Correlated Sampling in a Reference System

The correlated sampling technique described in the previous section assumes that the actual simulation is done in the unperturbed system and the adjusting weight factors F_i^ps (= $\frac{\kappa_i^p}{\kappa_i^{up}}$) account for the simulation of the correlated perturbed particles. The weight carried by the perturbed particles would be a product of these adjusting weight factors F_i^ps. This requires that the adjusting weight factors for the correlated game must always remain finite, which implies,

$$\kappa_i^p \gt 0, \kappa_i^{up} \gt 0.$$ (58)

In some actual simulations these conditions may not be satisfied. For example, in calculating perturbation effects due to voids, a particular phase-space transition probability may vanish in the perturbed system but exist in the unperturbed one. To account for this difficulty, a Monte Carlo perturbation approach has been developed [Rie88] where the actual simulation is done in an artificial reference system different from both the unperturbed and perturbed systems. The probability density functions describing the transition process in the reference system are a weighted mean of the probability density functions of the unperturbed and perturbed systems. The important characteristic of the reference system is that all particle reactions that can take place in the unperturbed and perturbed systems will also have a non-zero probability in the reference system. Thus, for example, a voided region would not be admissible in the reference system if the unperturbed or perturbed systems were not voided. However, the transpose is acceptable, i.e. one could have a voided region in the perturbed system, in which case the weights would be appropriately modified if a collision occurred in the ``voided'' region. The responses for the unperturbed and perturbed systems are calculated by correlating them to the reference system's simulation. Also by the proper choice of the reference (ref) system, it is possible to avoid large fluctuations of the adjusting weight factors, $\frac{\kappa_i^{up}}{\kappa_i^{ref}}$ and $\frac{\kappa_i^p}{\kappa_i^{ref}}$,and reduce uncertainty of the differential effect. The reference system [Maj94] may be chosen freely, but it also depends on the type of problem solved [Lux89]. Since the choice of the reference system will have an effect on the variance, it may be subject to optimization methods[Lar95]. This by itself leads to a topic of future research. In appendix A, some preliminary computational parametric studies are performed to observe the effect of different reference systems on the result of a perturbation problem. In this research work, the arithmetic average of the parameters of the unperturbed and perturbed systems has been used as the parameters for the reference system. One of the requirements of the reference system is that $\frac{\kappa_i^{up}}{\kappa_i^{ref}}$ and $\frac{\kappa_i^p}{\kappa_i^{ref}}$ should be as constant as possible. We will now explicitly derive the adjusting weight factors for the unperturbed and perturbed systems and introduce the concept of a $\delta$-scatter in the forward direction to avoid large fluctuations in the adjusting weight factors.

The cross sections for the reference, unperturbed and perturbed systems are $\Sigma_x^{ref}$, $\Sigma_x^{up}$ and $\Sigma_x^p$,respectively, where x denotes a different cross section type. According to our choice of the reference system, we have,

$$\Sigma_x^{ref} = \frac{1}{2}[\Sigma_x^{up} + \Sigma_x^p].$$ (59)

The spatial collision distance, d, is sampled from $\Sigma_t^{ref}exp(-\Sigma_t^{ref}d)$ in the reference system. Then the adjusting weight factors (WF) for the unperturbed and perturbed systems are given by,

$$WF^{up}=\frac{{\Sigma_t^{up}}exp(-\Sigma_t^{up}d)} {{\Sigma_... ...a_t^{up}}{\Sigma_t^{ref}}exp [(\Sigma_t^{ref}-\Sigma_t^{up})d],$$ (60)

and

$$WF^{p}=\frac{{\Sigma_t^{p}}exp(-\Sigma_t^{p}d)} {{\Sigma_t^{... ...gma_t^{p}}{\Sigma_t^{ref}}exp [(\Sigma_t^{ref}-\Sigma_t^{p})d],$$ (61)

respectively. The weight which is carried along by the unperturbed and perturbed particles is the product of these adjusting weight factors, WF^up and WF^p, respectively.

Now, assuming $\Sigma_t^p \gt \Sigma_t^{up}$, the adjusting weight factor for the unperturbed history has a positive exponent and may assume large values if d is large. Similarly, the factor for the perturbed history may assume a very small value. To avoid these large fluctuations, a $\delta$-scatter in the forward direction is added to all three total cross sections [Rie88]. Introduction of a pseudo-collision with a $\delta$-scatter in the forward direction increases the number of collisions without changing the expectation values. For each of these three cases the $\delta$-scatter is chosen to yield the same total cross section plus $\delta$-scatter :

$$\Sigma_t^{ref} + \delta^{ref} = \Sigma_t^{up} + \delta^{up} = \Sigma_t^p + \delta^p.$$ (62)

Using these $\delta$-scatters,

$$WF^{up}=\frac{\Sigma_t^{up}+\delta^{up}}{\Sigma_t^{ref}+\del... ...p[( \Sigma_t^{ref}+\delta^{ref}-\Sigma_t^{up}-\delta^{up})d]=1,$$ (63)

and

$$WF^{p}=\frac{\Sigma_t^{p}+\delta^{p}}{\Sigma_t^{ref}+\delta^... ...exp[( \Sigma_t^{ref}+\delta^{ref}-\Sigma_t^{p}-\delta^{p})d]=1.$$ (64)

Thus the large fluctuations in WF^up and WF^p are avoided. The reference system's cross sections are chosen using equation (3.27) and a $\delta$-scatter for the reference system ($\delta^{ref}$) is selected. Next equation (3.30) can be used to determine the $\delta$-scatters for the unperturbed and perturbed systems.