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s — Paper presents a new approach to investigate convergence of semistatistical and projection-statistical methods developed by authors in papers [l]-[3]. These are statisical methods based on approximation of integral included in the integral equation by Monte Carlo stochastic quadrature rule. Alternative treatment of methods convergence is introduced which allows to simplify convergence conditions essentially s compared to those given in previous papers. 1. Semi-statistical rnethod 1.1. Basic relations of the rnethod. Let us consider the Fredholm integral equation of the second kind: φ(χ) xjK(x,y)<p(y}dy = /(z), χ G D C IR, (1) D where f(x) G Li(D}, K(x,y) 6 Li(D χ D), λ Ε 1R and investigate the semi-statistical method of its solution [1], [2]. Let x\, χι,..., XN £ D be independent samples of a random variable distributed in D with density function p(x) such that p(x) > 0 for any χ 6 D, J p(x)dx = 1. D Application of the simplest Monte Carlo method [4] for estimation of the integral included in the equation (1) leads to the following equations in points χ and X{ respectively: Brought to you by | University of Queensland UQ Library Authenticated Download Date | 6/17/15 12:04 PM 298 Vladimir M. Ivanov and Maxim L. Korenevski λ *Κ(χ,χΐ)φ(χϊ) % / λ ./ χ ψ(χ] ~ Ν — -Λρ(χ) = f(x}] Λ , χ f, } Λΐ ° ~ /( ι ) > where i = l, 2, . . . , JV, and / χ /· ^x χ / Ν τ Ι ^ p(x) = / fT(x, y)^(y)dy g ^ Pi(x] = u Ijti are random errors of the Monte Carlo evaluation of the relevant integrals. After neglecting of values Pi(xi) in the equation (3) we obtain the linear System of the semi-statistical method equations: As it follows from (2), the exact solution of the equation (1) has the form: λ ^ Κ(χ,χι)φ(χι) Therefore after the System (4) is solved, the solution of the integral equation (1) can be estimated at any point χ G D by the relation: . (5) Let us introduce the matrices * = { «}5-i> ^« = {^ifl^· and H = -N^> and vectors Brought to you by | University of Queensland UQ Library Authenticated Download Date | 6/17/15 12:04 PM On convergence of semi-statistical and projection-statistical methods 299 = fo>(zi), y»(x2), · · ·, vM]; 7 = [/(«O, /te), · · ·, /(ZJV)]; rn Γ(χ,Χΐ) J^(x,X2) ^(^ p(xi) ' p(x2) ''"' In this notation the equations (2-5) can be written s follows: = /(χ); (2') Χρ = /; (3') = 7; (4') ^(x) = /(*) + ̂ (x)^. (5') If the matrix H of the linear System (4') is invertible then the relations (2'-5') lead to formulas φ(χ) = /(χ) + λρ(χ) + k(*)H(J + λρ) ; Hence the error of the method at a given point of domain is expressed s follows: φ(χ) φ(χ) = \p(x) + k(x)H~p. (6) 1.2. Auxiliar propositions This section contains several simple assertions which are widely used below, sometimes even without explicit indications. Proposition 1. Assume A and B are random events and the following inequalities are valid: P{A}>l-pA; P{B}>1-PB. Then P {AB} >1-ρΑ-ΡΒ· Proposition 2. Lei A and B are random events and the following inequalities are valid: Brought to you by | University of Queensland UQ Library Authenticated Download Date | 6/17/15 12:04 PM 300 Vladimir M. Ivanov and Maxim L. Korenevski P{A}>1-PA] P{B\A}>l-pB. Then P {AB} >1-ρΑ-ΡΒProposition 3 (Chebyshev inequality). Lei X be a substantially non-negative random variable, i.e. P{X < 0} = 0. Then for any positive quantity a a The inequality can be also written s follows: a Proposition 4. Lei X\ and X^ be substantially non-negative random vanables and P{Xi < αϊ} > l pi; P{X2 < a2} > l p2. Then Proposition 5. Let A and B be square matrices of the same dimension. Assume there exist inverse matrices (AB)" and (BA)~. Then the matrix A is invertible and A~ = B(AB)~ = (BA)-B. 1.3. Invertibility of the method matrix. It is assumed here and below that the numerical parameter λ of the equation (1) differs from characteristic values (inverses of eigenvalues) of integral operator with the kernel K(x,y). In this case the integral equation has the unique solution for any right term f(x) [5]. When the integral operator is approximated by a finite-dimensional operator depending on Brought to you by | University of Queensland UQ Library Authenticated Download Date | 6/17/15 12:04 PM On convergence of semi-statistical and projection-statistical methods 301 the sample, the spectrum is expected to be biased. In this Situation one of the eigenvalues of matrix K can become equal to (N — 1)/λ and the matrix H of semi-statistical method can become Singular. However it is clearly that the greater N (i.e., the more exact the approximation of the integral operator), the less the probability of this event. Below this fact is proved strictly. Let us introduce the set Ω = = D N times of all samples ω = (EI, £2> · · · ? ZN) with independent components. Evidently the matrix ff, the vectors ^, p, and the functions p(x) and φ(χ) depend on ω implicitly. The joint density function of a sample, N t=l defines the probability measure on the set Ω. Let us show that the probability of matrix H invertibility tends to l s the sample size N increases. For the further convenience we introduce in ΈΙ the special norm depending on the sample. For the fixed sample ω = (&ι, Ζ2» · · · » XN) we define the norm of a vector V from R by the formula N W Since for any square matrix A the inequality α·.υ} <Σ — \\AV\\l = V N N Σ ΦΟ*) Σ j=iP(xj). N N t=lj=l is valid, the corresponding matrix norm is estimated by the expression: ^ N N 1=1 j=i τ Λ ^ Δ It is well-known from the theory of integral equations [5] that if λ is not a characteristic value of the integral operator in the equation (1), then there exists the resolvent integral operator with the kernel K~(x, y] which satisfies the equality Brought to you by | University of Queensland UQ Library Authenticated Download Date | 6/17/15 12:04 PM 302 Vladimir M. Ivanov and Maxim L. Korenevski \/K(x, z)K~(z, y)dz = λ / K' (χ, z)K(z, y}dz = K~(x, y) K(x, y), D D (7) called a resolvent integral equation. We now construct the matrices and consider the matrix Here the matrix A has the following entries: tfi^j then Λ K-fax^-Kfaxj) Λ ^(χ,, i13 N-l p(Xj) (ATl ) 2 f e i P(XI)P(XJ) and λ Ν K(XI Let us consider the value and calculate its expectation with respect to ω. Lemma. Let the integrals f f [ x , z z , y J J J . f r / / / — Tl~^ -dzdxdy and / / έ έ ρ Ρ() D U exwi. T/ien ί/ie value M {(E(A))} has the order O(l/N) s N -> oo. Brought to you by | University of Queensland UQ Library Authenticated Download Date | 6/17/15 12:04 PM On convergence of semi-statistical and projection-statistical methods 303 Proof: M f Κ^χύΚ-^ έί P(xl}p(xj) Α γ Μ [ K ( X J , x t ) K ( x , , X i ) K ( x i , x m ) K ( x m > X j ) } 1 = 1 W M ( N M f f ^ fs /5έί,> m/t.j,/ •i,»,-) Kfr.xj) K(XJ,xt)K-(x,,Xj)}

On convergence of semi-statistical and projection-statistical methods for integral equations