Current methods for stochastic media transport are either computationally expensive or, by nature, approximate. Moreover, none of the well-developed, benchmarked approximate methods can compute the variance caused by the stochastic mixing, a quantity especially important to safety calculations. Therefore, we derive and apply a new conditional probability function (CPF) for use in the recently developed stochastic media transport algorithm Conditional Point Sampling (CoPS), which 1) leverages the full intra-particle memory of CoPS to yield errorless computation of stochastic media outputs in 1D, binary, Markovian-mixed media, and 2) leverages the full inter-particle memory of CoPS and the recently developed Embedded Variance Deconvolution method to yield computation of the variance in transport outputs caused by stochastic material mixing. Numerical results demonstrate errorless stochastic media transport as compared to reference benchmark solutions with the new CPF for this class of stochastic mixing as well as the ability to compute the variance caused by the stochastic mixing via CoPS. Using previously derived, non-errorless CPFs, CoPS is further found to be more accurate than the atomic mix approximation, Chord Length Sampling (CLS), and most of memory-enhanced versions of CLS surveyed. In addition, we study the compounding behavior of CPF error as a function of cohort size (where a cohort is a group of histories that share intra-particle memory) and recommend that small cohorts be used when computing the variance in transport outputs caused by stochastic mixing.
Conditional Point Sampling (CoPS) is a recently developed stochastic media transport algorithm that has demonstrated a high degree of accuracy in 1-D and 3-D calculations for binary mixtures with Markovian mixing statistics. In theory, CoPS has the capacity to be accurate for material structures beyond just those with Markovian statistics. However, realizing this capability will require development of conditional probability functions (CPFs) that are based, not on explicit Markovian properties, but rather on latent properties extracted from material structures. Here, we describe a first step towards extracting these properties by developing CPFs using deep neural networks (DNNs). Our new approach lays the groundwork for enabling accurate transport on many classes of stochastic media. We train DNNs on ternary stochastic media with Markovian mixing statistics and compare their CPF predictions to those made by existing CoPS CPFs, which are derived based on Markovian mixing properties. We find that the DNN CPF predictions usually outperform the existing approximate CPF predictions, but with wider variance. In addition, even when trained on only one material volume realization, the DNN CPFs are shown to make accurate predictions on other realizations that have the same internal mixing behavior. We show that it is possible to form a useful CoPS CPF by using a DNN to extract correlation properties from realizations of stochastically mixed media, thus establishing a foundation for creating CPFs for mixtures other than those with Markovian mixing, where it may not be possible to derive an accurate analytical CPF.
Conditional Point Sampling (CoPS) is a newly developed Monte Carlo method for computing radiation transport quantities in stochastic media. The algorithm involves a growing list of point-wise material designations during simulation that causes potentially unbounded increases in memory and runtime, making the calculation of probability density functions (PDFs) computationally expensive. In this work, we adapt CoPS by omitting material points used in the computation from being stored in persisting memory if they are within a user-defined “amnesia radius” from neighboring material points already defined within a realization. We conduct numerical studies to investigate trade-offs between accuracy, required computer memory, and computation time. We demonstrate CoPS's ability to produce accurate mean leakage results and PDFs of leakage results while improving memory and runtime through use of an amnesia radius. We show that a limit on required computer memory per cohort of histories and average runtime per history is imposed as a function of a non-zero amnesia radius. We find that, for the benchmark set investigated, using an amnesia radius of ra = 0.01 introduces minimal error (a 0.006 increase in CoPS3PO root mean squared relative error) in results while improving memory and runtime by an order of magnitude for a cohort size of 100.
Radiation transport in stochastic media is a problem found in a multitude of applications, and the need for tools that are capable of thoroughly modeling this type of problem remains. A collection of approximate methods have been developed to produce accurate mean results, but the demand for methods that are capable of quantifying the spread of results caused by the randomness of material mixing remains. In this work, the new stochastic media transport algorithm Conditional Point Sampling is expanded using Embedded Variance Deconvolution such that it can compute the variance caused by material mixing. The accuracy of this approach is assessed for 1D, binary, Markovian-mixed media by comparing results to published benchmark values, and the behavior of the method is numerically studied as a function of user parameters. We demonstrate that this extension of Conditional Point Sampling is able to compute the variance caused by material mixing with accuracy dependent on the accuracy of the conditional probability function used.
Radiation transport in stochastic media is a challenging problem type relevant for applications such as meteorological modeling, heterogeneous radiation shields, BWR coolant, and pebble-bed reactor fuel. A commonly cited challenge for methods performing transport in stochastic media is to simultaneously be accurate and efficient. Conditional Point Sampling (CoPS), a new method for transport in stochastic media, was recently shown to have accuracy comparable to the most accurate approximate methods for a common 1D benchmark set. In this paper, we use a pseudo-interface-based approach to extend CoPS to application in multi-D for Markovian-mixed media, compare its accuracy with published results for other approximate methods, and examine its accuracy and efficiency as a function of user options. CoPS is found to be the most accurate of the compared methods on the examined benchmark suite for transmittance and comparable in accuracy with the most accurate methods for reflectance and internal flux. Numerical studies examine accuracy and efficiency as a function of user parameters providing insight for effective parameter selection and further method development. Since the authors did not implement any of the other approximate methods, there is not yet a valid comparison for efficiency with the other methods.
Radiation transport in stochastic media is a problem found in a multitude of applications, and the need for tools that are capable of thoroughly modeling this type of problem remains. A collection of approximate methods have been developed to produce accurate mean results, but the demand for methods that are capable of quantifying the spread of results caused by the randomness of material mixing remains. In this work, the new stochastic media transport algorithm Conditional Point Sampling is expanded using Embedded Variance Deconvolution such that it can compute the variance caused by material mixing. The accuracy of this approach is assessed for 1D, binary, Markovian-mixed media by comparing results to published benchmark values, and the behavior of the method is numerically studied as a function of user parameters. We demonstrate that this extension of Conditional Point Sampling is able to compute the variance caused by material mixing with accuracy dependent on the accuracy of the conditional probability function used.