Информационная система "Конференции"

Abstracts

Approximation of functions and quadrature formulas

Cubature formulas for approximate calculation of multidimensional integrals are considered. Let us $Omega $ denotes the integration range from $R^n$; $k = left( {k_1 ,...,k_n } right)$ is multiindex; $h$ denotes step of lattice pitch. It is supposed that $hk$ nodes may reside outside $Omega $ domain. Let us consider cubature formulas with $c_k $ coefficients equal to one within $Omega $ domain and to zero outside $Omega $ domain on condition when the distance between the $Omega $ boundary domain and the corresponding nodes is more $hl$. The cubature formula quality is determined by the accuracy on polynomials of the type: $x^alpha = x_1^{alpha _1 } x_2^{alpha _2 } ...x_n^{alpha _n } $ where $alpha in Theta = left{ {alpha = left( {alpha _1 ,...,alpha _n } right):,0 le alpha _i le m,,i = 1,...,n} right}$. Having supposed this we can consider $l$ minimal value estimation task for which uniform inequality on $k$ is valid: $left| {c_k - 1} right| le C$ where $C$ is given constant. It has been proved that in condition of sufficiently small $h$ values inequality the $l le Lleft( C right)m^2$ is valid where $Lleft( C right)$ constant depend on $C$ and $Omega $, but independent from $m$ and $h$. $Lleft( C right)$ constant is estimated precisely. Formulas on which this estimation can be achieved are created constructively. Namely, $Omega $ domain is divided into $Delta _i $ elementary cells according to the introduced lattice. For each cell the $T_i $ cloud of nodes is selected; this cloud of nodes makes up a cube with $hl$ edges. Further for $Delta _i $ the special cubature formula is created on these cloud of nodes. After summation prime formulas on elementary cells the cubature formula with the desired properties is resulted. Let's remark, that till now from $Omega $ domain it was required existence of integrals from $x^alpha ,alpha in Theta $. For domain with smooth boundary it is possible to achieve, that all nodes is inside $Omega $ domain. If the domain satisfies to a weak condition of a cone it is possible to achieve that all nodes is inside $Omega $ domain, however in this case the condition of lattice dispositions of nodes is violated. This research was partially supported by the RAS grant of 6-th competition-expertise of young scientist projects, 1999 (grant number 3).

Note. Abstracts are published in author's edition

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