Adaptive Importance Sampling Monte Carlo

Another procedure to overcome the inefficiency of Metropolis Monte Carlo is adaptive importance sampling.194-196 In this technique, the partition function (and quantities derived from it, such as the probability of a given conformation) is evaluated by continually upgrading the distribution function (ultimately to the Boltzmann distribution) to concentrate the sampling in the region (s) where the probabilities are highest. 

The partition function for a polypeptide chain with a specific amino acid sequence can be written as194 

The quantity U J , i , x is computed by summing over the ECEPP energies of all pairs of nonbonded atoms of the whole molecule. Equations [11] and [12] are used to compute conditional free energy (4 ,i]i) contour maps, similar to standard conformational energy ( , ) contour maps, for each residue of the polypeptide. The probable conformation of the /th residue is taken as the one of lowest conditional free energy, and the probable conformation of the whole polypeptide chain is assumed to be the combination of the probable conformations of the individual residues. 

Besides calculating the probable conformation, it is also possible to compute the average conformation195 over the whole ensemble. For this purpose, we first compute the average values of f , and i i( for the /th residue as 

The average conformation of the oligopeptide chain is then taken as the combination of the average values of 4 and i / for all individual residues. 

---

The statistical error of such sampling is proportional to 1 / /N. Therefore, the accuracy of the method increases with the sample size. More advanced adaptive Monte Carlo methods use, for instance, importance sampling to enhance the convergence and the accuracy over that of the basic Monte Carlo integration. 

This Iterative Fast Monte Carlo (IFM) procedure [20] utilizes the results obtained from Importance sampling to adapt the sampling density to the specific problem. As stated above, the standard error of the estimated failure probability reduces to zero, If the sampHng density hytii) Is chosen to be the original density conditional on the failure domain Of... 

This section discussed a variety of simulation techniques to estimate structural reliability. Topics such as Monte Carlo sampling, importance sampling, and stratified sampling were discussed in detail, and some other advanced topics such as adaptive sampling, variance... 

---