Publications

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We compare two methods for generating Galerkin quadratures. In method 1, the standard SN method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard SN method is used to generate the discreteto-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwise sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding SN equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.

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This paper presents the theoretical development and numerical demonstration of a moment-preserving Monte Carlo electron transport method. Foremost, a full implementation of the moment-preserving (MP) method within the Geant4 particle simulation toolkit is demonstrated. Beyond implementation details, it is shown that the MP method is a viable alternative to the condensed history (CH) method for inclusion in current and future generation transport codes through demonstration of the key features of the method including: systematically controllable accuracy, computational efficiency, mathematical robustness, and versatility. A wide variety of results common to electron transport are presented illustrating the key features of the MP method. In particular, it is possible to achieve accuracy that is statistically indistinguishable from analog Monte Carlo, while remaining up to three orders of magnitude more efficient than analog Monte Carlo simulations. Finally, it is shown that the MP method can be generalized to any applicable analog scattering DCS model by extending previous work on the MP method beyond analytical DCSs to the partial-wave (PW) elastic tabulated DCS data.

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We present an improved deterministic method for analyzing transport problems in random media. In the original method realizations were generated by means of a product quadrature rule; transport calculations were performed on each realization and the results combined to produce ensemble averages. In the present work we recognize that many of these realizations yield identical transport problems. We describe a method to generate only unique transport problems with the proper weighting to produce identical ensemble-averaged results at reduced computational cost. We also describe a method to ignore relatively unimportant realizations in order to obtain nearly identical results with further reduction in costs. Our results demonstrate that these changes allow for the analysis of problems of greater complexity than was practical for the original algorithm.

A Monte Carlo solution method for the system of deterministic equations arising in the application of stochastic collocation (SCM) and stochastic Galerkin (SGM) methods in radiation transport computations with uncertainty is presented for an arbitrary number of materials each containing two uncertain random cross sections. Moments of the resulting random flux are calculated using an intrusive and a non-intrusive Monte Carlo based SCM and two different SGM implementations each with two different truncation methods and compared to the brute force Monte Carlo sampling approach. For the intrusive SCM and SGM, a single set of particle histories is solved and weight adjustments are used to produce flux moments for the stochastic problem. Memory and runtime scaling of each method is compared for increased complexity in stochastic dimensionality and moment truncation. Results are also compared for efficiency in terms of a statistical figure-of-merit. The memory savings for the total-order truncation method prove significant over the full-tensor-product truncation. Scaling shows relatively constant cost per moment calculated of SCM and tensor-product SGM. Total-order truncation may be worthwhile despite poorer runtime scaling by achieving better accuracy at lower cost. The figure-of-merit results show that all of the intrusive methods can improve efficiency for calculating low-order moments, but the intrusive SCM approach is the most efficient for calculating high-order moments.

Stochastic media transport problems have long posed challenges for accurate modeling. Brute force Monte Carlo or deterministic sampling of realizations can be expensive in order to achieve the desired accuracy. The well-known Levermore-Pomraning (LP) closure is very simple and inexpensive, but is inaccurate in many circumstances. We propose a generalization to the LP closure that may help bridge the gap between the two approaches. Our model consists of local calculations to approximately determine the relationship between ensemble-averaged angular fluxes and the corresponding averages at material interfaces. The expense and accuracy of the method are related to how "local" the model is and how much local detail it contains. We show through numerical results that our approach is more accurate than LP for benchmark problems, provided that we capture enough local detail. Thus we identify two approaches to using ensemble calculations for stochastic media calculations: direct averaging of ensemble results for transport quantities of interest, or indirect use via a generalized LP equation to determine those same quantities; in some cases the latter method is more efficient. However, the method is subject to creating ill-posed problems if insufficient local detail is included in the model.

A Monte Carlo solution method for the system of deterministic equations arising in the application of stochastic collocation (SCM) and stochastic Galerkin (SGM) methods in radiation transport computations with uncertainty is presented for an arbitrary number of materials each containing two uncertain random cross sections. Moments of the resulting random flux are calculated using an intrusive and a non-intrusive Monte Carlo based SCM and two different SGM implementations each with two different truncation methods and compared to the brute force Monte Carlo sampling approach. For the intrusive SCM and SGM, a single set of particle histories is solved and weight adjustments are used to produce flux moments for the stochastic problem. Memory and runtime scaling of each method is compared for increased complexity in stochastic dimensionality and moment truncation. Results are also compared for efficiency in terms of a statistical figure-of-merit. The memory savings for the total-order truncation method prove significant over the full-tensor-product truncation. Scaling shows relatively constant cost per moment calculated of SCM and tensor-product SGM. Total-order truncation may be worthwhile despite poorer runtime scaling by achieving better accuracy at lower cost. The figure-of-merit results show that all of the intrusive methods can improve efficiency for calculating low-order moments, but the intrusive SCM approach is the most efficient for calculating high-order moments.

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