Consistent Adjoint Driven Importance Sampling Research Articles

Validation of the MS-CADIS Method for Full-Scale Shutdown Dose Rate Analysis

Fusion energy systems present increasingly significant computational challenges as they grow in size and complexity. Once constructed, ITER will be a full-size nuclear facility with highly complicated structures and support systems, with an array of scientific equipment in close proximity to the neutron-emitting deuterium-tritium plasma. Characterization of shutdown dose rate (SDDR) distributions caused by the neutron activation of these structures is important to the final design and full-power operation of the device. This work summarizes the theoretical basis and parallel implementation of the Multi-Step Consistent Adjoint-Driven Importance Sampling (MS-CADIS) method designed specifically for highly efficient execution of multistep activation problems. Fusion SDDR benchmark problems have been solved with these new tools, and the results have been compared to experimental and other computational results to establish their validation basis.

Radiation dose rate distributions of spent fuel dry casks estimated with MAVRIC based on detailed geometry and continuous-energy models

This study presents a detailed comparison of the dose rate distributions of a dry TN-32 fuel cask with two geometry models and two cross-sectional datasets. The accuracies of radiation dose rate estimation and computational efficiencies of each geometry model with two cross-sectional data-sets are compared. The use of automated variance reduction techniques can significantly improve the computational efficiency of such a realistic, deep penetration problem that involves radiation transport from different volumetric sources, thereby eliciting only a small statistical error. Monaco with Automated Variance Reduction using Importance Calculations (MAVRIC) is a computational sequence within the SCALE 6.2 code package based on consistent adjoint driven importance sampling (CADIS), a type of automated variance reduction technique. Homogenous and full fuel assembly models are built herein, and two nuclear cross-section libraries (V7-200N47G and continuous energy) are applied in this work. Based on the detailed comparisons, we found that neutron dose rate estimation is more dependent on geometry modeling than on cross-section data. For neutron-induced gamma rays, the dose rate distribution depends on both the spatial self-shielding effect and the cross-section library. The primary gamma rays respectively contribute to the total dose rate by ∼91% and ∼99% on the side and top surfaces, and the dose rate accuracy is more dependent on the cross-section library than on geometry modeling. In terms of the computation efficiency and efforts spent on geometry modeling, the homogenous fuel assembly model with the MG library can produce an acceptable dose rate distribution, but the detailed fuel assembly model with the continuous-energy library is required for more precise dose rate estimation.

Analysis of inconsistent source sampling in monte carlo weight-window variance reduction methods

Abstract The application of Monte Carlo (MC) to large-scale fixed-source problems has recently become possible with new hybrid methods that automate generation of parameters for variance reduction techniques. Two common variance reduction techniques, weight windows and source biasing, have been automated and popularized by the consistent adjoint-driven importance sampling (CADIS) method. This method uses the adjoint solution from an inexpensive deterministic calculation to define a consistent set of weight windows and source particles for a subsequent MC calculation. One of the motivations for source consistency is to avoid the splitting or rouletting of particles at birth, which requires computational resources. However, it is not always possible or desirable to implement such consistency, which results in inconsistent source biasing. This paper develops an original framework that mathematically expresses the coupling of the weight window and source biasing techniques, allowing the authors to explore the impact of inconsistent source sampling on the variance of MC results. A numerical experiment supports this new framework and suggests that certain classes of problems may be relatively insensitive to inconsistent source sampling schemes with moderate levels of splitting and rouletting.

