a new castep and onetep geometry optimiser

8/13/2019 A New Castep and Onetep geometry optimiser

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N2

N


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(6 + 3N)


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O(N2)

f

x

f(x + x) =f(x) + J[f(x)]x +1

2xTH[f(x)]x . . .

J H

x

0 = J([f(x)]) + H[f(x)]x

f

x= H[f(xn)]1J([f(xn)])

xn+1= xn xn

N2

Hn+1sn = yn

sn yn

H1

n+1= (I

kyk

sT

k)H1

n (I

kyk

sT

k) +

ksk

sT

k


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k = 1yTksk

H1 B

pn = Bnf(xn)

(A + BC D)1 =A1 A1B(C1 + DA1B)1DA1

xn+1= xn+ npn

f(xn+ npn) f(xn) + w1nfTnpn

f(xn+ npn)Tpn w2f

Tnpn

w1 w2


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Bn = B0+

S B0Y

W

ST

(B0Y)T

Wn =

RTn (Dn+ Y

Tn B0Yn)R

1n R

Tn

R1n 0

B0

R1n+1=

R1n nR

1n S

Tn yn

0 n

Dn+1=

Dn 0

0 yTn sn

n = 1

yTn sn

Sn =

s0, . . . , sn1

, Yn =

y0, . . . , yn1

sn yn nth

nth R1n

Wn

m2

Rn Tn = Dn+ YTn B0Yn m n Sn Yn

v

p

f : Rn Rn, p Bnv

p= B0v

w1:m = Y

T

n p

wm+1:2m = STn v

w1:m R1n Y

Tn B0v

wm+1:2m R1n S

Tn v

= Tn

R1n

w1:m w1:m wm+1:2m

p p + STn w1:m

t= YTn wm+1:2m

p p + B0t


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w

2m

m m

n

k


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B0=

(3B0)1 0

(3B0)1

g10M120

0 g10M120

B0 M 20 g0

. . . . . . . . . . . . . . . . . .

. . . . . . . . .

B0

Bn

Bn = DTBnD

1

D=

(1 + ) 0(1 + )

(1 + )1

0 1

.

ndim2

Indim

f= (1 + ) B


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Bn

TfB(1:31:3)n f

TfB(4:64:6)n f B

(1:910:n)n

TfB(7:97:9)n f

B(10:n1:9)n B

(10:n10:n)n

B =V W

B Bo

B

B

N2

N

N2

B

B =

S B0Y

W

ST

(B0Y)T

B =QRW RTQT

RW RT

B =QPPTQT

Q P

QP

Z

Z

B

B :

cell cell 0

0 ion ion

B


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Z

RW RT

Z

B

9 9

N2

=

I6 0M1

M1M1

M2

0 MN

,

Z

A= (ZT1/2B1/2Z)1

Z

N

N

MM A B0

A= (ZT(B0+

S B0Y

W

ST

(B0Y)T)

)Z)1

Z=1/2Z

cell cell cell ion

ion

cell ion

ion


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O(N2)

A

N2

A m

Bn

Bn

6 6 9 9

=

=


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5 6 7System Size (Supercells of Quartz in 3 Coordinate Dimensions)

0

50

100

150

PercentageofBFGSbeforeBLAS

Memory

Time

2 2

W

n m

R m m

R

m n ST

R

O(N2)

O(M2

N)

O(N2) 2N M2+2M N+3M3+2M2 + 6N

10N2


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10 100 1000 10000Number of Degrees of Freedom

1000

10000

1e+05

1e+06

1e+07

1e+08

1e+09

Operations

BFGSLBFGS (15 update history)

LBFGS (30)

LBFGS (45)

LBFGS (60)

LBFGS (100)

N2


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f : Rn Rn, Z basis(B)

U1:n1:2m=

Sn B0Yn

Q1:n1:2m R1:2m1:2m

= qr(U)

C1:2m1:2m= RW

C CR P1:2m1:2m 1:2m

= spectral(C)

v= pivot()

i 2m: (i) (v(i))

i 2m (i)> 0 :P1:n(i) P1:n(v(i))

Z1:n1:l = U1:n1:2mP1:2m1:l

A l 2m

B0 B

n l


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10 100 1000 10000System Size (Atoms)

1e+05

1e+06

1e+07

1e+08

Memory

(Bytes)

+


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N2

2.5 24 = 60


1e+05

1e+06

1e+07

1e+08

1e+09

Memory(B

ytes)

+


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1

10

100

1000

Time(Seconds)

LBFGS (15 Updates)

LBFGS (30)

LBFGS (45)LBFGS (60)

LBFGS (100)

BFGS equivalent


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O(N)

O(N)


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a new castep and onetep geometry optimiser

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