South China University of Technology. Ising model and Stochastic optimization. Xiao Bao Yang Department of Physics. www.compphys.cn. Ordered Structures. AuCu Alloy. LiC 6. BC 3. H x C y. Ising model. The simplest Ising model assumes an interaction only between nearest neighbors:.
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South China University of Technology
Ising model and Stochastic optimization
XiaoBaoYang
Department of Physics
www.compphys.cn
Ordered Structures
AuCu Alloy
LiC6
BC3
HxCy
Ising model
The simplest Ising model assumes
an interaction only between
nearest neighbors:
Ernst Ising
For a certain J,
the lowest energy?
the corresponding structure?
Lattice gas model
Absorption: 1, 0 for atom, Vacancy
AB binary Alloy:1,0 for A,B
Ground state and ordered structures
For a certain J,
the lowest energy?
the corresponding structure?
onesearch.m & calha.m
Optimization and Curse of Dimensionality
Simulated Annealing
Genetic Algorithm
Particle Swarm Optimization
wolpertMacerday(No Free LunchNFL)
s s0; e E(s) // Initial state, energy.
sbest s; ebest e // Initial "best" solution
k 0 // Energy evaluation count.
while k < kmax and e > emax // While time left & not good enough:
T temperature(k/kmax) // Temperature calculation.
snew neighbour(s) // Pick some neighbour.
enew E(snew) // Compute its energy.
if P(e, enew, T) > random() then // Should we move to it?
s snew; e enew // Yes, change state.
if e < ebest then // Is this a new best?
sbest snew; ebest enew // Save 'new neighbour' to 'best found'.
k k + 1 // One more evaluation done
return sbest // Return the best solution found.
Metropolis algorithm(1953)
importance sampling
detail balance
Metropolis algorithm
Nicholas Metropolis
W(12)=1 E1>E2
Flipping rate
W(12)=exp[(E2E1)/kT] E1<E2
Reaches thermal equilibrium
Direct sampling
Markov chain sampling
Spin flip
A(ii,jj)
 A(ii,jj)
A(ii,jj)=sign(rand0.5);
Periodic Boundary Condition
Free Boundary Condition
Energy difference
dE=2*J*S(sf(1),sf(2))*
(S(sf(1)+1,sf(2))+S(sf(1)1,sf(2))
+S(sf(1),sf(2)+1)+S(sf(1),sf(2)1));
S=[A(:,n) A A(:,1)];
S=[S(end,:)
S
S(1,:)];
Accept or Reject?
S=[zeros(n,1) A zeros(n,1)];
S=[zeros(1,n+2)
S
zeros(1,n+2)];
if dE<0
S(sf(1),sf(2))=S(sf(1),sf(2));
end
if dE>0&&rand<exp(dE/T)
S(sf(1),sf(2))=S(sf(1),sf(2));
end
clear
n=10;
J=1;
nmax=10^6;
for ii=1:n
for jj=1:n
A(ii,jj)=sign(rand0.5);
end
end
S=[zeros(n,1) A zeros(n,1)];
S=[zeros(1,n+2)
S
zeros(1,n+2)];
TS=0.2:0.03:0.01
tmag=zeros(1,length(TS));
for it=1:length(TS)
T=TS(it)
for ii=1:nmax
sf=ceil(rand(1,2)*n)+1;
dE=2*J*S(sf(1),sf(2))*(S(sf(1)+1,sf(2))+S(sf(1)1,sf(2))+S(sf(1),sf(2)+1)+S(sf(1),sf(2)1));
if dE<0
S(sf(1),sf(2))=S(sf(1),sf(2));
end
if dE>0&&rand<exp(dE/T)
S(sf(1),sf(2))=S(sf(1),sf(2));
end
end
end
gsising.m & phase.m
Zungers research field is the condensed matter theory of real materials. He developed the firstprinciples pseudopotentialsfor the density functional theory (1977)
In recent years, Zunger has focused on developing the Inverse Band Structure concept, whereby one uses ideas from quantum mechanics as well as genetic algorithmsto search for atomic configurations that have a desired target property.
The simulatedannealing technique is an efficient algorithm for finding the global minimum of a multivariable, multivalley function.
Nature 402, 60 (1999)
(selectionreproduction), ,, NS,xiSP(xi), NSN, P(xi)
, ,
,
(crossover),,s1=01001011, s2=10010101, 4,
s1=01000101,s2=10011011s1s2s1s2
(mutation),(),s=1100110101, s=11101101
0, 14:
r1=0.450126, r2=0.110347, r3=0.572496, r4=0.98503
s1=1100024, s2=0110113, s3=1100024, s4=1001119
pc=100%S1s1s2s2s4
s1=1100125, s2=0110012,
s3=1101127, s4=1000016
pm=0.001S1540.001=0.020.021
s1=1100024, s2=0110113, s3=1100024, s4=1001119
S244
s1=1100125, s2=0110012, s3=1101127, s4=1000016
s1s2s3s4
s1=1110028, s2=010019, s3=1100024, s4=1001119
S3

s1=1110028, s2=1110028, s3=1100024, s4=1001119
s1s4s2s3
s1=1111131, s2=1110028, s3=1100024, s4=1000016
S4
s1=1111131, s2=1110028, s3=1100024, s4=1000016
Genomic Design of in Si/Ge films and Nanowires
much larger absorption from the magic sequence
NanoLett,12,984(2012);PRL,108,027401(2012)
Nat. Mater. 4, 391 (2005)
Divide people into
several teams;
Move towards the place
for lower cost.
PSO(fitness value)
PSO()pBestgBest
,V , Present ,pBest gBestrand ( )(0 ,1),c1c2 ,c1 = c2 = 2w , 0. 10. 9 , , gBest , Vmax ,VmaxVmax
function y=myfit(x)
y=1cos(3*x)*exp(x);
ClearN=4
re=zeros(N,5);
% x y pbest v ybest
w=0.5; re(:,1)=linspace(0,4,N);
re(:,4)=rand(N,1);
for ii=1:N
re(ii,2)=myfit(re(ii,1));
end
re(:,5)=re(:,2);re(:,3)=re(:,1);
[q,qq]=max(re(:,2));gbest=re(qq,3);
%
N
w
1); 2);3)pbestgbest
1) re(:,4)=re(:,4)*w+2*rand*(re(:,3)re(:,1))+rand*(gbest*ones(N,1)re(:,1))*2;
re(:,1)=re(:,1)+re(:,4);
3) for ii=1:N
re(ii,2)=myfit(re(ii,1));
if re(ii,5)<re(ii,2)
re(ii,5)=re(ii,2);
re(ii,3)=re(ii,1);
end
end
[q,qq]=max(re(:,2));
gbest=re(qq,3);
2) for ii=1:N
if re(ii,1)>4
re(ii,1)=4;
end
if re(ii,1)<0
re(ii,1)=0;
end
end
Artem R. Oganov
Yanming Ma
95 papers (including 5 in Nature,
1 in Nature Materials, 5 in PNAS, 6 in PRL)
2 in Nature and 9 in PRL
Find the minimum or maximum of a physical problem with one of the three typical optimization methods.