Transmutation Approximations for the Application of Hybrid Monte Carlo/Deterministic Neutron Transport to Shutdown Dose Rate Analysis

In fusion energy systems (FES) neutrons born from burning plasma activate system components. The photon dose rate after shutdown from resulting radionuclides must be quantified. This shutdown dose rate (SDR) is calculated by coupling neutron transport, activation analysis, and photon transport. The size, complexity, and attenuating configuration of FES motivate the use of hybrid Monte Carlo (MC)/deterministic neutron transport. The Multi-Step Consistent Adjoint Driven Importance Sampling (MS-CADIS) method can be used to optimize MC neutron transport for coupled multiphysics problems, including SDR analysis, using deterministic estimates of adjoint flux distributions. When used for SDR analysis, MS-CADIS requires the formulation of an adjoint neutron source that approximates the transmutation process. In this work, transmutation approximations are used to derive a solution for this adjoint neutron source. It is shown that these approximations are reasonably met for typical FES neutron spectra and materials over a range of irradiation scenarios. When these approximations are met, the Groupwise Transmutation (GT)-CADIS method, proposed here, can be used effectively. GT-CADIS is an implementation of the MS-CADIS method for SDR analysis that uses a series of single-energy-group irradiations to calculate the adjoint neutron source. For a simple SDR problem, GT-CADIS provides speedups of 200 100 relative to global variance reduction with the Forward-Weighted (FW)-CADIS method and 9 5 relative to analog. This work shows that GT-CADIS is broadly applicable to FES problems and will significantly reduce the computational resources necessary for SDR analysis.

A proposal on multi-response CADIS method to optimize reduction of regional variances in hybrid Monte Carlo simulation

For fixed source problems, it is known that the analog Monte Carlo simulation method can show low efficiencies. To increase the calculation efficiency, variance reduction techniques (VRT) have been introduced and utilized. To apply these techniques, parameters to control particle transport behaviors should be properly determined. The hybrid Monte Carlo methods were developed to automatically determine the VRT parameters using information estimated from a deterministic simulation. In the case of a single response, the Consistent Adjoint Driven Importance Sampling (CADIS) method is well derived to decide VRT parameters. Recently, for plural response problems, a variety of methods have been proposed. It is noted that Forward-Weighted CADIS (FW-CADIS) method gives the best performance to optimize the parameters for plural response problems. However, it can create inefficiencies due to an assumption that uniform particle density is used to uniformly obtain low variance in each response. In the Multi-Response CADIS (MR-CADIS) method proposed in this study, a new weight function is derived to effectively reduce plural relative errors. With the proposed method, VRT parameters are decided by that the sum of squared relative errors for plural responses has a minimum value. To verify the MR-CADIS method, simple shielding and duct problems are estimated and compared to those estimated with the FW-CADIS method. These results show that the MR-CADIS method can produce better efficiency and uniformity than the FW-CADIS method for plural response problems.

The Analysis of the External Neutron Monitor Responses in a Simplified JET-Like Tokamak Using ADVANTG

The application of the Automated Variance Reduction Generator (ADVANTG) code to accelerate MCNP neutron transport calculations in fusion-relevant geometries is presented. The ADVANTG code generates variance-reduction parameters using the Consistent Adjoint Driven Importance Sampling (CADIS) and Forward-Weighted Consistent Adjoint Driven Importance Sampling (FW-CADIS) methods based on deterministic transport calculations performed by the discrete ordinates code Denovo. The aim of ADVANTG is to reduce the MCNP computational time by automating the process of variance-reduction parameter generation. ADVANTG was tested on a simplified model of a JET-like tokamak that in spite of its simplicity retains all the major characteristics of such a tokamak. The performance of the nuclear data libraries provided with ADVANTG and of various other ADVANTG/Denovo settings on variance-reduction efficiency was tested. Several cases using deuterium-deuterium or deuterium-tritium (D-T) volumetric (plasma) sources and 252Cf or D-T point neutron sources were analyzed to find guidelines for successful use of the code for fusion applications. Additionally, the use of ADVANTG as a tool to identify major neutron pathways from the neutron source to the detector is demonstrated.

