Институт вычислительной математики и математической геофизики СОРАН

Abstracts

Function approximation and cubic formulas

In the late 60-ies, S.L. Sobolev ([1] [2]) proposed an algorithm formula of high precision, combining the approaches of functional analysis and algebraic. Namely, the smoothness of the integrants was set belonging to a particular Banach function space B (Omega), as a formula determined by the norm of the error functional with the optimization of the coefficients. And the algorithm was constructed as the sum of the local formulas for integration over the unit cell of the lattice by means of algebraic formulas with the same nodes, exactly integrating polynomials up to some extent related to the smoothness of the integrands. S.L. Sobolev established that the N, aspiring to infinity, the sequence of functional errors of these formulas has the property of asymptotic optimality in the spaces of integrands B (Omega) = L_2 ^ (m) (Omega).

Property of asymptotic optimality has been established for many other Banach spaces, usually used in computational mathematics. The most important feature of the algorithm is the property of Sobolev boundary layer: nodes removed from the border area to a distance of more than O (N ^ {-1 / n}) correspond to the same factors. This is an order of magnitude reduces the amount of computational work and makes it possible to establish such formulas actual equivalence and order asymptotic optimality in Sobolev spaces W_p ^ m.

Algebraic precision of the local formulas for all polynomials up to some extent manifested in the conditional M "asymptotic unsaturated" algorithm. It was built by a sequence of quadrature formulas is asymptotically (N, tends to infinity) optimal for each space $ W_ {p} ^ {m} $ with m of (n / p, M), that is in a weakened form satisfies introduced K.I. Babenko definition of unsaturation. (K.I. Babenko ([3], [4]) uses universal ordinal optimality without upper bounds on the smoothness of integrands). Unsaturation - an important property of numerical algorithms, allowing you to set them on the program will automatically adjust to exhibit the characteristics of smoothness parameters of tasks, ensuring the best possible rate of convergence of approximations.

We restrict ourselves to a cubic lattice of nodes and propose a new algorithm for lattice rules having the property of the limited boundary layer and is asymptotic to an insatiable in all the spaces $ W_ {2} ^ {m} ( mathbb {R} ^ {n}) $ with m> n / 2.

First, we derive the formula for the coefficients of optimal cubature formulas for the integrals of more general and more general spaces $W_ {2} ^ { mu} ( mathbb {R} ^ {n })$.

Then simplify the expression of the optimal coefficients, cutting out the terms whose orders are negligible in comparison with the principal term. Thus, we arrive at the coefficients are asymptotically optimal cubature formula with a limited boundary layer.

Finally, simplifying the formula for the coefficients in the boundary layer, we obtain the coefficients of the asymptotic formula unsatiated.

Literature.

1. Sobolev, S.L. Introduction to the Theory of cubature formulas, S.L. Sobolev. Moscow: Nauka, 1974. - 808 p.

2. Sobolev, S.L. The Theory of Cubature Formulas (Mathematics and Its Applications) / S.L. Sobolev, V.L. Vaskevich. - 1st edition. - Springer, 1997. - URL: http://www.amazon.com/Theory-Cubature-Formulas-Mathematics-Applications/dp/0792346319/ref=sr_1_1?ie=UTF8&s=books&qid=1274770504&sr=8-1. - ISBN 0792346319.

3. Ramazanov, M.D., Theory of lattice rules with a limited boundary layer / M.D. Ramazanov; IMVTS USC RAS. - Ufa: DizaynPoligrafServis, 2009. - 178 p. - ISBN 978-5-94423-172-7.

4. Babenko, K.I. Foundations of numerical analysis / K.I. Babenko. - 2 ed. - M.; Izhevsk: RKhD, 2002. - 848 p. - ISBN 5-93972-162-1.

5. Belykh, V.N. On the best approximation properties of {$ C ^ infty $}-smooth functions on an interval of the real axis (to the phenomenon of unsaturated numerical methods) / V.N. Belykh // Siberian Mathematical Journal. - 2005. - Vol. 46, No. 3. - P. 373-385. - URL: http://dx.doi.org/10.1007/s11202-005-0040-z.

Note. Abstracts are published in author's edition

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