Remarks about discrete Young measures and their Monte Carlo simulation

• Autorzy: Grzybowski A. , Puchała P.

Identyfikatory

DOI

10.17512/jamcm.2015.2.02

• Języki publikacji: EN

Abstrakty

EN

This article is devoted to the problem of simulation of random variables distributed according to Young measures associated with piecewise affine functions determined on bounded intervals. We start with simple functions which can take on a finite number of different values with inverse images being the intervals or their unions. We present some formal results connected with related discrete Young measures and propose an algorithm for generating random variables having such distributions. Next, based on these results we introduce an algorithm designed for approximation of Young measures in various, more general situations. We also present an example where a Young measure associated with a piecewise affine function is approximated with the help of computer simulations. In this benchmarking problem the theoretical results are compared with the ones obtained in the Monte Carlo experiment.

Słowa kluczowe

EN

Young measures; Monte Carlo simulation; simple function; piecewise affine function

PL

miara Younga; symulacja Monte Carlo; funkcja prosta

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2015
• Tom: Vol. 14, nr 2
• Strony: 13--20
• Opis fizyczny: Bibliogr. 9 poz., rys.

Twórcy

autor

Grzybowski A.

• Institute of Mathematics, Czestochowa University of Technology Częstochowa, Poland

autor

Puchała P.

• Institute of Mathematics, Czestochowa University of Technology Częstochowa, Poland

Bibliografia

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• [2] Puchała P., An elementary method of calculating an explicit form of Young measures in some special cases, Optimization 2014, 63, 9, 1419-1430.
• [3] Bielski W., Kruglenko E., Telega J.J., Miary Younga i ich zastosowania w mikromechanice i optymalizacji. Część I. Podstawy matematyczne, Matematyka Stosowana 2003, 4, 90-138.
• [4] Nicolaides R.A., Walkington N.J., Computation of microstructure utilizing Young measure representations, J. Intelligent Materials Systems and Structures 1993, 4, 795-800.
• [5] Spall J.C., Introduction to Stochastic Search and Optimization; Estimation, Simulation, and Control, A John Wiley & Sons. Inc. Publication, 2003.
• [6] Grzybowski A.Z., Simulation approach to optimal stopping in some blackjack type problems, Scientific Research of the Institute of Mathematics and Computer Science 2011, 10, 2, 75-86.
• [7] Bartłomiejczyk K., Chain-like structures' motion - metamodeling of the translocation time in some specific situations, Journal of Applied Mathematics and Computational Mechanics 2014, 13, 3, 5-12.
• [8] Pedregal P., Variational Methods in Nonlinear Elasticity, SIAM, Philadelphia 2000.
• [9] Roubíček T., Relaxation in Optimization Theory and Variational Calculus, Walter de Gruyter, Berlin, New York 1997.