Application of Variational Variance Reduction for Source-Detector Problems in Nuclear Non-proliferation

ABSTRACTThe VVR (Variational Variance Reduction) method has been tested extensively on criticality problems and has proven to be effective, but its application for shielding problems has been limited. In this work, the VVR method is tested on source-detector type problems that are of interest in nonproliferation applications. A 3D multigroup Monte Carlo code that uses parts of the framework of Attila, a production deterministic code for radiation transport problems, was developed to prototype hybrid (Deterministic/Monte Carlo) variance reduction techniques. The VVR method is applied to the forward Monte Carlo simulations based on a deterministically calculated adjoint solution using Attila. The performance of the VVR method when applied with 3 different configurations of the forward simulation are tested: (1) Monte Carlo calculation with implicit absorption, (2) Monte Carlo calculation with the Consistent Adjoint Driven Importance Sampling (CADIS) based weight windows method and (3) Monte Carlo calculation with the Local Importance Functional Transform (LIFT) method. The simulations were performed on 2 benchmark test problems for nonproliferation applications, the “Mulch Box” photon test problem and the PANDA neutron test problem. The angular domain was discretized for efficient evaluation of the VVR functional using a discrete ordinates based approach and a spherical harmonics based approach. For the mulch test problem, the VVR method provided only marginal improvements in the figure of merit (FOM) when applied to the Monte Carlo calculation with only implicit absorption, but provided up to two times improvement in FOM when applied with the CADIS based weight windows method. For the PANDA neutron problem, VVR method did not provide improvements in FOM due to the highly scattering nature of the problem. It was also observed that the VVR method when combined with the LIFT method is detrimental to the overall performance.

Shutdown dose rate analysis with CAD geometry, Cartesian/tetrahedral mesh, and advanced variance reduction

In fusion energy systems (FES) high-energy neutrons born from burning plasma activate system components to form radionuclides. The biological dose rate that results from photons emitted by these radionuclides after shutdown—the shutdown dose rate (SDR)—must be quantified for maintenance planning. This can be done using the Rigorous Two-Step (R2S) method, which involves separate neutron and photon transport calculations, coupled by a nuclear inventory analysis code. The geometric complexity and highly attenuating configuration of FES motivates the use of CAD geometry and advanced variance reduction for this analysis.An R2S workflow has been created with the new capability of performing SDR analysis directly from CAD geometry with Cartesian or tetrahedral meshes and with biased photon source sampling, enabling the use of the Consistent Adjoint Driven Importance Sampling (CADIS) variance reduction technique. This workflow has been validated with the Frascati Neutron Generator (FNG)-ITER SDR benchmark using both Cartesian and tetrahedral meshes and both unbiased and biased photon source sampling. All results are within 20.4% of experimental values, which constitutes satisfactory agreement. Photon transport using CADIS is demonstrated to yield speedups as high as 8.5·105 for problems using the FNG geometry.

Analysis of the LIFT Variance-Reduction Method Applied to Monte Carlo Radiation Transport Simulations of a Realistic Nonproliferation Test Problem

The Local Importance Function Transform (LIFT) method is a sophisticated automated variance-reduction technique for Monte Carlo simulation of radiation transport problems. In previous publications, the LIFT method was tested on geometrically simple problems with a coarse representation of radiation energy dependence, and the performance of the method was found to be promising when compared to traditional weight windows–based variance-reduction techniques. In this work, the LIFT method is tested on a spatially complex benchmark test problem with a more realistic representation of energy dependence (50 energy groups) and heterogeneous materials. The performance of the method in comparison with a CADIS (Consistent Adjoint Driven Importance Sampling)–based weight windows method and an analog Monte Carlo simulation is studied. A multigroup Monte Carlo code that utilizes portions of the framework of the deterministic tool Attila has been developed such that the overhead time in implementing the variance-reduction techniques is minimal. The Monte Carlo simulations are performed on an arbitrary tetrahedral mesh created by the mesh generator in Attila. A method to transfer the deterministic solution generated on a finer mesh to a coarser mesh for implementing the hybrid simulations has been developed, and the results are quantified.

