The iDTMC method is a hybrid approach that couples MC neutron transport with deterministic solutions to improve computational efficiency while preserving solutions that are consistent with MC transport physics. Unlike conventional two-step methods, which rely on diffusion-based nodal approximations to obtain assembly-level solutions from MC-generated group constants, iDTMC directly solves the pin-wise neutron balance equation without diffusion assumptions. As a result, iDTMC enables faster simulations while retaining high-fidelity, pin-wise solutions that are consistent with the underlying transport physics.

In eigenvalue problems, conventional MC simulations require a large number of inactive cycles to achieve convergence of the FSD, followed by additional active cycles to accumulate tallies, leading to high computational cost. The iDTMC method addresses this limitation by introducing deterministic feedback during both inactive and active cycles. Its overall concept, in comparison with conventional MC and MC accelerated by the partial current-based Coarse Mesh Finite Difference method (MC-CMFD), is illustrated in Figure 1.

The iDTMC method addresses this limitation by accelerating FSD convergence with partial current-based Coarse Mesh Finite Difference (p-CMFD) solutions during inactive cycles, and by deterministically obtaining pin-wise reactor solutions with partial current-based Fine Mesh Finite Difference (p-FMFD) solutions during active cycles. As a result, the number of both inactive and active cycles can be reduced compared to conventional MC, allowing accurate pin-wise reactor solutions to be obtained more efficiently. The iDTMC method has been implemented in KAIST’s in-house MC code, iMC, and extended to depletion and transient calculations.

The p-CMFD method solves the one-group neutron balance equation (NBE) at each inactive cycle using parameters tallied over coarse meshes, such as assembly-sized nodes, to update the FSD weight (Cho, 2012). The one-group NBE is given in Equation 1: ∑ n A V n J n + 1 2 − J n − 1 2 + Σ a , n M C ϕ n = 1 k e f f ν Σ f , n M C ϕ n (1)

Here, n is the node index, A is the surface area, V is the node volume, J is the net neutron current, Σ a M C is the one-group absorption cross section tallied from MC, ν Σ f M C is the one-group production cross section tallied from MC, k e f f is the effective multiplication factor, and ϕ is the neutron flux. The net current is expressed in terms of partial currents, which are further related to the neutron flux, diffusion coefficients, and correction factors, as in Equation 2: J n + 1 2 = J n + 1 2 + − J n + 1 2 − = − D ∼ n   + 1 2 ϕ n + 1 − ϕ n + D ^ n   + 1 2 + ϕ n − D ^ n + 1 2 − ϕ n   + 1 (2)

Here, D ∼ n + 1 2 is the interface diffusion coefficient defined as Equation 3, and D ^ n + 1 2 ± are correction factors defined as Equation 4: D ∼ n + 1 2 = 2 D n   D n + 1 D n + D n + 1 Δ n (3) D ^ n + 1 2 ± = J n + 1 2 ± , M C ± 1 2 D ∼ n + 1 2 ϕ n + 1 − ϕ n ϕ n + 1 2 ∓ 1 2 (4)

Here, D is the diffusion coefficient estimated from the tallied total cross section, i.e., 1 / 3 Σ t M C , Δ is the node width, and J ± , M C are the partial currents tallied from MC. At each inactive cycle, the p-CMFD equation is formed from each inactive cycle’s tallied parameters and its solution is used to update the FSD weights according to Equation 5: S n i + 1 = S n i + 1 2 × p n p − C M F D p n M C (5)

Here, S i + 1 2 is the FSD weight after the i ^th inactive cycle, S i + 1 is the weight for the i + 1 ^th inactive cycle, and p is the fission neutron probability defined as Equation 6: p n X = ν Σ f , n M C ϕ n X V n ∑ n ′ ν Σ f , n ′ M C ϕ n ′ X V n ′ (6) where X denotes either MC or p-CMFD. At the p-FMFD cycles, the p-FMFD method solves the same NBE as p-CMFD but on a fine mesh, such as pin-sized nodes, using accumulated parameters from active cycles. The p-FMFD solution provides deterministic, pin-wise reactor solutions.

