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 x old is the current state, define x new to be a neighboring state and delE=E(x new )-E(x old ) then If delE<0 replace x old by x new else choose x new with probability P(x new )= and x old with probability P(x old )=1- P(x new )

The probability to accept an increasing energy move

The Metropolis Algorithm  As T 0 and when delE>0 : P(x new ) 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 defectscrystalsdefects

Fast cooling – amorphous solid

Slow cooling - crystalline solid

SA for the 2D Ising E=-  ij s i s j, i and j are nearest neighbors E old =-2

SA for the 2D Ising E=-  ij s i s j, i and j are nearest neighbors E old =-2E new =2

SA for the 2D Ising E=-  ij s i s j, i and j are nearest neighbors E old =-2E new =2 delE=E new - E old =4>0 P(E new )=exp(-4/T)

SA for the 2D Ising E=-  ij s i s j, i and j are nearest neighbors E old =-2E new =2 delE=E new - E old =4>0 P(E new )=exp(-4/T) =0.3 => T=-4/ln0.3 ~ 3.3 Reduce T by a factor , 0<  <1: T n+1 =  T n

Exc#7: SA for the 2D Ising (see Exc#1) Consider the following cases: 1. For h 1 = h 2 =0 set a stripe of width 3,6 or 12 with opposite sign 2. For h 1 =-0.1, h 2 =0.4 set -1 at h 1 and +1 at h 2 3. Repeat 2. with 2 squares of 8x8 plus spins with h 2 =0.4 located apart from each other Calculate T 0 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 T 0, E in each case b. How was T 0 decreased, how many sweeps were employed c. What was the final configuration, was the global minimum achievable? If not try different T 0 d. Is it harder to flip a wider stripe? e. Is it harder to flip 2 squares than just one?