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Stochastic Relaxation, Simulating Annealing, Global Minimizers

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

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Stochastic Relaxation, Simulating Annealing, Global Minimizers

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  1. Stochastic Relaxation,Simulating Annealing,Global Minimizers

  2. 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

  3. Complex landscape of E(X)

  4. 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 xnewto be a neighboring state and delE=E(xnew)-E(xold) then IfdelE<0 replace xold by xnew else choose xnew with probability P(xnew)= and xoldwith probability P(xold)=1- P(xnew)

  5. The probability to accept an increasing energy move

  6. 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 minima • Starting 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

  7. Fast cooling – amorphous solid

  8. Slow cooling - crystalline solid

  9. SA for the 2D IsingE=-Sijsisj , i and j are nearest neighbors + + + + Eold=-2

  10. SA for the 2D IsingE=-Sijsisj , i and j are nearest neighbors + + + + + + + Eold=-2 Enew=2

  11. SA for the 2D IsingE=-Sijsisj , i and j are nearest neighbors + + + + + + + Eold=-2 Enew=2 delE=Enew- Eold=4>0 P(Enew)=exp(-4/T)

  12. SA for the 2D IsingE=-Sijsisj , 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 a, 0<a<1: Tn+1=aTn

  13. 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?

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