Probabilistic Error Estimation in Approximate Integration Formulas for Multivariable Functions

The study examines the problem of approximate integration of multivariable functions. These functions are taken from the space with Gaussian measure. According to it, we calculated the average value of the integral standard deviation from the integral sum. The paper gives the vanishing order for the standard deviation depending on the parameters that define the integral sum. We obtained probabilistic estimates of approximate integration errors.

References

[1] Ermakov S.M. Metody Monte-Karlo i smezhnye voprosy [Monte-Carlo methods and neighbouring problems]. Moscow, Nauka Publ., 1971. 472 p.

[2] Sul’din A.V. Wiener measure and its applications to approximation methods. I. Izvestiya vysshikh uchebnykh zavedeniy. Matematika, 1959, no. 6, pp. 145-158 (in Russ.).

[3] Smale S. On the efficiency of algoritms of analysis. Bulletin of the AMS, 1985, vol. 13, no. 2, pp. 87-121. Available at: http://projecteuclid.org/euclid.bams/1183552689

[4] Larkin F.M. Gaussian measures in Hilbert space and applications in numerical analysis. Rocky Mountain Journal Math, 1972, vol. 2, no. 3, pp. 372-421. DOI: 10.1216/RMJ-1972-2-3-379 Available at: https://projecteuclid.org/euclid.rmjm/1250131560

[5] Shilov G.E., Fan Dyk Tin’. Integral, mera, proizvodnaya na lineynykh prostranstvakh [Integral, measure, derivative in linear space]. Moscow, Nauka Publ., 1967. 220 p.

[6] Gel’fand I.M., Vilenkin N.Ya. Nekotorye primeneniya garmonicheskogo analiza. Osnashchennye gil’bertovy prostanstva [Some applications of Fourier analysis. Rigged Hilbert spaces]. Moscow, Fizmatgiz Publ., 1961. 472 p.

[7] Birman M.Sh., Solomyak M.Z. Spektral’naya teoriya samosopryazhennykh operatorov v gil’bertovom prostranstve [Spectral theory of self-adjoint operators in Hilbert space]. Sankt-Petersburg, Lan’ Publ., 2010. 464 p.