As discussed above, the performance advantages of the iDTMC method over conventional MC simulations arise from two distinct mechanisms that operate at different stages of the calculation. The first mechanism accelerates the convergence of the FSD during inactive cycles, while the second is the variance reduction in the final solution during active cycles through p-FMFD solution.

The first performance benefit is reflected in the Shannon entropy convergence (Ellis, 1985). Figure 2 compares the entropy convergence of the conventional MC and iDTMC methods, showing that the p-CMFD feedback in iDTMC effectively accelerates fission source convergence and reduces the required number of inactive cycles.

The second contribution arises from variance reduction during the active cycles enabled by the deterministic p-FMFD solution. Unlike conventional MC simulations, where statistical uncertainty is reduced solely by increasing the number of particle histories, iDTMC reconstructs the final solution by solving a deterministic p-FMFD system that enforces neutron balance using accumulated MC tallies. As a result, statistical fluctuations in quantities of interest can be reduced without proportionally increasing computational cost.

To quantify the overall computational efficiency, the Figure of Merit (FOM) is employed and defined as Equation 7: F O M = 1 T R 2 (7) where T is the computing time and R is the relative standard deviation of the quantity of interest. A higher FOM indicates improved efficiency, resulting from reduced computational time, reduced statistical uncertainty, or both.

In this study, FOM comparisons are performed using single-batch calculations in conjunction with the proposed uncertainty estimation framework. The efficiency improvement of the iDTMC method is evaluated for both the eigenvalue and the pin-wise power distribution, thereby consistently demonstrating the benefits of variance reduction during active cycles.

Although the iDTMC solution is obtained by solving a deterministic equation, the coefficients used to construct this system are derived from MC tallies and therefore contain inherent statistical uncertainty. In this study, the uncertainty of interest is exclusively the statistical uncertainty of the iDTMC solution induced by MC tallies. Other sources of uncertainty, such as nuclear data or modeling uncertainties, are not considered.

Because the iDTMC solution is not obtained directly from stochastic particle histories, conventional MC variance estimators are not applicable. Instead, the uncertainty is estimated by sampling multiple sets of p-FMFD parameters from the accumulated tallies, constructing a corresponding ensemble of deterministic p-FMFD problems, and evaluating the variance of the resulting solutions as an estimate of the real variance of the iDTMC solution.

In the sampling process, the parameters required to construct the NBE–the pin-wise Σ t , Σ a , ν Σ f , J ± , and ϕ as shown in Equation 1 – are sampled. For the estimation to be both accurate and physically meaningful, it is essential to preserve not only the distributions but also the correlations of the parameters accumulated from active cycles.

For example, an increase in the absorption cross section within a node naturally leads to an increase in the total cross section, reflecting parameter-wise correlation. Likewise, when the neutron flux in a node changes, the interface current to its neighboring nodes also changes in a consistent manner, illustrating node-wise correlation. Ignoring such dependencies may produce unphysical sampled parameter sets and result in unreliable variance estimates.

To preserve correlations among parameters, one might attempt to explicitly model both parameter-wise and node-wise correlations. However, constructing and applying correlation matrices for all fine meshes is nearly impossible. For instance, consider a hexahedral mesh. If only parameter-wise correlations are included, the correlation matrix must capture three cross sections, one neutron flux, and twelve surface partial currents, resulting in a 16 × 16 matrix per mesh. Extending this to include node-wise correlations with six neighboring nodes would require a 100 × 100 correlation matrix for each fine mesh. Such explicit modeling is computationally intensive and time-consuming.

For this reason, we instead account for correlations implicitly. Let P i = Σ a i , ν Σ f i , Σ t i , J ± , i , ϕ i , tallied from the i ^th active cycle of the MC simulation. Because each P i is obtained directly from MC tallies, it inherently contains all parameter-wise and node-wise correlations. Thus, by directly using accumulated parameter sets to generate new sets of p-FMFD parameters, these correlations can be preserved implicitly. Based on this assumption, the proposed sampling method consists of two steps. First, new parameter sets are generated as weighted sums of the accumulated sets, where the weights are randomly sampled from a Dirichlet distribution. Second, since the linear transformations do not change the correlations, the generated parameters are transformed by shifting their mean and variance to match those of the accumulated sets. This procedure is expected to allow the generated parameter sets to retain the original correlations while also reflecting the correct statistical properties.

