Jeffrey A. Favorite

Surface and Volume Integrals of Uncollided Adjoint Fluxes and Forward-Adjoint Flux Products

Abstract Ray analysis techniques are standard for computing the uncollided component of a detector response to radiation. In this paper, uncollided adjoint flux integrals and forward-adjoint inner product integrals in volumes and on surfaces are derived for general geometries. In numerical test problems using a one-dimensional sphere and a two-dimensional (r-z) cylinder, deterministic and stochastic evaluations of the integrals yielded the same results. A semianalytic benchmark for the adjoint flux integral on a cylindrical surface is also used.

Using the Schwinger Variational Functional for the Solution of Inverse Transport Problems

Abstract A new iterative inverse method for gamma-ray transport problems is presented. The method, based on a novel application of the Schwinger variational functional, is developed as a perturbation problem in which the current model (in the iterative process) is considered the initial, unperturbed system, and the actual model is considered the perturbed system. The new method requires the solution of a set of uncoupled one-group forward and adjoint transport equations in each iteration. Four inverse problems are considered: determination of (a) interface locations in a multilayer source/shield system, (b) the isotopic composition of an unknown source (including inert elements), (c) interface locations and the source composition simultaneously, and (d) the composition of an unknown layer in the shield. Only the first two problems were actually solved in numerical one-dimensional (spherical) test cases. The method worked well for the unknown interface location problem and extremely well for the unknown source composition problem. Convergence of the method was heavily dependent on the initial guess.

An alternative implementation of the differential operator (Taylor series) perturbation method for Monte Carlo criticality problems

Abstract The standard implementation of the differential operator (Taylor series) perturbation method for Monte Carlo criticality problems has previously been shown to have a wide range of applicability. In this method, the unperturbed fission distribution is used as a fixed source to estimate the change in the keff eigenvalue of a system due to a perturbation. A new method, based on the deterministic perturbation theory assumption that the flux distribution (rather than the fission source distribution) is unchanged after a perturbation, is proposed in this paper. Dubbed the F-A method, the new method is implemented within the framework of the standard differential operator method by making tallies only in perturbed fissionable regions and combining the standard differential operator estimate of their perturbations according to the deterministic first-order perturbation formula. The F-A method, developed to extend the range of applicability of the differential operator method rather than as a replacement, was more accurate than the standard implementation for positive and negative density perturbations in a thin shell at the exterior of a computational Godiva model. The F-A method was also more accurate than the standard implementation at estimating reactivity worth profiles of samples with a very small positive reactivity worth (compared to actual measurements) in the Zeus critical assembly, but it was less accurate for a sample with a small negative reactivity worth.

USING THE LEVENBERG-MARQUARDT METHOD FOR SOLUTIONS OF INVERSE TRANSPORT PROBLEMS IN ONE- AND TWO-DIMENSIONAL GEOMETRIES

Abstract Determining the components of a radioactive source/shield system using the system’s radiation signature, a type of inverse transport problem, is one of great importance in homeland security, material safeguards, and waste management. Here, the Levenberg-Marquardt (or simply “Marquardt”) method, a standard gradient-based optimization technique, is applied to the inverse transport problems of interface location identification, shield material identification, source composition identification, and material mass density identification (both separately and combined) in multilayered radioactive source/shield systems. One-dimensional spherical problems using leakage measurements of neutron-induced gamma-ray lines and two-dimensional cylindrical problems using flux measurements of uncollided passive gamma-ray lines are considered. Gradients are calculated using an adjoint-based differentiation technique that is more efficient than difference formulas. The Marquardt method is iterative and directly estimates unknown interface locations, source isotope weight fractions, and material mass densities, while the unknown shield material is identified by estimating its macroscopic gamma-ray cross sections. Numerical test cases illustrate the utility of the Marquardt method using both simulated data that are perfectly consistent with the optimization process and realistic data simulated by Monte Carlo.

Application of the Differential Evolution Method to Solving Inverse Transport Problems

Abstract The differential evolution method, a powerful stochastic optimization algorithm that mimics the process of evolution in nature, is applied to inverse transport problems with several unknown parameters of mixed types, including interface location identification, source composition identification, and material mass density identification, in spherical and cylindrical radioactive source/shield systems. In spherical systems, measurements of leakages of discrete gamma-ray lines are assumed, while in cylindrical systems, measurements of scalar fluxes of discrete lines at points outside the system are assumed. The performance of the differential evolution algorithm is compared to the Levenberg-Marquardt method, a standard gradient-based technique, and the covariance matrix adaptation evolution strategy, another stochastic technique, on a variety of numerical test problems with several (i.e., three or more) unknown parameters. Numerical results indicate that differential evolution is the most adept method for finding the global optimum for these problems. In spherical geometry, differential evolution implemented serially is run-time competitive with gradient-based methods, while a parallel version of differential evolution would be run-time competitive with gradient-based techniques in cylindrical geometry. A hybrid differential evolution/Levenberg-Marquardt method is also introduced, and numerical results indicate that it can be a fast and robust optimizer for inverse transport problems.

