
X
i=1
qi
erf(αkxij+Bnk) kxij+Bnk − 2α
√πqj
• short range partcan be obtained via direct evaluation after truncation
• limr→0erf(αr)
r =√2απ ⇒substractself potential
• write thelong range partas a sum in Fourier space
→use the FFT for nonequispaced data (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=1 i6=jifn=0
qi
kxij+Bnk = X
n∈Z3 N
X
i=1 i6=jifn=0
qi
erfc(αkxij+Bnk) kxij+Bnk +
X
n∈Z3 N
X
i=1
qi
erf(αkxij+Bnk) kxij+Bnk − 2α
√πqj
• short range partcan be obtained via direct evaluation after truncation
• limr→0erf(αr)
r =√2απ ⇒substractself potential
• write thelong range partas a sum in Fourier space
→use the FFT for nonequispaced data (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=1 i6=jifn=0
qi
kxij+Bnk = X
n∈Z3 N
X
i=1 i6=jifn=0
qi
erfc(αkxij+Bnk) kxij+Bnk +
X
n∈Z3 N
X
i=1
qi
erf(αkxij+Bnk) kxij+Bnk − 2α
√πqj
• short range partcan be obtained via direct evaluation after truncation
• limr→0erf(αr)
r = √2απ ⇒substractself potential
• write thelong range partas a sum in Fourier space
→use the FFT for nonequispaced data (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⊂Z3
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]
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⊂Z3
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]
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⊂Z3
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]
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
3d Ewald Summation - long range part
Long range part of the potential
φLZ3(xj) := X
n∈Z3 N
X
i=1
qi
erf(αkxij+Bnk) kxij+Bnk is conditional convergent.
Representation in Fourier space
assume charge neutrality + spherical order of summation to obtain
φLZ3(xj) = 1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xij/B+ 4π 3B2
N
X
i=1
qixi
!
·xj
• Fourier coefficients tend to zero exponentially fast
• Fourier coefficients are singular ink=0
• thek=0termrepresents the order of summation
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
3d Ewald Summation - long range part
Long range part of the potential
φLZ3(xj) := X
n∈Z3 N
X
i=1
qi
erf(αkxij+Bnk) kxij+Bnk is conditional convergent.
Representation in Fourier space
assume charge neutrality + spherical order of summation to obtain
φLZ3(xj) = 1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xij/B+ 4π 3B2
N
X
i=1
qixi
!
·xj
• Fourier coefficients tend to zero exponentially fast
• Fourier coefficients are singular ink=0
• thek=0termrepresents the order of summation
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
3d Ewald Summation - long range part
Long range part of the potential
φLZ3(xj) := X
n∈Z3 N
X
i=1
qi
erf(αkxij+Bnk) kxij+Bnk is conditional convergent.
Representation in Fourier space
assumecharge neutrality + spherical orderof summation to obtain
φLZ3(xj) = 1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xij/B+ 4π 3B2
N
X
i=1
qixi
!
·xj
• Fourier coefficients tend to zero exponentially fast
• Fourier coefficients are singular ink=0
• thek=0termrepresents the order of summation
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast Algorithm: P
2NFFT O (N log N )
[Pippig, Potts 2011]O(NlogN)Algorithm:
• Fourier part:
truncate the infinite sum (Z3→ IM)
1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xij/B
= 1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xi/B
!
e−2πik·xj/B
• direct: short range part, self potentials,k=0terms
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast Algorithm: P
2NFFT O (N log N )
[Pippig, Potts 2011]O(NlogN)Algorithm:
• Fourier part: truncate the infinite sum (Z3→ IM)
1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xij/B
= 1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xi/B
!
| {z }
adjoint NFFT
e−2πik·xj/B
• direct: short range part, self potentials,k=0terms
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Fast Algorithm: P
2NFFT O (N log N )
[Pippig, Potts 2011]O(NlogN)Algorithm:
• Fourier part: truncate the infinite sum (Z3→ IM)
1 πB
X
k∈Z3\{0}
e−π2kkk2/(α2B2) kkk2
N
X
i=1
qie2πik·xij/B
= 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
• direct: short range part, self potentials,k=0terms
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 writexij= (˜xij, xij,3)and obtain forφLZ2×{0}(xj) 1
2B X
k∈Z2\{0}
N
X
i=1
qie2πik·˜xij/BΘp2(kkk, xij,3)−2√π B2
N
X
i=1
qiΘp20 (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)
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 writexij= (˜xij, xij,3)and obtain forφLZ2×{0}(xj) 1
2B X
k∈Z2\{0}
N
X
i=1
qie2πik·˜xij/BΘp2(kkk, xij,3)−2√π B2
N
X
i=1
qiΘp20 (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)
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 writexij= (˜xij, xij,3)and obtain forφLZ2×{0}(xj) 1
2B X
k∈Z2\{0}
N
X
i=1
qie2πik·˜xij/BΘp2(kkk, xij,3)−2√π B2
N
X
i=1
qiΘp20 (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)
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm I
The 2d Fourier sum
1 2B
P
k∈Z2\{0}
PN
i=1qie2πik·˜xij/BΘp2(kkk, xij,3) Properties ofΘp2(k, r)
• ∀r: Θp2(k, r)∼k−2e−k2 ask→ ∞
• ∀k: Θp2(k,·)∈C∞(R) Regularization:
• xij,3 are absolutely bounded
• we are just interested in Θp2(k,·)over a finite interval (white area)
• construct a smooth and periodic functionKR(k,·)
0
−h/2+hε h/2−hε
−h/2 h/2
∂j
∂rjKB(k,h/2−hε) = 1
2B∂j
∂rjΘp2(k,h/2−hε) 1
2BΘp2(k,·)
KB(k,·) KB(k,·)
.
