Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

97whereW 1 0= |dxQ(x)2. Select the (j + I)-St coordinate X 1 j t, from the conditional probabilityk(xjx,j)and compute the (j + I)-St weight factor3. Set j = j + 1 and return to step 2.KO^x 1 J + 1)U7 = - —Ktx^x,.,, ,)4. After N repetitions of the process the quantityis an unbiased estimate of L of Equation (4.28) if the W nweight is defined as... ,', ' K(X 1,; Xf 11)w n= ii w,t= w i0nA proof cm be obtained again by recursive application of the proof of Theorem 4.9.nTheorem 4.9 demonstrates that unbiased estimates of the functional (4.27) can be obtainedby proper weighting even if the kernel of the Fredholm equation (4.26) is distortedOne can raise, however, the question: what is the benefit of deviating from the straightforwardsampling? The answer is clear, the implementation of those, and only those distortions areto be considered proper which lead to variance reduction.Another approach to variance reduction, leading to similar results, is importance sampling.Here, the density function cp(x) given in Equation (4.26) is "multiplied by a chosenfunction V(x), which measures the importance of an event at x, on the quite reasonableground that important regions . . . should get intensified sampling attention." 5Let us now multiply each term in Equation (4.26) by V(x) > 0, with the result(p(x)V(x) - Q(X)V(X) + Jdx'K(x'.x) ^tPtV)V(X')functionThis equation is identical with Equation (4.26) if