Analysis of dpa Rates in the HFIR Reactor Vessel using a Hybrid Monte Carlo/Deterministic Method*

The Oak Ridge High Flux Isotope Reactor (HFIR), which began full-power operation in 1966, provides one of the highest steady-state neutron flux levels of any research reactor in the world. An ongoing vessel integrity analysis program to assess radiation-induced embrittlement of the HFIR reactor vessel requires the calculation of neutron and gamma displacements per atom (dpa), particularly at locations near the beam tube nozzles, where radiation streaming effects are most pronounced. In this study we apply the Forward-Weighted Consistent Adjoint Driven Importance Sampling (FW-CADIS) technique in the ADVANTG code to develop variance reduction parameters for use in the MCNP radiation transport code. We initially evaluated dpa rates for dosimetry capsule locations, regions in the vicinity of the HB-2 beamline, and the vessel beltline region. We then extended the study to provide dpa rate maps using three-dimensional cylindrical mesh tallies that extend from approximately 12 in. below to approximately 12 in. above the height of the core. The mesh tally structures contain over 15,000 mesh cells, providing a detailed spatial map of neutron and photon dpa rates at all locations of interest. Relative errors in the mesh tally cells are typically less than 1%.

Radiation Damage Assessment in the Reactor Pressure Vessel of the Integral Inherently Safe Light Water Reactor (I2S-LWR)

One of the major limiting factors to nuclear reactors lifetime is the radiation- induced material damage in the Reactor Pressure Vessel (RPV). While older reactors were designed assuming a 40-year operating lifetime, new reactor designs are expected to have lifetimes up to 100 years. For safe operation, the integrity of the RPV must be ensured against significant material property changes. In this work, typical neutron damage indicators are calculated in the RPV of the I 2 S-LWR (Integral Inherently Safe LWR) Power Plant, including DPA (displacements per atom) and fast neutron fluence (>1 MeV and >0.1MeV). I 2 S-LWR is a PWR of integral design, which means that its wider downcomer provides additional shielding to the vessel. However, its higher core power density and longer lifetime may offset this advantage. In order to accurately represent the neutron environment for RPV damage assessment, a detailed model based on the preliminary design specifications of the I 2 S-LWR was developed to be used in the MAVRIC (Monaco with Automated Variance Reduction using Importance Calculations) sequence of the Scale6.1 code package. MAVRIC uses the CADIS (Consistent Adjoint-Driven Importance Sampling) methodology to bias a fixed-source MC (Monte Carlo) simulation. To establish the upper limit of a bounding envelope, a flat-source distribution was used. For the low limit, a center- peaked source was generated using the KENO-VI criticality sequence assuming uniform fresh fuel core. Results based on the preliminary I 2 S-LWR model show that DPA rates and fast fluence rates are conservatively 75% lower than in typical PWRs being operated currently in the US.

Implementation, capabilities, and benchmarking of Shift, a massively parallel Monte Carlo radiation transport code

This work discusses the implementation, capabilities, and validation of Shift, a massively parallel Monte Carlo radiation transport package authored at Oak Ridge National Laboratory. Shift has been developed to scale well from laptops to small computing clusters to advanced supercomputers and includes features such as support for multiple geometry and physics engines, hybrid capabilities for variance reduction methods such as the Consistent Adjoint-Driven Importance Sampling methodology, advanced parallel decompositions, and tally methods optimized for scalability on supercomputing architectures. The scaling studies presented in this paper demonstrate good weak and strong scaling behavior for the implemented algorithms. Shift has also been validated and verified against various reactor physics benchmarks, including the Consortium for Advanced Simulation of Light Water Reactors' Virtual Environment for Reactor Analysis criticality test suite and several Westinghouse AP1000 ® problems presented in this paper. These benchmark results compare well to those from other contemporary Monte Carlo codes such as MCNP5 and KENO.

