Stochastic tunneling

In numerical analysis, stochastic tunneling (STUN) is an approach to global optimization based on the Monte Carlo method-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. In numerical analysis, stochastic tunneling (STUN) is an approach to global optimization based on the Monte Carlo method-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. 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 Δ E {displaystyle Delta E} . The acceptance probability of such a trial jump is in most cases chosen to be min ( 1 ; exp ⁡ ( − β ⋅ Δ E ) ) {displaystyle min left(1;exp left(-eta cdot Delta E ight) ight)} (Metropolis criterion) with an appropriate parameter β {displaystyle eta } . The general idea of STUN is to circumvent the slow dynamics of ill-shaped energy functions that one encounters for example in spin glasses by tunneling through such barriers. This goal is achieved by Monte Carlo sampling of atransformed function that lacks this slow dynamics. In the 'standard-form'the transformation reads f S T U N := 1 − exp ⁡ ( − γ ⋅ ( E ( x ) − E o ) ) {displaystyle f_{STUN}:=1-exp left(-gamma cdot left(E(x)-E_{o} ight) ight)} where E o {displaystyle E_{o}} is the lowest function value found so far. This transformation preserves the loci of the minima. f S T U N {displaystyle f_{STUN}} is then used in place of E {displaystyle E} in the original algorithm giving a new acceptance probability of min ( 1 ; exp ⁡ ( − β ⋅ Δ f S T U N ) ) {displaystyle min left(1;exp left(-eta cdot Delta f_{STUN} ight) ight)}