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Nonlinear Optimization
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
We might also consider a quasi-Monte Carlo method based on a quasi-random number generator that generates numbers that are quasi-random, meaning ”just random enough for our purposes”2. Quasi-random number generators generate deterministic sequences that cover a region with a non-uniform grid and they generate different sequences for each initial condition, but these sequences will not pass nearly as many tests of randomness as a PRNG would. For example, typically the quasi-random number generators ensure that every sub-region is sampled a minimum number of times. For a PRNG more deviation would be expected. It's also possible to have the quasi-random numbers appear in order, as though a pseudo-random sequence had been sorted. Methods based on a PRNG are more commonly employed, partly because the PRNG is generally provided as a built-in function of the language; the error estimates are typically considerably better for quasi-Monte Carlo methods, called quasi-random searches in the optimization context.
A modified Monte Carlo approach to the approximation of invariant measures
Published in B Bertram, C Constanda, A Struthers, Integral methods in science and engineering, 2019
The original Monte Carlo method first proposed in [1] selected the K numbers randomly. That is, the points zi,k are obtained from a random number generator. Numerical experiments of [1] indicate that the resulting error may be relatively large compared with the exact method. To get a better approximation in (8), we use the idea of the quasi Monte Carlo method. Here the test points zi,k are chosen deterministically. In other words, if Ii = [xi−1, xi−1 + h], then zi,k=xi−1+kKh,k=1,2,…,K.
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Published in Ilya M. Sobol’, A Primer for the Monte Carlo Method, 2018
The corresponding quasi-Monte Carlo method Mξ≈IN=1N∑j=1N1+2qj122+2qj23…9+2qj910
Statistical Modeling of the Effectiveness of Preventive Maintenance for Repairable Systems
Published in Technometrics, 2023
Xin YE, Jiaxiang CAI, Loon Ching TANG, Zhi-Sheng YE
When the random effect Ak follows a gamma or lognormal distribution, neither the conditional distribution in (5) nor the Q-function in (4) has a closed-form expression. The Monte Carlo integration method is a useful tool for evaluating such multidimensional integrations. The law of large numbers guarantees the convergence of the Monte Carlo approximation (Mendenhall et al. 2012). However, one shortcoming is that the rate of convergence is slow because the size of the Monte Carlo samples should be large enough to achieve acceptable convergence precision. To reduce the computational burden, the quasi-Monte Carlo integration method is adopted here (Morokoff and Caflisch 1995). In contrast to regular Monte Carlo methods, the quasi-Monte Carlo method does not draw sampling points in a pseudorandom way. The latter produces a sequence of deterministic numbers with the best-possible spread in the sampling space, and sampling points are more uniformly distributed than the regular Monte Carlo method. This sequence can also be referred to as the low discrepancy sequence (Fang et al. 2000). Smaller errors and improved convergence rates can be obtained by the more evenly distributed sampling points. Detailed discussions on the Monte Carlo method and quasi-Monte Carlo method can be found in Supplement Section S4.
The pinpointing of the most prominent parameters on the energy performance for optimal passive strategies in ecological buildings based on bioclimatic, sensitivity and uncertainty analyses
Published in International Journal of Ambient Energy, 2022
The sampling techniques that can be used are numerous, among these we find: The Monte Carlo method: is based on a random sampling in the variation range of the input factors according to their distributions. It requires a large number of simulations to arrive at the output results.Quasi-Monte Carlo method (Niederreiter 1992): is a deterministic version of the Monte Carlo method. The convergence is faster than that of the previous technique for the same number of points.Sampling of Latin Hypercube Sampling (McKay, Beckman, and Conover 1979): its sampling consists in cutting the space of the input variables into small equiprobable spaces (Jacques 2011).
Uncertainty quantification of offshore wind farms using Monte Carlo and sparse grid
Published in Energy Sources, Part B: Economics, Planning, and Policy, 2022
Pascal Richter, Jannick Wolters, Martin Frank
The presented results show that Monte Carlo is the least accurate method in terms of the observed relative error across all sample sizes which is to be expected from its theoretical convergence rate. The sparse grid Stochastic Collocation method shows a higher convergence rate compared to the quasi-Monte Carlo method, despite having a higher initial error for lower grid levels. Both methods achieve similar relative errors of for the expectation and for the variance with respect to the reference quasi-Monte Carlo solution. The dashed lines in Figures 10 and 11 show the theoretical convergence rates of the classical Monte Carlo and the quasi-Monte Carlo methods. While the classical Monte Carlo method shows the expected convergence rate, it is worth noting that the quasi-Monte Carlo method, shows a comparable error decline despite the much higher dimensionality of the problem. This implies that the probability space of the underlying problem might be dominated by a single dimension and that the low discrepancy sequence of the method can successfully exploit this fact for an accelerated convergence behavior. This observation becomes even more apparent in the sensitivity analysis presented in Section 5.