In the iCS process, the Dirichlet distribution is used to generate the weights for the weighted sum (Kotz et al., 2000). The Dirichlet distribution is the continuous multivariate generalization of the beta distribution. Since the beta distribution describes random variables constrained to the limited interval and is commonly used for sampling random percentages or proportions, its multivariate extension is well suited for generating normalized weights. The probability density function (PDF) of a Dirichlet distribution of order K with parameter α = α 1 , … , α K is defined as Equation 8: f x 1 , … , x K ; α 1 , … , α K = 1 B α ∏ i = 1 K x i α i − 1 (8) where x i ∈ 0 , 1 for all i ∈ 1 , … , K , ∑ i = 1 K x i = 1 , and B α is given in Equation 9: B α = ∏ i = 1 K Γ α i Γ ∑ i = 1 K α i (9)

For a random variable X = X 1 , … , X K ∼ D i r α , the mean and variance of X i are defined as Equations 10, 11, respectively: E X i = α i α 0 (10) V a r X i = α i α 0 − α i α 0 2 α 0 + 1 (11) where α 0 is defined as Equation 12: α 0 = ∑ i = 1 K α i (12)

The PDF of the Dirichlet distribution of order 3 with various values of α is illustrated in Figure 3.

The Dirichlet distribution is called symmetric when all elements of α are equal. In a symmetric Dirichlet distribution, as illustrated in the left and middle plots of Figure 3, increasing the α reduces the variance of each random variable, leading to more uniform weights across the samples. In contrast, if the elements of α are not equal, as shown in the right plot of Figure 3, the resulting distribution produces biased weights that favor certain parameter sets. Because the MC method solves the same neutron transport equation with a converged FSD at each active cycle, no prior preference should be given to any specific cycle. Therefore, in this study we adopt a symmetric Dirichlet distribution with α = 1.0 .

Based on the Dirichlet distribution, we now describe how it is incorporated into the sampling process for variance estimation in iDTMC. Suppose we have accumulated N sets of p-FMFD parameters, P = P 1 , … , P N , from the active cycles of MC and aim to sample M new parameter sets. First, for each m ∈ 1 , … , M , we draw a weight vector ω m = ω m , 1 , … , ω m , N from the Dirichlet distribution of order N , as shown in Equation 13: ω m ∼ D i r α (13)

Second, we generate M sampled parameter sets, P s = P 1 s , … , P M s , by taking the weighted sum of the accumulated sets, as defined in Equation 14: P m S = ω m · P (14)

At this stage, we obtain normally distributed parameter sets that implicitly preserve correlations. Finally, to ensure that the sampled sets match the original statistical properties, we shift and rescale their mean and variance to those of the accumulated parameters, as shown in Equation 15: P m = P m s − E P s V a r P s × V a r P + E P (15)

Through this process, we obtain M correlated p-FMFD parameter sets, P m , that follow the mean and variance of the accumulated sets. Left schematic of Figure 4 illustrates the variance estimation procedure of the iDTMC method using the iCS method.

As a simple example, the right schematic in Figure 4 shows the case where only cross sections are considered, with five accumulated p-FMFD parameter sets and ten sampled parameter sets generated through this procedure. This example is presented solely to illustrate the part of sampling procedure.

The OECD/NEA 3 × 3 benchmark problem was first selected to preliminarily assess the effectiveness of the iCS method (Daeubler et al., 2015). The detailed geometry of the model is illustrated in the top panel of Figure 5.

The model consists of nine fuel assemblies (FAs), including six uranium oxide (UOX) FAs and three mixed oxide (MOX) FAs. Each UOX FA contains fuel rods with 4.2 w/o enrichment and guide tubes, while each MOX FA includes fuel rods with enrichments of 2.5, 3.0, and 5.0 w/o, together with Wet Annular Burnable Absorbers (WABAs) and guide tubes. All fuel rods are arranged in a 17 × 17 lattice. In total, the model comprises 2,376 fuel rods, 153 guide tubes, and 72 WABAs. Reflective boundary conditions are applied in the radial direction, while vacuum boundary conditions are imposed axially. Each FA has a width of 21.42 cm and a height of 365.76 cm, resulting in an overall model width of 64.26 cm with the same axial height. The nominal thermal power of the system is 166.24 MW.

