@article{hoagland_azmy_2021, title={Hybrid approaches for accelerated convergence of block-Jacobi iterative methods for solution of the neutron transport equation}, volume={439}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2021.110382}, abstractNote={The Parallel Block Jacobi (PBJ) [1], [2] scheme is an iterative method with application in neutron transport solution which requires asynchronous sub-domain interfaces, where asynchronous here means flux continuity is not imposed across sub-domain interfaces. In this work, we present the development, analysis, and testing of hybrid methods which utilize both PBJ-type methods and synchronous sweep-based methods in a single iterative step. These methods are developed to reduce the penalty to iterative performance of the former incurred in problems containing optically thin cells, while maintaining the PBJ method's applicability to massively parallel solution on unstructured grids. As a prelude to this ultimate goal, this study is conducted in serial operation on 2-D Cartesian grids to facilitate implementation of the various tested iterative methods, to enable the reported spectral analysis, and for shorter computational execution times allowing the large amounts of performance data to be collected. The analysis consists of two Fourier analyses, demonstrating the sequential execution of a PBJ method and SI to be iteratively robust with respect to optical thickness. The results of these analyses and preliminary computational experimentation demonstrate the effectiveness of methods that execute PBJ and SI only in optically thick and thin regions of a problem, respectively. Further experimentation, including a parametric study, then demonstrates this approach to be effective when the thin region is solved using Inexact Parallel Block Jacobi (IPBJ), with sub-domains of a size such that the applicability to massively parallel solution on unstructured grids is not diminished. Finally, testing on realistic test problems containing large void regions confirms the viability of this approach. Performance comparisons include the traditional iterative acceleration methods Diffusion Synthetic Acceleration, Partial-current Nonlinear Diffusion Acceleration, and Adjacent-cell Preconditioner. The results show that while the hybrid PBJ methods generally consume more iterations than these traditional methods, our hybrid approach maintains an iteration count on the same order of magnitude as these acceleration methods, which are not fully developed for massively parallel solution on unstructured grids.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Hoagland, Dylan S. and Azmy, Yousry Y.}, year={2021}, month={Aug} }
 @article{hu_azmy_2021, title={On the Regularity Order of the Pointwise Uncollided Angular Flux and Asymptotic Convergence of the Discrete Ordinates Approximation of the Scalar Flux}, volume={195}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2020.1860634}, abstractNote={Abstract To determine the angular discretization error asymptotic convergence rate of the uncollided scalar flux computed with the discrete ordinates (S ) method, a comprehensive theory of the regularity order with respect to the azimuthal angle of the exact pointwise SN uncollided angular flux is derived based on the integral form of the transport equation in two-dimensional Cartesian geometry. With this theory, the regularity order of the pointwise uncollided angular flux can be estimated for a given problem configuration. Our new theory inspired a novel Modified Simpson’s (MS) quadrature that converges the uncollided scalar flux faster than any of the traditional quadratures by avoiding integration across points of irregularity in the azimuthal angle. Numerical results successfully verify our new theory in four variants of a test configuration, and the angular discretization errors in the corresponding scalar flux computed with conventional angular quadrature types and with our new quadrature types are found to converge with different orders. The error convergence rates obtained with traditional quadrature types are limited by the regularity order of the exact angular flux and the quadrature’s integration intervals while our new MS quadrature types converge with order two to four times higher than traditional quadratures. A detailed study of oscillations observed in certain quadrature errors is provided by introducing the effective length of the irregular interval and the associated oscillating function.}, number={6}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hu, Xiaoyu and Azmy, Yousry Y.}, year={2021}, month={Jun}, pages={598–613} }
 @article{hart_azmy_2021, title={Solution Irregularity Remediation for Spatial Discretization Error Estimation for S-N Transport Solutions}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2021.1982548}, abstractNote={Abstract The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error. In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. A metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity. Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hart, Nathan H. and Azmy, Yousry Y.}, year={2021}, month={Nov} }