The Multi-Step CADIS Method for Shutdown Dose Rate Calculations and Uncertainty Propagation

Shutdown dose rate (SDDR) analysis requires (a) a neutron transport calculation to estimate neutron flux fields, (b) an activation calculation to compute radionuclide inventories and associated photon sources, and (c) a photon transport calculation to estimate final SDDR. In some applications, accurate full-scale Monte Carlo (MC) SDDR simulations are needed for very large systems with massive amounts of shielding materials. However, these simulations are impractical because calculation of space- and energy-dependent neutron fluxes throughout the structural materials is needed to estimate distribution of radioisotopes causing the SDDR. Biasing the neutron MC calculation using an importance function is not simple because it is difficult to explicitly express the response function, which depends on subsequent computational steps. Typical SDDR calculations do not consider how uncertainties in MC neutron calculation impact SDDR uncertainty, even though MC neutron calculation uncertainties usually dominate SDDR uncertainty.The Multi-Step Consistent Adjoint Driven Importance Sampling (MS-CADIS) hybrid MC/deterministic method was developed to speed SDDR MC neutron transport calculation using a deterministically calculated importance function representing the neutron importance to the final SDDR. Undersampling is usually inevitable in large-problem SDDR simulations because it is very difficult for the MC method to simulate particles in all space and energy elements of the neutron calculation. MS-CADIS can assess the degree of undersampling in SDDR calculations by determining the fraction of the SDDR response in the space and energy elements that did not have any scores in the MC neutron calculation. It can also provide estimates for upper and lower limits of SDDR statistical uncertainties resulting from uncertainties in MC neutron calculation.MS-CADIS was applied to the ITER SDDR benchmark problem that resembles the configuration and geometrical arrangement of an upper port plug in ITER. Without using the hybrid MC/deterministic methods to speed MC neutron calculations, SDDR calculations were significantly undersampled for all tallies, even when MC neutron calculation computational time was 32 CPU-days. However, all SDDR tally results with MC neutron calculations of only 2 CPU-days converged with the standard Forward-Weighted CADIS (FW-CADIS) method and the MS-CADIS method. Compared to the standard FW-CADIS approach, MS-CADIS decreased the undersampling in the calculated SDDR by factors between 0.9% and 0.3% for computational times between 4 and 32 CPU-days, and it increased the computational efficiency of the SDDR neutron MC calculation by factors between 43% and 69%.

Shutdown Dose Rate Analysis Using the Multi-Step CADIS Method

The Multi-Step Consistent Adjoint Driven Importance Sampling (MS-CADIS) hybrid Monte Carlo (MC)/deterministic radiation transport method was proposed to speed up the shutdown dose rate (SDDR) neutron MC calculation using an importance function that represents the neutron importance to the final SDDR. In this work, the MS-CADIS method was applied to the ITER SDDR benchmark problem. The MS-CADIS method was also used to calculate the SDDR uncertainty resulting from uncertainties in the MC neutron calculation and to determine the degree of undersampling in SDDR calculations because of the limited ability of the MC method to tally detailed spatial and energy distributions. The analysis that used the ITER benchmark problem compared the efficiency of the MS-CADIS method to the traditional approach of using global MC variance reduction techniques for speeding up SDDR neutron MC calculation. Compared to the standard Forward-Weighted-CADIS (FW-CADIS) method, the MS-CADIS method increased the efficiency of the SDDR neutron MC calculation by 69%. The MS-CADIS method also increased the fraction of nonzero scoring mesh tally elements in the space-energy regions of high importance to the final SDDR.

Automatic Mesh Adaptivity for Hybrid Monte Carlo/Deterministic Neutronics Modeling of Difficult Shielding Problems

The well-established Consistent Adjoint Driven Importance Sampling (CADIS) and the Forward Weighted Consistent Adjoint Driven Importance Sampling (FW-CADIS) hybrid Monte Carlo/deterministic techniques have dramatically increased the efficiency of neutronics simulations, yielding accurate solutions for increasingly complex problems through full-scale, high-fidelity simulations. However, for full-scale simulations of very large and geometrically complex nuclear energy systems, even the CADIS and FW-CADIS techniques can reach the CPU and memory limits of all but the very powerful supercomputers. In this work, three mesh adaptivity algorithms were developed to reduce the computational resource requirements of CADIS and FW-CADIS without sacrificing their efficiency improvements. First, a macromaterial approach was developed to enhance the fidelity of the deterministic models without changing the mesh. Second, a deterministic mesh refinement algorithm was developed to generate meshes that capture as much geometric detail as possible without exceeding a specified maximum number of mesh elements. Finally, a weight window (WW) coarsening (WWC) algorithm was developed to decouple the WW mesh and energy bins from the mesh and energy group structure of the deterministic calculations. By removing the memory constraint of the WW map from the resolution of the mesh and the energy group structure of the deterministic calculations, the WWC algorithm allows higher-fidelity deterministic calculations that, consequently, increase the efficiency and reliability of the CADIS and the FW-CADIS simulations. The three algorithms were used to enhance an FW-CADIS calculation of the prompt dose rate throughout the ITER experimental facility. Using these algorithms increased both the number of mesh tally elements in which nonzero results were obtained (+23.3%) and the overall efficiency of the calculation (a factor of >3.4). The three algorithms enabled this difficult calculation to be accurately solved using an FW-CADIS simulation on a 94-CPU computer cluster, eliminating the need for a world-class supercomputer.