Within the iDTMC framework, each fuel assembly was treated as a coarse mesh, and each fuel pin was modeled as a fine mesh. Axially, the fine and coarse meshes were subdivided into 20 and 10 equal-height regions, respectively.

To further evaluate the proposed uncertainty estimation framework under a more realistic and spatially complex configuration, a full-core small modular reactor (SMR) model was also considered (Kim et al., 2022). The detailed assembly configurations and core layouts are shown in the bottom panel of Figure 5.

The SMR core consists of 37 fuel assemblies, including 16 FA1 assemblies and 21 FA2 assemblies. FA1 assemblies contain 3.8 w/o enriched UO₂ fuel rods and guide tubes, while FA2 assemblies additionally include gadolinia-mixed UO₂ fuel rods. All assemblies follow a 17 × 17 lattice arrangement, resulting in a total of 9,264 fuel rods, 925 guide tubes, and 504 gadolinia-bearing rods in the core. The assemblies are surrounded by water reflectors in both the radial and axial directions. Each FA has a width of 21.42 cm and a height of 100 cm, yielding an overall core radius of 96.39 cm and a total core height of 140 cm. The nominal thermal power of the SMR core is 100 MW.

Similar to the OECD/NEA 3 × 3 problem, each fuel assembly in the SMR model was treated as a coarse mesh and each fuel pin as a fine mesh in the iDTMC calculations. Axially, the fine and coarse meshes were divided into 10 and 5 equal-height regions, respectively.

All neutronic analyses were performed using the iDTMC method implemented in the in-house Monte Carlo code iMC, employing the ENDF/B-VII.1 nuclear data library (Chadwick et al., 2011). The number of inactive cycles was determined based on the convergence of the fission source distribution, monitored using the Shannon entropy.

As an initial assessment of the proposed iCS method, eigenvalue uncertainty estimation was first performed for the OECD/NEA 3 × 3 model. The iDTMC calculation employed 1,500,000 particle histories per cycle, with 25 inactive cycles followed by 35 active cycles. Deterministic p-FMFD calculations were performed for the final 10 active cycles using the reactor parameters accumulated up to each cycle. The real variance was obtained from 90 independent iDTMC batch calculations using different random number seeds to ensure statistical independence. For both the iCS and CS methods, 30 independent batch estimations were conducted, each using 100 sampled reactor parameter sets. All error bars represent 99.75% confidence intervals based on the corresponding distributions. The top panel of Figure 6 shows the real variance (red) with the estimates obtained using the iCS (blue) and CS (green) methods.

Due to the small problem size and the application of reflective boundary conditions in the radial directions, spatial correlations in the OECD/NEA 3 × 3 model are relatively weak. Consequently, both the iCS and CS methods yield similar variance estimates, and no significant discrepancy from the reference variance is observed. In addition, the limited number of batch calculations used to estimate the real variance further reduces the ability to resolve subtle differences between the two methods for this simplified configuration.

To more rigorously evaluate the effectiveness of the iCS method under realistic and strongly correlated conditions, eigenvalue uncertainty estimation was next performed for the SMR problem. The iDTMC calculation again used 1,500,000 histories per cycle, with 15 inactive cycles and 115 active cycles. Deterministic p-FMFD calculations were carried out for the final 100 active cycles to investigate the consistency of the uncertainty estimation as the amount of accumulated statistical information increases. The real variance was obtained from 300 independent batch calculations. For each estimation method, 30 independent batch estimations were performed using 100 sampled parameter sets, and 99.75% confidence intervals were constructed. The bottom panel of Figure 6 compares the real eigenvalue variance with the estimates from the iCS and CS methods.