 @article{hoagland_yessayan_azmy_2021, title={Solution of the Neutron Transport Equation on Unstructured Grids Using the Parallel Block Jacobi-Integral Transport Matrix Method via the Novel Green's Function ITMM Construction Algorithm on Massively Parallel Computer Systems}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2021.1898309}, abstractNote={The Parallel Block Jacobi (PBJ) spatial domain decomposition is well suited for implementation on massively parallel computers to solve the neutron transport equation on unstructured grids due to the simple scheduling policy that arises from the PBJ’s iterative asynchronicity. The Parallel Block Jacobi-Integral Transport Matrix Method (PBJ-ITMM) is an iterative method that utilizes the PBJ decomposition and resolves local within-group scattering in a single iteration, but requires a matrix-vector iterative solution. This work details the development, implementation, and testing of the novel Green’s Function ITMM Construction (GFIC) algorithm. The GFIC constructs the matrices required for the PBJ-ITMM’s iterative solution on unstructured grids, utilizing the physical interpretation of these matrices as discretized response functions to create a local problem with a Green’s Function–like source. Conducting a set of mesh sweeps over all angles on this local problem yields the ITMM matrix elements. On unstructured grids, this approach utilizes the kernel calculation and fundamental solution algorithm present in an existing transport code, thus avoiding reimplementation of code functionality. Using the GFIC, the PBJ-ITMM is implemented in THOR, a tetrahedral mesh transport code, along with the Inexact Parallel Block Jacobi (IPBJ) method for performance comparison. This comparison involves strong and weak scaling studies of the Godiva and C5G7 benchmark problems using up to 32 768 processors. These studies establish that the PBJ-ITMM executes faster than the IPBJ when the number of cells per subdomain falls below a problem-dependent threshold, ~128 cells for Godiva, >256 cells for C5G7. The largest problem tested, comprising more than 6.8 billion unknowns, solves in <30 min with the IPBJ and <20 min with the PBJ-ITMM, using 32 768 processors. These results demonstrate the PBJ-ITMM as a viable approach for solving neutron transport problems on unstructured grids using massively parallel computers. Additionally, this study illustrates the range of number of cells per subdomain over which this method is favorable.}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hoagland, Dylan S. and Yessayan, Raffi A. and Azmy, Yousry Y.}, year={2021}, month={May} }
 @article{hu_azmy_2020, title={Asymptotic convergence of the angular discretization error in the scalar flux computed from the particle transport equation with the method of discrete ordinates}, volume={138}, ISSN={["0306-4549"]}, DOI={10.1016/j.anucene.2019.107199}, abstractNote={The asymptotic convergence of the angular discretization error in the scalar flux solution of the particle transport equation computed with the Discrete Ordinates (SN) method with increasing quadrature order is examined. Five selected angular quadrature types are considered: Level Symmetric (LS), Legendre-Chebyshev Quadrangular (LCQ), Legendre-Chebyshev Triangular(LCT), Quadruple Range (QR) and Quadruple Range Spence-type (QRS) quadrature sets. We relate the SN angular discretization error to the quadrature error, and split the total flux into the uncollided flux and the fully collided flux, and then we verify the uncollided scalar flux error and the fully collided scalar flux error separately. After developing the theoretical basis for the relationship between true solution regularity and quadrature rule error, we employ a two-dimensional problem to verify the observed and theoretical convergence orders for the region-averaged uncollided and fully collided scalar flux errors. Numerical results show that the angular discretization errors in the region-averaged scalar flux obtained by different quadrature types converge with different rates, that are commensurate with the regularity order of the exact angular flux within the quadrature-designed integration interval. The angular discretization error in the uncollided region-averaged scalar flux obtained by LC class quadratures converges linearly, and the error obtained by QR class quadrature sets converges quadratically. The angular discretization error in the fully collided region-averaged scalar flux converges linearly for LC class quadratures, and faster-than-second order for QR class quadrature sets.}, journal={ANNALS OF NUCLEAR ENERGY}, author={Hu, Xiaoyu and Azmy, Yousry Y.}, year={2020}, month={Apr} }
