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Monte carlo sampling4/3/2023 ![]() ![]() Caflisch, Quasi-Monte Carlo integration, J. Glynn, Stochastic Simulation: Algorithms and Analysis, Springer, 2007, 476 pages Compared to standard Monte-Carlo, the variance and the computation speed are slightly better from the experimental results in Tuffin (2008) See also Compared to pure quasi Monte-Carlo, the number of samples of the quasi random sequence will be divided by R for an equivalent computational cost, which reduces the theoretical convergence rate. Randomization allows to give an estimate of the variance while still using quasi-random sequences. If we have R replications for Monte Carlo, sample s-dimensional random vector U for each replication. ![]() ![]() ∫ s f ( u ) d u ≈ 1 N ∑ i = 1 N f ( x i ). The problem is to approximate the integral of a function f as the average of the function evaluated at a set of points x 1. Monte Carlo and quasi-Monte Carlo methods are stated in a similar way. This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers. In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences). Points from Sobol sequence are more evenly distributed. 256 points from a pseudorandom number source, Halton sequence, and Sobol sequence (red=1.,10, blue=11.,100, green=101.,256). ![]()
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