A Monte Carlo method for high dimensional integration
type: Item
instance of: scholarly article
title: A Monte Carlo method for high dimensional integration
zbMATH Open document ID: 0669.65011
DOI: 10.1007/BF01406511
author: Yosihiko Ogata
published in: Numerische Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133348
review text: The problem to compute the multiple integral \(Z_ K=\int^{b}_{a}...\int^{b}_{a}f(x_ 1,...,x_ K)dx_ 1...dx_ K\) is considered for some constants a and b, and large K, the multiplicity of the integral. The crude Monte Carlo method does not work well for large K. For this reason, the investigation is interested in estimating \(\log (Z_ K)\) directly rather than \(Z_ K\) itself. This way is motivated by a previous paper of the author [A Monte Carlo method for the objective Bayesian procedure, Res. Memorandum No.347, Inst. Statistical Math., Tokyo (1988)] which provides a solution to the objective Bayesian procedure. A new numerical integration method is proposed. This method is an appropriate one for very high dimensional functions, while its implementation is based on the Metropolis Monte Carlo algorithm. The computation of \(\log (Z_ K)\) is reduced to a simple integration of a certain statistical function with respect to a scale parameter over the range of unit interval. This new method ensures a substantial improvement in the accuracy comparing to the crude Monte Carlo integration. Results of some numerical experiments are given. A numerical example illustrates how the high dimensional integration on the infinite domain can be reasonably calculated. A FORTRAN program for estimating \(\log (Z_ K)\) is also presented.
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 26-04
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 4096666
zbMATH Keywords: crude Monte Carlo method
zbMATH Keywords: numerical integration
zbMATH Keywords: high dimensional functions
zbMATH Keywords: Metropolis Monte Carlo algorithm
zbMATH Keywords: numerical experiments
zbMATH Keywords: numerical example
zbMATH Keywords: high dimensional integration
zbMATH Keywords: FORTRAN program
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4096666