In contrast to the OECD/NEA 3 × 3 problem, the SMR model exhibits strong spatial and parameter-wise correlations due to its realistic core geometry. Under these conditions, a clear difference in prediction accuracy is observed. As the number of active cycles increases, the eigenvalue variance estimated by the iCS method converges toward the real variance, whereas the CS method consistently and significantly underestimates the uncertainty. This behavior reflects the inherent limitation of the CS method, which accounts only for parameter-wise correlations among a limited set of cross sections within individual nodes. In realistic reactor problems, node-wise correlations and the coupled effects of neutron flux and interface partial currents play a critical role in uncertainty estimation. Neglecting these effects leads to unreliable estimation of the eigenvalue variance, as observed for the CS method.

To further isolate the impact of non–cross-section parameters, an additional iCS-based estimation was performed in which sampling was restricted to cross sections only (black). Although this restricted iCS approach yields variance estimates closer to the reference than the CS method, it still deviates from the full iCS results. This comparison demonstrates that the improved performance of the iCS method is not solely attributable to the inclusion of additional parameters, but primarily to its ability to implicitly preserve node-wise correlations.

While the previous analysis focused on convergence behavior with a large number of active cycles, the applicability of the iCS method in early active-cycle regimes was also investigated, which is particularly relevant for efficient iDTMC applications. For this purpose, the SMR problem was reanalyzed using only 15 active cycles, with deterministic p-FMFD calculations performed for the final 14 cycles. The real variance was obtained from 90 independent batch calculations, and iCS-based estimations were performed using 30 independent batches. The results are shown in the top panel of Figure 7.

Because the iCS method constructs sampled parameter sets as weighted linear combinations of accumulated data, at least two active cycles are required for sampling. When only three to four active cycles are available, the variance estimates exhibit large fluctuations and poor consistency, reflecting insufficient statistical stabilization of the underlying reactor parameters. As the number of active cycles increases beyond approximately five to six cycles, the iCS estimates stabilize and provide physically meaningful uncertainty predictions.

To further examine the effect of statistical stabilization, an additional calculation was performed using an increased number of particle histories per cycle, 6,000,000 histories per cycle. The corresponding results, shown in the bottom panel of Figure 7, indicate that parameter variances stabilize more rapidly and the reference eigenvalue variance is reduced by approximately a factor of two. Under these conditions, reliable and consistent eigenvalue uncertainty estimates are obtained with as few as three to four active cycles.

Overall, these results demonstrate that the iCS method provides reliable eigenvalue uncertainty estimates for iDTMC calculations, with particularly strong advantages for large, realistic reactor problems and acceptable performance even in early active-cycle regimes, provided that a minimal level of statistical stabilization is achieved.

As a preliminary assessment of the iCS method for pin power distribution uncertainty estimation, the OECD/NEA 3 × 3 benchmark problem was first analyzed. The iDTMC calculation employed 1,500,000 neutron histories per cycle, with 25 inactive cycles followed by 35 active cycles. A deterministic p-FMFD calculation was performed at the final active cycle using the reactor parameters accumulated up to that point to obtain the pin-wise power distribution. The real variance of the pin power was evaluated from 90 independent batch calculations using different random number seeds. For each uncertainty estimation method, 30 independent batch estimations were conducted, each using 100 sampled reactor parameter sets.

Figure 8 compares the reference pin power variance with the estimates obtained using the iCS and CS methods. To quantitatively assess the estimation accuracy, two global metrics were employed. The average standard deviation of the pin power over all fine-mesh nodes is defined as Equation 16: σ P ¯ = 1 N f i n e ∑ i = 1 N f i n e σ P , i (16) where N f i n e denotes the number of fine-mesh nodes with nonzero power and σ P , i is the standard deviation of the pin power at node i . In addition, the mean absolute relative error of the estimated standard deviations is defined as Equation 17: ε P ¯ = 1 N f i n e ∑ i = 1 N f i n e σ P , i r e f − σ P , i X σ P , i r e f (17) where superscripts r e f and X denote the real and estimated method, respectively. Table 1 summarizes the average standard deviation and mean absolute relative error.

As in the eigenvalue analysis, the OECD/NEA 3 × 3 problem exhibits weak spatial correlations, resulting in similar pin power variance estimates from both the iCS and CS methods.

