A New Castep and Onetep geometry optimiser
Transcript of 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)
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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
10 100 1000 10000System Size (Atoms)
1e+05
1e+06
1e+07
1e+08
1e+09
Memory(B
ytes)
+
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10 100 1000 10000System Size (Atoms)
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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