By Frank Ayres Jr.

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3 Quasi-Monte Carlo Methods Quasi-Monte Carlo methods seek to construct a sequence of points that perform significantly better than Monte Carlo, which has an average case of complexity of the order of 12 . For a suitably chosen set of samples, the quasi-Monte Carlo method 20 2 Uncertainty Analysis and Sampling Techniques provides a deterministic error bound of the order n−1 (log n)k−1 without any strong assumptions about the integrand. Some well-known quasi-Monte Carlo sequences are Halton, Hammersley, Sobol, Faure, Korobov and Neiderreiter [35].

D) Calculate i ωi . e) Estimate the probabilistic objective function and constraints values: i. Set i = 1, J k = 0. ii. While i < Nsamp , calculate: J k = Jik ∗ ωi / iii. i = i + 1. Go to step ii. i ωi . 2. 2. While d ≤ D, perturb one decision variable θdk to find θdk,Δ . Reset deterministic decision variable counter d = 1. a) Generate (i = 1 to Nsamp ) samples with the appropriate distributions at θdk,Δ for all variables uik . 2, using Eq. 6 in step ii instead. c) Determine the weights ωi from the product of ratios, ΠS fs (uik )/fˆs (ui ).

Frequently used variance reduction sampling methods are importance sampling, Latin Hypercube Sampling, descriptive sampling and Hammersley Sequence Sampling (HSS). HSS is based on quasi-random numbers generated using Hammersley sequences. HSS is found to be 3 to 100 times faster than other sampling techniques. 4 Summary Pm scaled probabilities R integer in R-radix notation xi , Xi random number → xk (n) Um Hammersley points samples from uniform distribution (U(0,1)) Greek letters error σ standard deviation ϕ(n) inverse radix function for n 25 Chapter 3 Probability Density Functions and Kernel Density Estimation Stochastic modeling loop in the stochastic optimization framework involves dealing with evaluation of a probabilistic objective function and constraints from the output data.