To evaluate the performance of the iCS method under more realistic and strongly correlated conditions, the SMR problem was next analyzed. The iDTMC calculation employed 1,500,000 neutron histories per cycle, with 15 inactive cycles and 115 active cycles, and a deterministic p-FMFD calculation was performed at the final active cycle using the accumulated reactor parameters. The real variance of the pin power distribution was obtained from 240 independent batch calculations. As in the previous case, 30 independent batch estimations were performed for each uncertainty estimation method, each using 100 sampled parameter sets. Figure 8 presents the corresponding pin power variance distributions, and Table 1 summarizes the average standard deviation and mean absolute relative error.

In contrast to the OECD/NEA 3 × 3 problem, the SMR model exhibits strong parameter-wise and node-wise correlations arising from its realistic geometry and heterogeneous core configuration. Under these conditions, a clear difference in prediction accuracy is observed. The iCS method produces pin power variance distributions that closely match the real variance across the core, whereas the CS method systematically underestimates the variance. This discrepancy reflects the fundamental limitation of the CS method, which neglects node-wise correlations that are essential for pin-wise power uncertainty estimation. Neglecting these effects leads to the observed unreliable estimation of uncertainty by the CS method.

Overall, these results demonstrate that the iCS method, by implicitly preserving both parameter-wise and node-wise correlations across all reactor parameters, provides significantly improved accuracy for pin power distribution uncertainty estimation compared to CS approaches. The advantage of the iCS method is particularly pronounced for realistic reactor configurations with strong spatial correlations.

A key feature of the iCS method is its ability to preserve the correlation structure inherently embedded in MC–tallied reactor parameters. For uncertainty estimation within the iDTMC framework to be physically meaningful, sampled reactor parameter sets must retain not only parameter-wise correlations within individual nodes but also node-wise correlations that naturally arise from neutron transport physics.

To quantitatively assess correlation preservation, a dedicated analysis was performed using the SMR problem. The MC simulation employed 1,500,000 neutron histories per cycle, with 15 inactive cycles followed by 25 active cycles. Reactor parameter sets tallied during the active cycles were treated as reference data. Based on these tallies, 100 sampled parameter sets were generated using both the iCS and the CS methods. The correlation structures of the sampled parameter sets were then compared against those of the reference tallies.

Node-wise correlations were first examined for all sampled reactor parameters. Pearson correlation coefficients were used to quantify correlations, and only correlations between directly adjacent nodes were considered, as spatial correlations are strongest over nearest-neighbor distances. For each spatial direction, the average node-wise correlation coefficient was defined as Equation 18: ρ ¯ i ^ X = 1 N i ^ ∑ i ^ ρ i ^ , i ^ − 1 X (18) where N i ^ denotes the number of fine-mesh nodes in direction i ^ , ρ i ^ , i ^ − 1 X is the Pearson correlation coefficient adjacent nodes and the superscript X denotes the either the reference tallies or a sampling method. In addition, the mean absolute correlation error relative to the reference tallies was evaluated as Equation 19: ε ¯ i ^ X = 1 N i ^ ∑ i ^ ρ i ^ , i ^ − 1 r e f − ρ i ^ , i ^ − 1 X (19) where the superscript r e f denotes the reference tallies. Table 2 summarizes the average node-wise correlation coefficients and the corresponding mean absolute errors for all reactor parameters.

The results show that the iCS method reproduces node-wise correlation levels that are in close agreement with the reference tallies. Moreover, the mean absolute correlation errors remain smaller than the corresponding average correlation magnitudes, indicating that spatial correlations are effectively preserved.

In contrast, the CS method yields node-wise correlation values close to zero in all spatial directions, with mean absolute errors approximately two to three times larger than those obtained using iCS. This behavior clearly indicates that the CS method fails to preserve spatial correlations. The underlying reason is that CS performs sampling independently at each node and does not account for spatial coupling between neighboring nodes. In addition, CS does not sample neutron fluxes or interface partial currents, and therefore cannot represent the associated uncertainties or their correlations. This limitation is particularly critical in realistic reactor problems, where spatial power distributions and their uncertainties are strongly influenced by flux and current coupling between neighboring nodes.