Improved Monte Carlo Variance Reduction for Space and Energy Self-Shielding

Continued demand for accurate and computationally efficient transport methods to solve optically thick, fixed-source transport problems has inspired research on variance-reduction (VR) techniques for Monte Carlo (MC). Methods that use deterministic results to create VR maps for MC constitute a dominant branch of this research, with Forward Weighted–Consistent Adjoint Driven Importance Sampling (FW-CADIS) being a particularly successful example. However, locations in which energy and spatial self-shielding are combined, such as thin plates embedded in concrete, challenge FW-CADIS. In these cases the deterministic flux cannot appropriately capture transport behavior, and the associated VR parameters result in high variance in and following the plate.This work presents a new method that improves performance in transport calculations that contain regions of combined space and energy self-shielding without significant impact on the solution quality in other parts of the problem. This method is based on FW-CADIS and applies a Resonance Factor correction to the adjoint source. The impact of the Resonance Factor method is investigated in this work through an example problem. It is clear that this new method dramatically improves performance in terms of lowering the maximum 95% confidence interval relative error and reducing the compute time. Based on this work, we recommend that the Resonance Factor method be used when the accuracy of the solution in the presence of combined space and energy self-shielding is important.

Novel hybrid Monte Carlo/deterministic technique for shutdown dose rate analyses of fusion energy systems

The rigorous 2-step (R2S) computational system uses three-dimensional Monte Carlo transport simulations to calculate the shutdown dose rate (SDDR) in fusion reactors. Accurate full-scale R2S calculations are impractical in fusion reactors because they require calculating space- and energy-dependent neutron fluxes everywhere inside the reactor. The use of global Monte Carlo variance reduction techniques was suggested for accelerating the R2S neutron transport calculation. However, the prohibitive computational costs of these approaches, which increase with the problem size and amount of shielding materials, inhibit their ability to accurately predict the SDDR in fusion energy systems using full-scale modeling of an entire fusion plant. This paper describes a novel hybrid Monte Carlo/deterministic methodology that uses the Consistent Adjoint Driven Importance Sampling (CADIS) method but focuses on multi-step shielding calculations. The Multi-Step CADIS (MS-CADIS) methodology speeds up the R2S neutron Monte Carlo calculation using an importance function that represents the neutron importance to the final SDDR. Using a simplified example, preliminary results showed that the use of MS-CADIS enhanced the efficiency of the neutron Monte Carlo simulation of an SDDR calculation by a factor of 550 compared to standard global variance reduction techniques, and that the efficiency enhancement compared to analog Monte Carlo is higher than a factor of 10,000.