| {z }
KR(k,·) :hT→R
→ε∈(0,1/2)determines the size of the areas at the boundaries
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm I
The 2d Fourier sum
1 2B
P
k∈Z2\{0}
PN
i=1qie2πik·˜xij/BΘp2(kkk, xij,3) Properties ofΘp2(k, r)
• ∀r: Θp2(k, r)∼k−2e−k2 ask→ ∞
• ∀k: Θp2(k,·)∈C∞(R) Regularization:
• xij,3 are absolutely bounded
• we are just interested in Θp2(k,·)over a finite interval (white area)
• construct a smooth and periodic functionKR(k,·)
0
−h/2+hε h/2−hε
−h/2 h/2
∂j
∂rjKB(k,h/2−hε) = 1
2B∂j
∂rjΘp2(k,h/2−hε) 1
2BΘp2(k,·)
KB(k,·) KB(k,·)
.
| {z }
KR(k,·) :hT→R
→ε∈(0,1/2)determines the size of the areas at the boundaries
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm II
The 2d Fourier sum(∗)
1 2B
P
k∈Z2\{0}
PN
i=1qie2πik·˜xij/BΘp2(kkk, xij,3)
• truncate the infinite sum,Z2→ IM˜ :={−M˜/2. . . ,M˜/2−1}2
• forM ∈2Nlarge enough we find for eachk=kkk 6= 0 (k∈ IM˜) Θp2(k, xij,3) =KR(k, xij,3)≈
M/2−1
X
l=−M/2
bk,le2πilxij,3/h
• we obtain withxˆi:= xBi,1,xBi,2,xi,3h
the approximation
• analog for thek=0term
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm II
The 2d Fourier sum(∗)
1 2B
P
k∈Z2\{0}
PN
i=1qie2πik·˜xij/BΘp2(kkk, xij,3)
• truncate the infinite sum,Z2→ IM˜ :={−M˜/2. . . ,M˜/2−1}2
• forM ∈2Nlarge enough we find for eachk=kkk 6= 0 (k∈ IM˜) Θp2(k, xij,3) =KR(k, xij,3)≈
M/2−1
X
l=−M/2
bk,le2πilxij,3/h
• we obtain withxˆi:= xBi,1,xBi,2,xi,3h
the approximation
(∗)≈ X
k∈IM˜\{0} M/2−1
X
l=−M/2
bkkk,l
N
X
i=1
qie
2πi
k1
k2
l
·ˆxi
e
−2πi
k1
k2
l
·xˆj
• analog for thek=0term
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm II
The 2d Fourier sum(∗)
1 2B
P
k∈Z2\{0}
PN
i=1qie2πik·˜xij/BΘp2(kkk, xij,3)
• truncate the infinite sum,Z2→ IM˜ :={−M˜/2. . . ,M˜/2−1}2
• forM ∈2Nlarge enough we find for eachk=kkk 6= 0 (k∈ IM˜) Θp2(k, xij,3) =KR(k, xij,3)≈
M/2−1
X
l=−M/2
bk,le2πilxij,3/h
• we obtain withxˆi:= xBi,1,xBi,2,xi,3h
the approximation
(∗)≈ X
k∈IM˜\{0}
M/2−1
X
l=−M/2
bkkk,l
N
X
i=1
qie
2πi
k1
k2
l
·ˆxi
e
−2πi
k1
k2
l
·xˆj
• analog for thek=0term
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm II
The 2d Fourier sum(∗)
1 2B
P
k∈Z2\{0}
PN
i=1qie2πik·˜xij/BΘp2(kkk, xij,3)
• truncate the infinite sum,Z2→ IM˜ :={−M˜/2. . . ,M˜/2−1}2
• forM ∈2Nlarge enough we find for eachk=kkk 6= 0 (k∈ IM˜) Θp2(k, xij,3) =KR(k, xij,3)≈
M/2−1
X
l=−M/2
bk,le2πilxij,3/h
• we obtain withxˆi:= xBi,1,xBi,2,xi,3h
the approximation
(∗)≈ X
k∈IM˜\{0}
M/2−1
X
l=−M/2
bkkk,l
N
X
i=1
qie
2πi
k1
k2
l
·ˆxi
| {z }
3d adj. NFFT
e
−2πi
k1
k2
l
·xˆj
| {z }
3d NFFT
• analog for thek=0term
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
A fast algorithm II
The 2d Fourier sum(∗)
1 2B
P
k∈Z2\{0}
PN
i=1qie2πik·˜xij/BΘp2(kkk, xij,3)
• truncate the infinite sum,Z2→ IM˜ :={−M˜/2. . . ,M˜/2−1}2
• forM ∈2Nlarge enough we find for eachk=kkk 6= 0 (k∈ IM˜) Θp2(k, xij,3) =KR(k, xij,3)≈
M/2−1
X
l=−M/2
bk,le2πilxij,3/h
• we obtain withxˆi:= xBi,1,xBi,2,xi,3h
the approximation
(∗)≈ X
k∈IM˜\{0}
M/2−1
X
l=−M/2
bkkk,l
N
X
i=1