Next, parameter-wise correlations among cross sections within individual nodes were examined. To assess how well these correlations are preserved, the average parameter-wise correlation over all fine-mesh nodes was defined as Equation 20: ρ ¯ X Q 1 , Q 2 = 1 N f i n e ∑ i = 1 N f i n e ρ n X Q 1 , Q 2 (20) where N f i n e is the total number of fine-mesh nodes and ρ n X Q 1 , Q 2 is the Pearson correlation coefficient between parameters Q 1 and Q 2 at node n . The corresponding mean absolute correlation error was defined as Equation 21: ε ¯ X Q 1 , Q 2 = 1 N f i n e ∑ i = 1 N f i n e ρ n r e f Q 1 , Q 2 − ρ n X Q 1 , Q 2 (21)

Table 2 also presents the average parameter-wise correlation coefficients and the associated mean absolute errors.

The iCS method reproduces parameter-wise correlation levels that are in close agreement with the reference values, with mean absolute errors that remain small relative to the average correlations. As expected, the CS method exhibits slightly lower errors for parameter-wise correlations among cross sections, owing to its explicit construction of a correlation matrix for these quantities.

Nevertheless, despite not explicitly modeling parameter-wise correlation matrices, the iCS method achieves comparable accuracy while simultaneously preserving node-wise correlations and enabling sampling over a broader set of reactor parameters, including neutron fluxes and interface partial currents. This combined capability is beyond the scope of the CS approach.

The combined node-wise and parameter-wise analyses demonstrate that the iCS method successfully preserves the essential correlation structures inherent in MC tallies. This correlation preservation provides a consistent statistical foundation for the improved uncertainty estimation performance of the iDTMC method observed in the preceding sections.

In the proposed iCS method, the only user-specified constant is the Dirichlet distribution parameter α , which controls the sampling weights assigned to cycle-wise reactor parameter sets. The Dirichlet distribution determines how strongly individual active-cycle tallies are emphasized during sampling. As α increases above unity, the sampled weights become more uniformly distributed across all cycles, resulting in nearly equal contributions from each cycle. Conversely, when α is smaller than unity, the weights become increasingly concentrated on a small subset of cycles, leading to stronger preference for specific cycle-wise parameter sets.

It is important to note that, regardless of the value of α , a symmetric Dirichlet distribution guarantees identical expected values for all weights, equal to the inverse of the number of active cycles. Therefore, variations in α affect only the dispersion of the weights, not their mean. From the perspective of uncertainty estimation, however, it is necessary to examine whether changes in weight dispersion influence correlation preservation and, consequently, the accuracy of uncertainty estimates.

The fundamental assumption underlying the iCS method is that accurate uncertainty estimation can be achieved as long as the sampling procedure preserves the distribution and correlation structures of the tallied reactor parameters. Accordingly, the sensitivity analysis in this study focuses on node-wise correlation preservation.

Using the SMR problem, reactor parameters tallied with 1,500,000 histories per cycle over 15 inactive and 25 active cycles were treated as reference data. Based on these tallies, parameter sets were sampled using the iCS method for 69 different values of α , ranging from 0.8 34 to 0.8 − 34 . For each value of α , the mean absolute node-wise correlation error ε ¯ i ^ i C S was evaluated for all reactor parameters and spatial directions. The results are presented in Figure 9.

No systematic dependence on α is observed in any spatial direction. The fluctuations remain small, and all values of ε ¯ i ^ i C S are bounded below 0.1, indicating that node-wise correlation preservation is largely insensitive to the choice of α over a wide range of values.

To further examine the impact of α on uncertainty estimation, eigenvalue uncertainty analyses were also performed for different values of α using the same SMR model. The iDTMC calculations employed 1,500,000 histories per cycle, with 15 inactive cycles and 115 active cycles. Deterministic p-FMFD calculations were performed for the final 100 active cycles using the accumulated reactor parameters. Eigenvalue uncertainty estimates were obtained for three representative values of α, and the results are shown in Figure 10.

The estimated eigenvalue uncertainties exhibit no significant dependence on α . This behavior can be interpreted by noting that reactor parameter sets tallied at different active cycles are independent and identically distributed and already contain the relevant correlation structures. Under these conditions, the Dirichlet-sampled weights primarily control the relative preference among cycle-wise parameter sets and do not materially affect the resulting uncertainty estimates.

