Stochastic Relaxation,

Simulating Annealing,Global Minimizers

Different types of relaxation

Variable by variable relaxation – strict minimization

Changing a small subset of variables simultaneously – Window strict minimization relaxation

Stochastic relaxation – may increase the energy – should be followed by strict minimization

Complex landscape of E(X)

How to escape local minima? First go uphill, then may hit a lower basin In order to go uphill should allow increase in E(x) Add stochasticity: allow E(x) to increase with

probability which is governed by an external temperature-like parameter T

The Metropolis Algorithm (Kirpartick et al. 1983)

Assume xold is the current state, define xnew to be a neighboring state and delE=E(xnew)-E(xold) then

If delE<0 replace xold by xnew

else choose xnew with probability P(xnew)=

and xold with probability P(xold)=1- P(xnew)

TdelEe /

The probability to accept an increasing energy move

The Metropolis Algorithm As T 0 and when delE>0 : P(xnew) 0 At T=0: strict minimization High T randomizes the configuration away

from the minimum Low T cannot escape local minimaStarting from a high T, the slower T is

decreased the lower E(x) is achieved The slow reduction in T allows the material to

obtain a more arranged configuration: increase the size of its crystals and reduce their defects

Fast cooling – amorphous solid

Slow cooling - crystalline solid

SA for the 2D Ising

E=-ijsisj , i and j are nearest neighbors

++

++

Eold=-2

SA for the 2D Ising

E=-ijsisj , i and j are nearest neighbors

++

++

+

++

Eold=-2 Enew=2

SA for the 2D Ising

E=-ijsisj , i and j are nearest neighbors

++

++

+

++

Eold=-2 Enew=2

delE=Enew- Eold=4>0

P(Enew)=exp(-4/T)

SA for the 2D Ising

E=-ijsisj , i and j are nearest neighbors

++

++

+

++

Eold=-2 Enew=2

delE=Enew- Eold=4>0

P(Enew)=exp(-4/T) =0.3

=> T=-4/ln0.3 ~ 3.3

Reduce T by a factor , 0<<1: Tn+1=Tn

Exc#7: SA for the 2D Ising (see Exc#1)Consider the following cases:

1. For h1= h2=0 set a stripe of width 3,6 or 12 with opposite sign

2. For h1=-0.1, h2=0.4 set -1 at h1 and +1 at h2

3. Repeat 2. with 2 squares of 8x8 plus spins with h2=0.4 located apart from each other

Calculate T0 to allow 10% flips of a spin surrounded by 4 neighbors of the same sign

Use faster / slower cooling scheduling

a. What was the starting T0 , E in each case

b. How was T0 decreased, how many sweeps were employed c. What was the final configuration, was the global minimum

achievable? If not try different T0

d. Is it harder to flip a wider stripe?e. Is it harder to flip 2 squares than just one?