qie
2πi
k1
k2
l
·ˆxi
| {z }
3d adj. NFFT
e
−2πi
k1
k2
l
·xˆj
| {z }
3d NFFT
• analog for thek=0term
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Some numerical results
• optimal choice of the parameters? (especiallyε)
• to obtain first numerical results: tunedεby hand
• we obtain acceptable errors in comparison to the 3d-periodic method
0.5 1 1.5 2 2.5
10−13 10−10 10−7 10−4 10−1
splitting parameterα
∆EZ2×{0}
M= 16, ε= 0.115 M= 32, ε= 0.115 M= 64, ε= 0.145 M= 128, ε= 0.170 M= 256, ε= 0.190
0.5 1 1.5 2 2.5
10−13 10−10 10−7 10−4 10−1
splitting parameterα
∆EZ3
Figure :Achieved rms errors for a 2d-periodic (left) and the corresponding 3d-periodic computation (right) withN= 300particles.
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Structure is the same as for 3d-periodic systems
• Near field, self interactions
• Far field
1 adjoint 3d NFFT
2 Multiplikation with the Fourier coefficients
3 3d NFFT
Problems
• for eachkkkwe have to construct the regularization and compute the Fourier coefficients via the FFT
→many precomputation steps
• more parameters: degree of smoothness, regularization parameterε, . . .
→optimal choice? exact error control?
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
How to reduce precomputation costs?
Fork large enough:
−h/2 h/2
Θp2(k, r)≈ X∞ n=−∞
Θp2(k, r+nh) Θp2(k,·)
h-periodization
1
Poisson summation:
∞
X
n=−∞
Θp2(k, r+nh) = 1 h
∞
X
l=−∞
2Bh2
π(h2k2+B2l2)e−π2l2/(α2h2)−π2k2/(α2B2)e2πilr/h
→Fourier coefficientsare given analytically
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
How to reduce precomputation costs?
Fork large enough:
−h/2 h/2
Θp2(k, r)≈ X∞ n=−∞
Θp2(k, r+nh) Θp2(k,·)
h-periodization
1
Poisson summation:
∞
X
n=−∞
Θp2(k, r+nh) = 1 h
∞
X
l=−∞
2Bh2
π(h2k2+B2l2)e−π2l2/(α2h2)−π2k2/(α2B2)e2πilr/h
→Fourier coefficientsare given analytically
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Ewald summation s.t. 1d periodic boundary conditions
Long range part under 1d-periodic b.c. [Porto 2000]
We writexij= (xij,1,x˜ij)and obtain forφLZ×{0}2(xj) 2
B X
k∈Z\{0}
N
X
i=1
qie2πikxij,1/BΘp1(|k|,kx˜ijk)− 1 B
N
X
i=1
qiΘp10 (kx˜ijk)
1d Fourier series k= 0part
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Θp1(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 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Θp1(k, r) = Zα
0 1 texp
−πB22kt22−r2t2
dt Θp10 (r) =γ+ Γ(0, α2r2) + ln(α2r2)
Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion
Ewald summation s.t. 1d periodic boundary conditions
Long range part under 1d-periodic b.c. [Porto 2000]
We writexij= (xij,1,x˜ij)and obtain forφLZ×{0}2(xj) 2
B X
k∈Z\{0}
N
X
i=1
qie2πikxij,1/BΘp1(|k|,kx˜ijk)− 1 B
N
X
i=1
qiΘp10 (kx˜ijk)
1d Fourier series k= 0part
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Θp1(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 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Θp1(k, r) = Zα
0 1 texp
−πB22kt22 −r2t2 dt
Θp10 (r) =γ+ Γ(0, α2r2) + ln(α2r2)