3.4.1 Simulated Annealing

Consider a Metropolis walk through the space of ``states'' $\mbox{$\bf x$}_{\alpha}$ with

$$p_{\alpha} = A \exp{- \beta U(\mbox{$\bf x$})}$$

where $U(x_{1},\dots x_{M})$ is a ``cost function'' to be minimized, and $\beta$ a tunable parameter (a reciprocal ``temperature''.)

- Low $\beta$ $\Longrightarrow$smaller variation of $p_{\alpha}$; higher $U(\mbox{$\bf x$})$ are accessible

- High $\beta$ $\Longrightarrow$ $\mbox{$\bf x$}$ will tend to go ``downhill''





EXERCISE: Create (fake!) a table of ``measured values with errors'' according to

$$y_{i}=f(x_{i};c_{1},\dots c_{6})+\xi_{i}\,,\;\;\; i=1,20$$ (3.2)

with $\xi_{i}$ coming from a Gauss distribution with suitable variance, and with the function $f$ defined by

$$f(x;\mbox{$\bf c$})\equiv c_{1}e^{\textstyle - c_{2}(x-c_{3})^{2}} +c_{4}e^{\textstyle - c_{5}(x-c_{6})^{2}}$$ (3.3)

( $c_{1} \dots c_{6}$ being a set of arbitrary coefficients).

Using these data, try to reconstruct the parameters $c_{1} \dots c_{6}$ by fitting the theoretical function $f$ to the table points $(x_{i},y_{i})$. The cost function is

$$U(\mbox{$\bf c$})\equiv \sum_{i}\left[ y_{i}-f(x_{i};\mbox{$\bf c$})\right]^{2}$$ (3.4)

Choose an initial vector $\mbox{$\bf c$}^{0}$ and perform an MC random walk through $\mbox{$\bf c$}$-space, slowly lowering the temperature.