Biomedical Engineering Reference
In-Depth Information
The value of using the Metropolis Algorithm is that it offers a means of maximizing the odds of
jumping out of a local minima and into the global minimum—and staying there. In nature, molten
materials, such as quartz, when allowed to cool slowly, find the global local minimum state—they
crystallize. However, when the material is cooled quickly, the material ends up in local minima—an
amorphous state.
Algorithmically, the probability that a system at temperature
T
is in a state of energy
E
(not at the
global minimum) appears as:
Assuming a temperature of
T
is assigned to the system, the probability of changing from state 1 to
state 2 is:
The initial value of
T
should be great enough to allow all local minima and the global minimum to be
evaluated. As long as
T
is greater than zero, there is a probability of a jump from a local to a global
minimum. Consider the method graphically in
Figure 9-9
.
Figure 9-9. The Metropolis Algorithm Applied in a Simulated Annealing
Method. As the temperature (
T
) of the system decreases, the number of
local minima available to evaluate
f(x)
decreases. Whereas three minima
are available at the initial temperature (left), at a lower temperature
(right) only two minima are available.
As the temperature is decreased, the maximum value of the function
f(x)
decreases as well.
f(x)
is
represented by the black circle in
Figure 9-9
, which can be thought of as a particle with kinetic
energy defined by the temperature. As random values of
x
are evaluated by the annealing function,
the function locates local minima and maxima by chance. Assuming enough iterations of evaluating
