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a r t i c l e i n f o Article history: Available online xxxx Keywords: Optimal quadrature formulas Monte Carlo methods a b s t r a c tWhen approximately calculating integrals of high dimension with the Monte Carlo method, one uses fewer values of the integrand than when calculating with the help of classical deterministic quadrature formulas.However, the error estimation for the Monte Carlo method does not depend on the smoothness of the integrand. This suggests that it is possible to obtain methods that give a better order of convergence in case of smooth functions. © 2015 Published by Elsevier Inc. N.S. Bakhvalov / Journal of Complexity ( ) -Since |S t − S τ | ≥ 1 and the calculated approximate values S t and S τ of the arithmetic means are the same, then max(|S t − S t |, |S τ − S τ |) ≥ 1 2 . Definition 1. A method of evaluating an integral (arithmetic mean) is called non-deterministic if the coordinates of each successive knot are random functions that depend on the integration domain and all values obtained previously in the computational process (coordinates of the knots, function values and its derivatives, the number of knots, etc.), and the approximate value of the integral (arithmetic mean) is a random function depending on all the obtained values.This definition applies to the Monte Carlo method, the method proposed in [4], methods described in Sections 2 and 4 of the present paper, and many others. Lemma 2. For any nondeterministic method of evaluating the arithmetic mean that uses informationabout function values at no more than N points, there is a function in the given class for which the average of the absolute error is at least C 0 N −1/2 where C 0 is an absolute constant.We will say that an event K γ ,, took place in the process of evaluating the arithmetic mean, if information about values of the function at points β 1 , . . . , β m γ was used, andSince the approximate value of the arithmetic mean (integral) can be written using a finite number of binary digits, it can take no more than countably many values I s (s = 1, 2, . . .). Denote by π j γ the probability of the event K γ in case the integral is evaluated for the function f j .By σ γ s we denote the probability that the approximate evaluation of the integral equals I s under the condition that the event K γ took place.

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We consider a family of symmetric matrices A ω = A 0 + ωB, with a nonnegative definite matrix A 0 , a positive definite matrix B, and a nonnegative parameter ω ≤ 1. Small ω leads to a poor conditioned matrix A ω with jumps in the coefficients. For solving linear algebraic equations with the matrix A ω , we use standard preconditioned iterative methods with the matrix B as a preconditioner. We show that a proper choice of the initial guess makes possible keeping all residuals in the subspace Im(A 0). Using this property we estimate, uniformly in ω, the convergence rate of the methods. Algebraic equations of this type arise naturally as finite element discretizations of boundary value problems for PDE with large jumps of coefficients. For such problems the rate of convergence does not decrease when the mesh gets finer and/or ω tends to zero; each iteration has only a modest cost. The case ω = 0 corresponds to the fictitious component/capacitance matrix method.