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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

(2)

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

(3)

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)

(4)

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

(5)

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)

(6)

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)

(7)

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

(8)

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

(9)

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)

(10)

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 =π ⇒substractself potential

write thelong range partas a sum in Fourier space

→use the FFT for nonequispaced data (NFFT)

(11)

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 =π ⇒substractself potential

write thelong range partas a sum in Fourier space

→use the FFT for nonequispaced data (NFFT)

(12)

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 = π ⇒substractself potential

write thelong range partas a sum in Fourier space

→use the FFT for nonequispaced data (NFFT)

(13)

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

ke2πik·xj xj∈T3, j= 1, . . . , N

FFT: f(j) := X

k∈IM

ke2πik·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]

(14)

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

ke2πik·xj xj∈T3, j= 1, . . . , N FFT: f(j) := X

k∈IM

ke2πik·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]

(15)

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

ke2πik·xj xj∈T3, j= 1, . . . , N FFT: f(j) := X

k∈IM

ke2πik·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]

(16)

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+ 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

(17)

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+ 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

(18)

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+ 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

(19)

Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion

Fast Algorithm: P

2

NFFT 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

(20)

Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion

Fast Algorithm: P

2

NFFT 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

(21)

Coulomb interactions 3d Ewald 2d Ewald 1d Ewald Conclusion

Fast Algorithm: P

2

NFFT 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

(22)

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)

(23)

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)

(24)

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)

(25)

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+ h/2

h/2 h/2

j

∂rjKB(k,h/2hε) = 1

2Bj

∂rjΘp2(k,h/2hε) 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

(26)

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+ h/2

h/2 h/2

j

∂rjKB(k,h/2hε) = 1

2Bj

∂rjΘp2(k,h/2hε) 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

(27)

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

(28)

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

(29)

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

(30)

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

(31)

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

(32)

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.

(33)

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?

(34)

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

(35)

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

(36)

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

πB22kt22r2t2

dt Θp10 (r) =γ+ Γ(0, α2r2) + ln(α2r2)

(37)

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)

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