Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

96 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsI). KERNEL DISTORTION, IMPORTANCE SAMPLINGIn Section A the K(x',x) kernel was replaced by the conditional probability k(x|x') andthe statistical weight w(x') in the actual sampling, so as to work with a normalized PDF.The principle can be generalized for the sake of variance reduction. Let us define anotherkernel K(x',x) > 0 and assume that it is normalized:dxK(x',x) = 1that is the conditional probability of x is:k(x|x') = K(x',x)Let the weight now be(, . K(x',x)w(x ,X) =K(x',x)process:Theorem 4.9 — The integral I of Equation (4.24) can be estimated by the following1. Choose N values of \[ from tp M(x')2. Choose subsequent values of x,-s from the conditional PDF k(xjx|)3. Compute the averageI = -J- 2w( X;,x,)h(x,)Proof. The expected value of I is:(D = JJdx'dxw(x',x)h(x)k(x|x')( P]_ 1(x')= IdXh(X) Idx' K(x'.x)^ i(x')J J K(x ,x)= fdxh(x) CP^ 1(X) = ILet us now apply the kernel distortion to the Fredholm type of integral equationof the second kind given in Equation (4.26), and replace again K(x'.x) by K(x',x). LetK(x',x) be again a normalized p.d.f., thus the conditional probability of x is againk(x|x') - K(x',x).Theorem 4.10 — The integral I of Equation (4.27) can be estimated in the followingway:1. Set j = 0, select initial coordinate x ( 0from the PDF:, , QWq(x) =Win