Editing Stochastic tunneling

{{Short description|Stochastic method of global optimization}}
In [[numerical analysis]], '''stochastic tunneling''' (STUN) is an approach to [[global optimization]] based on the [[Monte Carlo method]]-[[Sampling (signal processing)|sampling]] of the function to be objective minimized in which the function is nonlinearly transformed to allow for easier tunneling among regions containing function minima. Easier tunneling allows for faster exploration of sample space and faster convergence to a good solution.

== Idea ==
[[image:stun.jpg|thumb|400px| Schematic one-dimensional test function (black) and  STUN effective potential (red & blue), where the minimum indicated by the arrows is the best minimum found so far. All [[Potential well|well]]s that lie above the best minimum found are suppressed. If the dynamical process can escape the well around the current minimum estimate it will not be trapped by other local minima that are higher. Wells with deeper minima are enhanced. The dynamical process is accelerated by that.]]

[[Monte Carlo method]]-based optimization techniques sample the [[objective function]] by randomly "hopping" from the current solution vector to another with a difference in the function value of <math>\Delta E</math>. The acceptance probability  of such a trial jump is in most cases chosen to be
<math> \min\left(1;\exp\left(-\beta\cdot\Delta E\right)\right)
</math> ([[Nicholas Metropolis|Metropolis]] criterion) with an appropriate parameter <math>\beta</math>.

The general idea of STUN is to circumvent the slow dynamics of ill-shaped energy functions that one encounters for example in [[spin glass]]es by tunneling through such barriers.

This goal is achieved by Monte Carlo sampling of a
transformed function that lacks this slow dynamics. In the "standard-form"
the transformation reads <math>f_{STUN}:=1-\exp\left(
-\gamma\cdot\left( E(x)-E_o\right) \right)</math> where <math>E_o</math>
is the lowest function value found so far. This transformation preserves the [[Locus (mathematics)|loci]] of the minima.

<math>f_{STUN}</math> is then used in place of <math>E</math> in the original algorithm giving a new acceptance probability of <math> \min\left(1;\exp\left(-\beta\cdot\Delta f_{STUN}\right)\right)
</math>

The effect of such a transformation is shown in the graph.

== Dynamically adaptive stochastic tunneling ==

A variation on always tunneling is to do so only when trapped at a local minimum. <math>\gamma</math> is then adjusted to tunnel out of the minimum and pursue a more globally optimum solution. [[Detrended fluctuation analysis]] is the recommended way of determining if trapped at a local minimum.

== Other approaches ==
* [[Simulated annealing]]
* [[Parallel tempering]]
* [[Genetic algorithm]]
* [[Differential evolution]]

== References ==
* {{Cite journal
 | author = K. Hamacher
 | title = Adaptation in Stochastic Tunneling Global Optimization of Complex Potential Energy Landscapes
 | journal = [[Europhys. Lett.]]  
 | volume = 74
 | issue = 6
 | pages = 944–950
 | year = 2006
 | doi = 10.1209/epl/i2006-10058-0
|bibcode = 2006EL.....74..944H | s2cid = 250761754
 }}
* {{Cite journal
 |author1=K. Hamacher  |author2=W. Wenzel
  |name-list-style=amp | title = The Scaling Behaviour of Stochastic Minimization Algorithms in a Perfect Funnel Landscape
 | journal = [[Phys. Rev. E]]
 | volume = 59
 | issue = 1
 | pages = 938&ndash;941
 | year = 1999
 | doi = 10.1103/PhysRevE.59.938
|arxiv = physics/9810035 |bibcode = 1999PhRvE..59..938H |s2cid=119096368
 }}
* {{Cite journal
 |author1=W. Wenzel  |author2=K. Hamacher
  |name-list-style=amp | title = A Stochastic tunneling approach for global minimization
 | journal = [[Phys. Rev. Lett.]]
 | volume = 82
 | issue = 15
 | pages = 3003&ndash;3007
 | year = 1999
 | doi = 10.1103/PhysRevLett.82.3003
| bibcode=1999PhRvL..82.3003W
|arxiv = physics/9903008 |s2cid=5113626
 }}
* {{Cite journal
 | author = [[Nicholas Metropolis]], Arianna W. Rosenbluth, [[Marshall N. Rosenbluth]], Augusta H. Teller and [[Edward Teller]]
 | title = Equation of State Calculations by Fast Computing Machines
 | journal = [[The Journal of Chemical Physics]]
 | volume = 21
 |date=June 1953
 | pages = 1087&ndash;1092
 | doi = 10.1063/1.1699114
 | url = http://scienze-como.uninsubria.it/bressanini/montecarlo-history/mrt2.pdf
 | issue = 6
 | bibcode=1953JChPh..21.1087M
| osti = 4390578
 | s2cid = 1046577
 }}
* {{Cite journal
 | author = Mingjie Lin
 | title = Improving FPGA Placement with Dynamically Adaptive Stochastic Tunneling
 | journal = [[IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems]]
 | volume = 29
 |date=December 2010
 | pages = 1858&ndash;1869
 | issue = 12
 | ref = lin
 | doi=10.1109/tcad.2010.2061670
| s2cid = 8706692
 | url = https://zenodo.org/record/1232243
 }}

[[Category:Stochastic optimization]]