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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

Xiao-BaoYang

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


heuristic algorithm

Simulated Annealing

Genetic Algorithm

Particle Swarm Optimization


No Free Lunch

wolpertMacerday(No Free LunchNFL)


SimulatedAnnealing

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[-(E2-E1)/kT] E1<E2

Reaches thermal equilibrium


The Monte Carlo method

Direct sampling

Markov chain sampling


Spin flip

A(ii,jj)

- A(ii,jj)

A(ii,jj)=sign(rand-0.5);

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(rand-0.5);

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 first-principles 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 inverse band-structure problem of finding an atomic configuration with given electronic properties

The simulated-annealing technique is an efficient algorithm for finding the global minimum of a multi-variable, multi-valley function.

Nature 402, 60 (1999)


Genetic Algorithm

  • (fitness),

  • (chromosome), ,

  • (population-),

  • : -,(genetic operator)


--(selectionreproduction), ,-, NS,xiSP(xi), NSN, P(xi)

, ,

,


(crossover),,s1=01001011, s2=10010101, 4,

s1=01000101,s2=10011011s1s2s1s2

(mutation),(),s=1100110101, s=11101101


0,31y=x2

  • 0,315x

  • 4,s1=01101(13),s2=11000(24),s3=01000(8), s4=10011(19)S1

-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)


Evolutionary approach for determining first-principles hamiltonians

Nat. Mater. 4, 391 (2005)


Divide people into

several teams;

Move towards the place

for lower cost.


Particle Swarm Optimization

PSO(fitness value)

PSO()pBestgBest


,V , Present ,pBest gBestrand ( )(0 ,1),c1c2 ,c1 = c2 = 2w , 0. 10. 9 , , gBest , Vmax ,VmaxVmax


Program(1)

function y=myfit(x)

y=1-cos(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


Program(2)

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


Famous scientists in High pressure

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


Materials under High Pressure


Homework

Find the minimum or maximum of a physical problem with one of the three typical optimization methods.


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