samples/texts/202179/page_5.md

a global minima statistically can be obtained. I.e.,

$$\sum _{k_0}^{\mathrm{\infty }}{g}_k\approx \sum _{k_0}^{\mathrm{\infty }}\left[\prod _{i=1}^D\frac{1}{2\mathrm{\mid }{y}^i\mathrm{\mid }{c}_i}\right]\frac{1}{k}=\mathrm{\infty }\mathrm{.}(18)\backslash sum\_\{k\_0\}^\{\backslash infty\}\; g\_k\; \backslash approx\; \backslash sum\_\{k\_0\}^\{\backslash infty\}\; \backslash left[\; \backslash prod\_\{i=1\}^\{D\}\; \backslash frac\{1\}\{2|y^i|c\_i\}\; \backslash right]\; \backslash frac\{1\}\{k\}\; =\; \backslash infty.\; \backslash quad\; (18)$$

It seems sensible to choose control over $c_i$, such that

$$T_{fi}=T_{0i}\exp (-m_i)\text{ when }{k}_f=\exp n_i,T\_\{fi\}\; =\; T\_\{0i\}\; \backslash exp(-m\_i)\; \backslash text\{\; when\; \}\; k\_f\; =\; \backslash exp\; n\_i,$$

$$\begin{array}{cccc}& c_i=m_i\exp (-n_i\mathrm{/}D)\mathrm{.}& & \text{(19)}\end{array}c\_i\; =\; m\_i\; \backslash exp(-n\_i\; /\; D).\; \backslash tag\{19\}$$

The cost-functions $\underline{L}$ we are exploring are of the form

$$h(M;\alpha )=\exp (-\underset{‾}{L}\mathrm{/}T),h(M;\; \backslash alpha)\; =\; \backslash exp(-\backslash underline\{L\}/T),$$

$$\underset{‾}{L}=L\mathrm{\Delta }t+\frac{1}{2}\ln (2\pi \mathrm{\Delta }t{g}_t^2),(20)\backslash underline\{L\}\; =\; L\backslash Delta\; t\; +\; \backslash frac\{1\}\{2\}\; \backslash ln(2\backslash pi\backslash Delta\; t\; g\_t^2),\; \backslash qquad\; (20)$$

where $L$ is a Lagrangian with dynamic variables $M(t)$, and parameter-coefficients $\alpha$ to be fit to data. $g_t$ is the determinant of the metric, defined below. It has proven fruitful to use the same annealing schedule for this acceptance function $h$ as used for the generating function $g$, i.e., Eqs. (17) and (19).

New parameters $\alpha_{k+1}^i$ are generated from old parameters $\alpha_k^i$ from

$${\alpha }_{k+1}^i={\alpha }_k^i+y^i(B_i-A_i),(21)\backslash alpha\_\{k+1\}^\{i\}\; =\; \backslash alpha\_\{k\}^\{i\}\; +\; y^\{i\}(B\_\{i\}\; -\; A\_\{i\}),\; \backslash qquad\; (21)$$

constrained by

$$\begin{array}{cccc}& {\alpha }_{k+1}^i\in [A_i,B_i]\mathrm{.}& & \text{(22)}\end{array}\backslash alpha\_\{k+1\}^i\; \backslash in\; [A\_i,\; B\_i].\; \backslash tag\{22\}$$

I.e., $y^i$'s are generated until a set of $D$ are obtained satisfying these constraints.

V. Re-Annealing

Whenever doing a multi-dimensional search in the course of a real-world nonlinear physi- cal problem, inevitably one must deal with different changing sensitivities of the $\alpha^i$ in the search. At any given annealing-time, it seems sensible to attempt to “stretch out” the range over which the relatively insensitive parameters are being searched, relative to the ranges of the more sensi- tive parameters.

It has proven fruitful to accomplish this by periodically rescaling the annealing-time k, essentially re-annealing, every hundred or so acceptance-events, in terms of the sensitivities sᵢ, calculated at the most current minimum value of L,

$$\begin{array}{cccc}& s_i=(A_i-B_i)\mathrm{\partial }\underset{‾}{L}\mathrm{/}\mathrm{\partial }{\alpha }^i\mathrm{.}& & \text{(23)}\end{array}s\_i\; =\; (A\_i\; -\; B\_i)\; \backslash partial\; \backslash underline\{L\}\; /\; \backslash partial\; \backslash alpha^i\; .\; \backslash tag\{23\}$$

In terms of the largest $s_i = s_{\max}$, it has proven fruitful to re-anneal by using a linear rescaling,

$$k_i^{\mathrm{\prime }}=((\ln [(T_{i0}\mathrm{/}{T}_{ik})(s_{\max }\mathrm{/}{s}_i)])\mathrm{/}{c}_i{)}^D\mathrm{.}(24)k\text{'}\_i\; =\; ((\backslash ln[(T\_\{i0\}/T\_\{ik\})(s\_\{\backslash max\}/s\_i)])/c\_i)^D\; .\; \backslash quad\; (24)$$

$T_{i0}$ is set to unity to begin the search, which is ample to span each parameter dimension.

The acceptance temperature is similarly rescaled. In addition, since the initial acceptance temperature is set equal to a trial value of $\underline{L}$, this is typically very large relative to the global min- imum. Therefore, when this rescaling is performed, the initial acceptance temperature is reset to the most current minimum of $\underline{L}$, and the annealing-time associated with this temperature is set to give a new temperature equal to the lowest value of the cost-function encountered to annealing- date.