Identification of an unknown material in a radiation shield using the schwinger inverse method

Abstract The Schwinger method for solving inverse gamma-ray transport problems was proposed in a previous paper. The method is iterative and requires a set of uncoupled forward and adjoint transport calculations in each iteration. In this paper, the Schwinger inverse method is applied to the problem of identifying an unknown material in a radiation shield by calculating its total macroscopic photon cross sections. The gamma source (its energy and spatial distribution as well as the composition of the material) is known and the total (angle-independent) gamma leakage is measured. In numerical one-dimensional spherical and slab test problems, the Schwinger inverse method successfully calculated the photon cross sections of an unknown material. Material identification was successfully achieved by comparing the calculated cross sections with those in a precomputed material cross-section library, although there was some ambiguity when realistic measurements were used. The Schwinger inverse method compared very favorably with the standard single-energy transmission technique.

Adjoint-based sensitivity and uncertainty analysis for density and composition: A user's guide

Abstract The evaluation of uncertainties is essential for criticality safety. This paper deals with material density and composition uncertainties and provides guidance on how traditional first-order sensitivity methods can be used to predict their effects. Unlike problems that deal with traditional cross-section uncertainty analysis, material density and composition-related problems are often characterized by constraints that do not allow arbitrary and independent variations of the input parameters. Their proper handling requires constrained sensitivities that take into account the interdependence of the inputs. This paper discusses how traditional unconstrained isotopic density sensitivities can be calculated using the adjoint sensitivity capabilities of the popular Monte Carlo codes MCNP6 and SCALE 6.2, and we also present the equations to be used when forward and adjoint flux distributions are available. Subsequently, we show how the constrained sensitivities can be computed using the unconstrained (adjoint-based) sensitivities as well as by applying central differences directly. Three distinct procedures are presented for enforcing the constraint on the input variables, each leading to different constrained sensitivities. As a guide, the sensitivity and uncertainty formulas for several frequently encountered specific cases involving densities and compositions are given. An analytic k∞ example highlights the relationship between constrained sensitivity formulas and central differences, and a more realistic numerical problem reveals similarities among the computer codes used and differences among the three methods of enforcing the constraint.

On the Accuracy of a Common Monte Carlo Surface Flux Grazing Approximation

Abstract Particle fluxes on surfaces are difficult to calculate with Monte Carlo codes because the score requires a division by the surface-crossing angle cosine, and grazing angles lead to inaccuracies. We revisit the standard practice of dividing by half of a cosine “cutoff” for particles whose surface-crossing cosines are below the cutoff. We concentrate on the flux crossing an external boundary, deriving the standard approach in a manner that explicitly points out three assumptions: (a) that the external boundary surface flux is isotropic or mostly isotropic, (b) that the cosine cutoff is small, and (c) that the minimum possible surface-crossing cosine is 0. We find that the requirement for accuracy of the standard surface flux estimate is more restrictive for external boundaries (a very isotropic surface flux) than for internal surfaces (an isotropic or linearly anisotropic surface flux). Numerical demonstrations involve analytic and semianalytic solutions for monoenergetic point sources irradiating surfaces with no scattering. We conclude with a discussion of potentially more robust approaches.

Variational Estimates of Internal Interface Perturbations and a New Variational Functional for Inhomogeneous Transport Problems

Abstract Variational perturbation theory is applied to internal interface perturbations in neutral-particle inhomogeneous transport problems. The leakage from a radioactive system is the quantity of interest. The Schwinger and Roussopolos variational functionals are used with volume- and surface-integral formulations of the integrals of perturbed quantities. In numerical one-dimensional spherical tests of source radius perturbations, the Roussopolos functional in the surface-integral formulation worked better when the source was large, and the Schwinger functional in the volume-integral formulation worked better when the source was small. A new variational functional is presented that formally allows a combination of the Schwinger and Roussopolos functionals; the contribution of each to the total estimate is adjusted with a parameter introduced in one of the trial functions. When the parameter is correctly chosen, the new functional is generally more accurate than either the Schwinger or Roussopolos functional alone. An analytic monodirectional slab transport problem is also considered.

Revisiting Boundary Perturbation Theory for Inhomogeneous Transport Problems

Abstract Adjoint-based first-order perturbation theory is applied again to boundary perturbation problems. Rahnema developed a perturbation estimate that gives an accurate first-order approximation of a flux or reaction rate within a radioactive system when the boundary is perturbed. When the response of interest is the flux or leakage current on the boundary, the Roussopoulos perturbation estimate has long been used. The Rahnema and Roussopoulos estimates differ in one term. This paper shows that the Rahnema and Roussopoulos estimates can be derived consistently, using different responses, from a single variational functional (due to Gheorghiu and Rahnema), resolving any apparent contradiction. In analytic test problems, Rahnema’s estimate and the Roussopoulos estimate produce exact first derivatives of the response of interest when appropriately applied. A realistic, nonanalytic test problem is also presented.