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By D. G. Arsenev, V. M. Ivanov, O. Iu Kulchitskii

An outline of the adaptive equipment of statistical numerical research utilizing evaluate of integrals, answer of quintessential equations, boundary worth difficulties of the idea of elasticity and warmth conduction as examples. the implications and methods supplied are various from these to be had within the literature, as exact descriptions of the mechanisms of variation of statistical review tactics, which speed up their convergence, are given.

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Adaptive Methods of Computing Mathematics and Mechanics: Stochastic Variant

An outline of the adaptive tools of statistical numerical research utilizing assessment of integrals, resolution of indispensable equations, boundary worth difficulties of the speculation of elasticity and warmth conduction as examples. the consequences and techniques supplied are varied from these on hand within the literature, as targeted descriptions of the mechanisms of model of statistical overview techniques, which speed up their convergence, are given.

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Extra info for Adaptive Methods of Computing Mathematics and Mechanics: Stochastic Variant

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4. 4-1- Stochastic computing algorithms as an object of control adaptive Introduction In this section computing algorithms are considered from the viewpoint of the theory of adaptive control. In these terms processes of evaluating multi-dimensional integrals and numerical solution of integral equations are the objects of control. Function of distribution density of nodes of a random integration grid is the control action for these objects. Evaluation accuracy criterion, which can be represented in analytic form for both objects of control, is the criterion of optimal functioning.

Such a selection of integrals for the numerical experiments is stipulated by the fact, that in the case of evaluating, the adaptation effect depends on the form (slight, steep) of the extremum of integrand, which modifies with value of parameter X. 20). 2 — the same information concerning integral J2 for various values of parameters. /VT2). NTe) and variances (DISP = 100 x {DN)/INTe, where DN is the variance of the estimate, were computed in these tables. Integration bounds are ( —1; 1). Analysis of data, obtained while performing experiment, demonstrates that adaptation effect becomes stable when the number of points N performed attains 400.

Let us assume that «the price* of obtaining a value of this function, corresponding to input value x €. D , is sufficiently high. 1) D can be obtained by the cost of the least computations of particular values. Such a problem may arise, for example, when analyzing the quality of a complex 21 22 PART I. Evaluation of integrals and solution of integral equations system. 2) Pi\xi) ,=1 where Xi are independent random variables, distributed on D with distribution densities p,(x) > 0, f Pi(x) dx = 1.

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