FW-CADIS Method for Global and Regional Variance Reduction of Monte Carlo Radiation Transport Calculations

This paper presents a hybrid (Monte Carlo/deterministic) method for increasing the efficiency of Monte Carlo calculations of distributions, such as flux or dose rate distributions (e.g., mesh tallies), as well as responses at multiple localized detectors and spectra. This method, referred to as Forward-Weighted CADIS (FW-CADIS), is an extension of the Consistent Adjoint Driven Importance Sampling (CADIS) method, which has been used for more than a decade to very effectively improve the efficiency of Monte Carlo calculations of localized quantities (e.g., flux, dose, or reaction rate at a specific location). The basis of this method is the development of an importance function that represents the importance of particles to the objective of uniform Monte Carlo particle density in the desired tally regions. Implementation of this method utilizes the results from a forward deterministic calculation to develop a forward-weighted source for a deterministic adjoint calculation. The resulting adjoint function is then used to generate consistent space-and energy-dependent source biasing parameters and weight windows that are used in a forward Monte Carlo calculation to obtain more uniform statistical uncertainties in the desired tally regions. The FW-CADIS method has been implemented and demonstrated within the MAVRIC (Monaco with Automated Variance Reduction using Importance Calculations) sequence of SCALE and the ADVANTG (Automated Deterministic Variance Reduction Generator)/MCNP framework. Application of the method to representative real-world problems, including calculation of dose rate and energy-dependent flux throughout the problem space, dose rates in specific areas, and energy spectra at multiple detectors, is presented and discussed. Results of the FW-CADIS method and other recently developed global variance-reduction approaches are also compared, and the FW-CADIS method outperformed the other methods in all cases considered.

Hybrid biasing approaches for global variance reduction

A new variant of Monte Carlo—deterministic (DT) hybrid variance reduction approach based on Gaussian process theory is presented for accelerating convergence of Monte Carlo simulation and compared with Forward-Weighted Consistent Adjoint Driven Importance Sampling (FW-CADIS) approach implemented in the SCALE package from Oak Ridge National Laboratory. The new approach, denoted the Gaussian process approach, treats the responses of interest as normally distributed random processes. The Gaussian process approach improves the selection of the weight windows of simulated particles by identifying a subspace that captures the dominant sources of statistical response variations. Like the FW-CADIS approach, the Gaussian process approach utilizes particle importance maps obtained from deterministic adjoint models to derive weight window biasing. In contrast to the FW-CADIS approach, the Gaussian process approach identifies the response correlations (via a covariance matrix) and employs them to reduce the computational overhead required for global variance reduction (GVR) purpose. The effective rank of the covariance matrix identifies the minimum number of uncorrelated pseudo responses, which are employed to bias simulated particles. Numerical experiments, serving as a proof of principle, are presented to compare the Gaussian process and FW-CADIS approaches in terms of the global reduction in standard deviation of the estimated responses.

Review of Hybrid (Deterministic/Monte Carlo) Radiation Transport Methods, Codes, and Applications at Oak Ridge National Laboratory

** This paper provides a review of the hybrid (Monte Carlo/deterministic) radiation transport methods and codes used at the Oak Ridge National Laboratory and examples of their application for increasing the efficiency of real-world, fixed-source Monte Carlo analyses. The two principal hybrid methods are (1) Consistent Adjoint Driven Importance Sampling (CADIS) for optimization of a localized detector (tally) region (e.g., flux, dose, or reaction rate at a particular location) and (2) Forward Weighted CADIS (FW-CADIS) for optimizing distributions (e.g., mesh tallies over all or part of the problem space) or multiple localized detector regions (e.g., simultaneous optimization of two or more localized tally regions). The two methods have been implemented and automated in both the MAVRIC sequence of SCALE 6 and ADVANTG, a code that works with the MCNP code. As implemented, the methods utilize the results of approximate, fast-running 3-D discrete ordinates transport calculations (with the Denovo code) to generate consistent space- and energy-dependent source and transport (weight windows) biasing parameters. These methods and codes have been applied to many relevant and challenging problems, including calculations of PWR ex-core thermal detector response, dose rates throughout an entire PWR facility, site boundary dose from arrays of commercial spent fuel storage casks, radiation fields for criticality accident alarm system placement, and detector response for special nuclear material detection scenarios and nuclear well-logging tools. Substantial computational speed-ups, generally O(10 2-4 ), have been realized for all applications to date. This paper provides a brief review of the methods, their implementation, results of their application, and current development activities, as well as a considerable list of references for readers seeking more information about the methods and/or their applications.