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

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

An outline of the adaptive tools of statistical numerical research utilizing review of integrals, answer of crucial equations, boundary price difficulties of the idea of elasticity and warmth conduction as examples. the consequences and ways supplied are varied from these on hand within the literature, as precise descriptions of the mechanisms of version of statistical review tactics, which speed up their convergence, are given.

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

Example text

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 )

CHAPTER 2. Evaluation of integrals by means of statistic simulation 41 2. 26) is performed by means of the method of the least squares: 0N = argmjn I ^^(ar,-) - ^ ( * , - ) ) +(0-0(O)fQ(o)(e-e(o)) , where art- are independent implementations of random variable f, distributed with density p(x) ; 0(0) is an a priori estimate of unknown parameters 0 ; Q(0) is an a priori variance matrix of estimates 0(0). Estimate of parameters 0 meets the following relation: 6N = QN*NGN . , QN = FN Ag(xN)}T ; ' N where FN = $N$JI — X) ^(xt)V,T(^«) ' s tne Fisher data matrix.

According to property 3. 41), expression for variance T(xi) is the Fisher data matrix. 1=1 _ Parallel with matrix FN, consider normed matrix F/v, defined as follows: FN = Af^N- For N -> oo, matrix FN converges in the sense of mean squares to matrix F^,, where Foo = / ip(x)ipT(x)p(x) dx. _ D _ _ Represent matrix FN as follows: F/v = Foo + AF. 44) 44 PART I. Evaluation of integrate and solution of integral equations where SN = T^AFT^h = F _ 1 A F e , and M{5%} -► 0 for N -> oo.

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