Stochastic methods

1
Stochastic methods

• Based on probability and chance
• Used for
• Percolation
• Statistical mechanics
• Quantum mechanics a la Feynman
• Brownian motion
• equations deterministic but forces random
• Diffusion
• Noise problem
• Certain transport problems

2
Deterministic problems

• Stochastic methods now used for deterministic
  problems with many degrees of freedom.
• To reduce the dimensionality of dynamics
• Heat bath particles exchange energy with bath
• replace with random force
• Mori theory
• For multidimensional integration

3
Multidimensional integral

• Phase space integral
• Quantum mechanical expectation value
• Functional integral for partition function

4
Variance of f
5
Comparison with trapezoidal rule
6
Elementary statistics

• Population Complete range - each member within
  range has a certain property.
• eg All the people in a country each has age
• all x within range a,bd in d - dimensions
• The property is the value f(x) and the average or
  mean value of f(x) relates to the integral.
• The various values of f will be spread around the
  population mean measured by the standard
  deviation s.

7
Sample means

• Because it is impractical to obtain the exact
  integral (the population (true) mean) we
  approximate with the sample mean.
• In general the means of sample will fluctuate
  about the true mean.
• The sample means will follow a normal
  distribution with a spread of s/sqrt(N)
• Normal distribution is characterised by its mean
  (the true mean) and its std deviation.

8
Probability distribution
Uniform distribution
1
0
1

• Normal distribution
• mean 3
• std dev 0.5

9
Probability distributions 2

• Uniform distribution each number within range
  0,1 has an equal chance of being chosen.
• non-uniform distribution the probability of
  choosing a number x in range x,xdx is p(x)dx.
  Note the choice is still random i.e. we dont
  know which but some are more likely.
• Normal or Gaussian p(x) exp(-ax2)

10
Increasing accuracy
11
Importance sampling
12
Example
13
Plots of f(x), f(x)/p(x)
f(x)
f(x)/p(x)
14
MC example
15
Numerical inversion

• Note inversion of () is not always possible. We
  can do it numerically.

16
Rejection method for p(x)dx
A
a
q(x0)
q(x)
p(x)
x0
0
1

• q(x) is ve function s.t. q(x) gt p(x)
• Pick uniform deviate,a, between 0 A (A is
  total areas under q(x)
• Find x0 s.t. area under q(x) between 0 and x0 is
  a
• Pick uniform deviate, y, between 0 and q(x0)
• Accept if y lt p(x)

17
Generating Gaussian distribution

• 1. use previous general method
• note previous equivalent to choosing uniform
  deviate h from 0,1 and accept if h lt p(x)/q(x)
  and reject otherwise
• 2. Inversion method - complicated
• 3. Central limit theorem
• The sum of a large number of uniformly
  distributed random numbers will approach Gaussian
  distrib.
• Mean and variance of uniform distribution is 1
  1/12 gt mean of 12 uniformly distributed numbers
  is 6 and variance 1.

18
Two Gaussians

• Hence if we generate u between 0 and with an
  exponential distribution and q between 0 and 2 p
  uniformly then and
  . will be distributed
  uniformly.
• To generate
• multiply x by s and increment by .

19
Importance sampling

• To minimise error in MC integration we choose a
  measure so that the integrand is as constant as
  possible i.e.
• we
  interpret choice of p(x) as sampling the most
  important parts of f(x)
• Previous methods of generating non-uniform
  distribution cannot apply to many dimensions.
• There is however the Metropolis algorithm

20
Metropolis algorithm

• Ref Metropolis, (Rosenbluth)2,(Teller)2, J.
  Chem. Phys 81 (1953) 1087
• Generate a set of points in multidimensional
  space distributed with prob .
• Points are generated via a random walk in
  space.
• As walk gets longer and longer the points move
  closer to the desired distribution.

21
Metropolis rules

• Suppose walker is at in the sequence
• To generate it makes a trial step to
• is chosen in a convenient manner eg
• at random within a multidimensional cube of side
  about
• Trial step is accepted or rejected according to
• If the step is accepted (i.e.
  ) whereas if the step is
  accepted with probability r

22
Rules continued

• Previous step gt generate another number h
  between 0,1 and accept if r gt h
• If trial step is rejected we put
  
• This generates n1 step, the next step n2 is
  generated by the same procedure
• Needs large n before distribution is obtained
• Size of d is critical. If too large only a small
  number of trial steps will be accepted
• If too small too many accepted and also
  inefficient. 1/3 to 1/2 should be accepted

23
Proof of Metropolis
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Proof continued