Download Adaptive Methods of Computing Mathematics and Mechanics: by D. G. Arsenev, V. M. Ivanov, O. Iu Kulchitskii PDF

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

An outline of the adaptive equipment of statistical numerical research utilizing overview of integrals, answer of critical equations, boundary price difficulties of the idea of elasticity and warmth conduction as examples. the consequences and methods supplied are diverse from these on hand within the literature, as particular descriptions of the mechanisms of version of statistical assessment strategies, which speed up their convergence, are given.

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

Example text

As was pointed out in [36], procedures of Monte-Carlo method can be organized in series. e. the evaluation process can be split into some stages, each of which ends with the correction of algorithm parameters on the basis of information on function obtained at points under performance. The idea of how to modify probability density, based on approximation of an integrand by means of Fourier series, was also proposed in [36]. However, an estimate of convergence of this successive procedure was not listed in the paper, such an effective technique of improving the convergence of estimates, as an introduction of optimal weight coefficients a*, was not considered.

Evaluation of integrals by means of statistic simulation 35 which splits each of DJ r _ I ) into two subdomains, while crossing them. Here (XJ/,2/,J) € D\r~ is a point of subdomain D, '. Thus, form a new system of subdomains Z>, , I = 1 , 2 , . . , L, where L = m(m+ 1) (m+1)2 for odd r ; for even r . 13. ' ( 1, if there exists index ji, such - I that (xih yii) € £>fr) , j , = 1 , . . , k ; 1 00 else. r 14. Evaluate density pr(x,y) > i V )

If Q > p{£i), then go to step 1 to perform the (i+l)th step of the algorithm. 4. If Q ^ p(£,), then set r/ = £,. End. It is obvious that points (£,p(£)) a r e distributed uniformly on set G={(x,z): x€[a,b}; z € [0,p] } . Therefore with accuracy of infinitesimal of degree Ay P{y <: n ^ y + ^y} = p(y)&y, thus distribution density of r; is p(y). Exercise 1. , which distribution density after truncation _ f f, \ undefined, if a < a ' < i < i i ' < 6 ; if 1 € [&', b] and x e [a, a1] , is p>,(y) = cp(y) for yeW,&']• 14 PART I.

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