Geodesic
On numerical methods for functions depending on a very large number of variables
Matematičeskoe modelirovanie, Tome 29 (2017) no. 2, pp. 135-138

Voir la notice de l'article provenant de la source Math-Net.Ru

The question under discussion is why do optimal algorithms on classes of functions sometimes become useless in practice. As an example let's consider the class of functions which satisfy a general Lipschitz condition. The methods of integral evaluation over a unit cube of $d$ dimensions, where $d$ is significantly large, are discussed. It is assumed that the integrand is square integrable. A crude Monte Carlo estimation can be used. In that case the probable error of estimation is proportional $1/\sqrt{N}$, where $N$ is the number of values of the integrand. If we use a quasi-Monte Carlo method instead of Monte Carlo one, then the error does not depend on the dimension $d$, numerous examples show that it depends on the average dimension $\hat{d}$ of the integrand. For small $\hat{d}$ the order of the error is close to $1/N$.
Mots-clés : optimal algorithm
Keywords: Lipschitz condition, Monte Carlo method, quasi-Monte Carlo method, the average dimension. 
@article{MM_2017_29_2_a10,
     author = {I. M. Sobol},
     title = {On numerical methods for functions depending on a very large number of variables},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {135--138},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2017_29_2_a10/}
}
TY  - JOUR
AU  - I. M. Sobol
TI  - On numerical methods for functions depending on a very large number of variables
JO  - Matematičeskoe modelirovanie
PY  - 2017
SP  - 135
EP  - 138
VL  - 29
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2017_29_2_a10/
LA  - ru
ID  - MM_2017_29_2_a10
ER  - 
%0 Journal Article
%A I. M. Sobol
%T On numerical methods for functions depending on a very large number of variables
%J Matematičeskoe modelirovanie
%D 2017
%P 135-138
%V 29
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2017_29_2_a10/
%G ru
%F MM_2017_29_2_a10
I. M. Sobol. On numerical methods for functions depending on a very large number of variables. Matematičeskoe modelirovanie, Tome 29 (2017) no. 2, pp. 135-138. http://geodesic.mathdoc.fr/item/MM_2017_29_2_a10/