Nevertheless, since the Dirichlet distribution is constructed from Gamma-distributed random variables, excessively large or small values of α may lead to numerical instability. Considering both numerical robustness and consistency of the results, the use of a symmetric Dirichlet distribution with α = 1 is recommended in this study.

As discussed in the previous section, the performance advantages of the iDTMC method over conventional MC arise from two distinct mechanisms. The first is the acceleration of FSD convergence during inactive cycles through p-CMFD feedback, which reduces the number of inactive cycles required for source convergence. The second is variance reduction during active cycles achieved by enforcing neutron balance through deterministic p-FMFD solutions.

To establish a reliable reference for efficiency comparison, statistically well-converged FOM values were first obtained using independent batch calculations. These reference calculations employed 1,500,000 particle histories per cycle, 15 inactive cycles, and 50 active cycles. A total of 119 independent simulations were performed to estimate the real variance with sufficient statistical confidence. The resulting reference FOM ratios for the eigenvalue, average power, and peak power are summarized in Table 3, where the FOM ratio is defined as the FOM of iDTMC divided by that of MC-CMFD. Based on these results, the iDTMC method demonstrates a clear efficiency advantage over MC-CMFD, with an improvement of approximately a factor of two for eigenvalue uncertainty estimation. These values represent the practical efficiency gain achievable by iDTMC under statistically stabilized conditions.

Using the reference results as a benchmark, additional single-batch FOM comparisons were conducted to examine whether similar efficiency trends could be observed without independent batch calculations. In all single-batch studies, MC-CMFD and iDTMC calculations were performed under identical numerical conditions.

In the first set of single-batch calculations, summarized in Table 4, 37,500,000 particle histories per cycle were used with 15 inactive cycles and various number of active cycles. When only two or five active cycles were employed, the resulting FOM ratios exhibited large discrepancies relative to the reference values and showed strong inconsistency across different quantities of interest. As the number of active cycles increased, the FOM ratios gradually approached the reference values and displayed more consistent trends. This behavior indicates that an insufficient number of active cycles leads to unstable reactor parameter estimates, which directly deteriorates the reliability of uncertainty estimation and FOM evaluation. In particular, under these early active-cycle conditions, the uncertainty of the peak power was occasionally observed to exceed that of the average power, which is physically inconsistent and indicative of poorly stabilized reactor parameters.

A second set of single-batch calculations was performed using a higher number of particle histories per cycle, 97,500,000, while maintaining the same inactive and active-cycle configurations. The results are summarized in Table 5. Increasing the number of particle histories improves the statistical quality of the tallied reactor parameters and leads to more consistent FOM behavior as the number of active cycles increases. Nevertheless, even under these high-statistics conditions, the single-batch FOM ratios remain noticeably lower than the reference values.

This discrepancy does not indicate a fundamental limitation of the iDTMC methodology itself, but rather reflects intrinsic challenges associated with uncertainty estimation in single-batch calculations. In particular, the apparent variance used in MC-CMFD is known to systematically underestimate the real variance due to cycle-wise correlations inherent in steady-state MC simulations. This underestimation persists regardless of the number of particle histories per cycle, leading to biased uncertainty estimates for MC-CMFD and, consequently, to an underestimation of the corresponding FOM ratios. In addition, uncertainty estimates obtained using the iCS method also exhibit statistical dispersion, which further complicates quantitative efficiency comparisons in early active-cycle regimes.

Overall, the numerical results indicate that, for reliable FOM comparisons in early active-cycle regimes, a methodology is required that can obtain statistically stabilized reactor parameters from the initial active cycles. Without such stabilization, single-batch FOM comparisons cannot accurately reflect the true efficiency differences between MC-CMFD and iDTMC.

To address this limitation, future work will focus on improving variance estimation for MC-CMFD and enabling statistically stabilized uncertainty estimates in early active-cycle regimes. In particular, a new approach inspired by the history-based batch method will be investigated to obtain reliable variance estimates within a single active cycle (Shim and Choi, 2012).