# zbMATH — the first resource for mathematics

Variance reduction in Monte Carlo methods and optimization problems in $${\mathbb{R}}^ n$$. (Italian. English summary) Zbl 0601.65003
In the framework of Monte Carlo methods a technique for reducing the variance is considered. Generalizing a procedure suggested by the second author, Probabilistic estimation of error in Richardson extrapolation. ibid. 20, 455-465 (Italian) (1983), we give a method which reduces the variance by solving an unconstrained optimization problem. Concerned with the approximate integration, some numerical tests are considered. The described technique is shown to allow more accurate results than some known ones.
##### MSC:
 65C05 Monte Carlo methods 65D32 Numerical quadrature and cubature formulas 65K10 Numerical optimization and variational techniques 62F10 Point estimation
Full Text:
##### References:
 [1] P. Marzulli,Stima probabilistica dell’errore nella estrapolazione di Richardson, CAL-COLO, in questo numero. [2] J. H. Halton–D. C. Handscomb,A method for increasing the efficiency of Monte Carlo integration, Journal ACM,4, 1957, 329–340. · doi:10.1145/320881.320889 [3] R. Cranley–T. N. L. Patterson,A regression method for Monte Carlo evaluation of multidimensional integrals, Numer. Math.,16, 1970, 58–72. · Zbl 0188.22402 · doi:10.1007/BF02162407 [4] M. Cugiani,Metodi Numerico Statistici, (1980), UTET, Torino. [5] Y. A. Shreider,The Monte Carlo Methods, (1966), Pergamon Press, Oxford. · Zbl 0139.35602 [6] C. W. Clenshaw–A. R. Curtis,A method for numerical integration on an automatic computer, Numer. Math.,2, 1960, 197–205. · Zbl 0093.14006 · doi:10.1007/BF01386223 [7] A. H. Stroud,Approximate Calculation of Multiple Integrals, (1971), Prentice-Hall, Englewood Cliffs, New Jersey. · Zbl 0379.65013 [8] H. Engels,Numerical Quadrature and Cubature, (1980), Academic Press, New York. · Zbl 0435.65013 [9] S. Haber,Numerical evaluation of multiple integrals, SIAM Rev.,12, 1970, 481–526. · Zbl 0206.46905 · doi:10.1137/1012102 [10] IBM Application Program,System/360 Scientific Subroutine Package, (1970), Rep. N. GH20-0205-4. [11] IBM Data Processing Techniques,Random Number Generation and Testing (1969), Rep. N. GC20-8011-0.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.