 @article{schmidt_smith_hite_mattingly_azmy_rajan_goldhahn_2019, title={Sequential optimal positioning of mobile sensors using mutual information}, volume={12}, ISSN={["1932-1872"]}, DOI={10.1002/sam.11431}, abstractNote={Abstract}, number={6}, journal={STATISTICAL ANALYSIS AND DATA MINING}, author={Schmidt, Kathleen and Smith, Ralph C. and Hite, Jason and Mattingly, John and Azmy, Yousry and Rajan, Deepak and Goldhahn, Ryan}, year={2019}, month={Dec}, pages={465–478} }
 @article{nelson_azmy_2017, title={Numerical convergence and validation of the DIMP inverse particle transport model}, volume={49}, ISSN={1738-5733}, url={http://dx.doi.org/10.1016/J.NET.2017.07.009}, DOI={10.1016/J.NET.2017.07.009}, abstractNote={The data integration with modeled predictions (DIMP) model is a promising inverse radiation transport method for solving the special nuclear material (SNM) holdup problem. Unlike previous methods, DIMP is a completely passive nondestructive assay technique that requires no initial assumptions regarding the source distribution or active measurement time. DIMP predicts the most probable source location and distribution through Bayesian inference and quasi-Newtonian optimization of predicted detector responses (using the adjoint transport solution) with measured responses. DIMP performs well with forward hemispherical collimation and unshielded measurements, but several considerations are required when using narrow-view collimated detectors. DIMP converged well to the correct source distribution as the number of synthetic responses increased. DIMP also performed well for the first experimental validation exercise after applying a collimation factor, and sufficiently reducing the source search volume's extent to prevent the optimizer from getting stuck in local minima. DIMP's simple point detector response function (DRF) is being improved to address coplanar false positive/negative responses, and an angular DRF is being considered for integration with the next version of DIMP to account for highly collimated responses. Overall, DIMP shows promise for solving the SNM holdup inverse problem, especially once an improved optimization algorithm is implemented.}, number={6}, journal={Nuclear Engineering and Technology}, publisher={Elsevier BV}, author={Nelson, Noel and Azmy, Yousry}, year={2017}, month={Sep}, pages={1358–1367} }
 @article{nelson_azmy_gardner_mattingly_smith_worrall_dewji_2017, title={Validation and uncertainty quantification of detector response functions for a 1″×2″ NaI collimated detector intended for inverse radioisotope source mapping applications}, volume={410}, ISSN={0168-583X}, url={http://dx.doi.org/10.1016/J.NIMB.2017.07.015}, DOI={10.1016/J.NIMB.2017.07.015}, abstractNote={Detector response functions (DRFs) are often used for inverse analysis. We compute the DRF of a sodium iodide (NaI) nuclear material holdup field detector using the code named g03 developed by the Center for Engineering Applications of Radioisotopes (CEAR) at NC State University. Three measurement campaigns were performed in order to validate the DRF’s constructed by g03: on-axis detection of calibration sources, off-axis measurements of a highly enriched uranium (HEU) disk, and on-axis measurements of the HEU disk with steel plates inserted between the source and the detector to provide attenuation. Furthermore, this work quantifies the uncertainty of the Monte Carlo simulations used in and with g03, as well as the uncertainties associated with each semi-empirical model employed in the full DRF representation. Overall, for the calibration source measurements, the response computed by the DRF for the prediction of the full-energy peak region of responses was good, i.e. within two standard deviations of the experimental response. In contrast, the DRF tended to overestimate the Compton continuum by about 45–65% due to inadequate tuning of the electron range multiplier fit variable that empirically represents physics associated with electron transport that is not modeled explicitly in g03. For the HEU disk measurements, computed DRF responses tended to significantly underestimate (more than 20%) the secondary full-energy peaks (any peak of lower energy than the highest-energy peak computed) due to scattering in the detector collimator and aluminum can, which is not included in the g03 model. We ran a sufficiently large number of histories to ensure for all of the Monte Carlo simulations that the statistical uncertainties were lower than their experimental counterpart’s Poisson uncertainties. The uncertainties associated with least-squares fits to the experimental data tended to have parameter relative standard deviations lower than the peak channel relative standard deviation in most cases and good reduced chi-square values. The highest sources of uncertainty were identified as the energy calibration polynomial factor (due to limited source availability and NaI resolution) and the Ba-133 peak fit (only a very weak source was available), which were 20% and 10%, respectively.}, journal={Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms}, publisher={Elsevier BV}, author={Nelson, N. and Azmy, Y. and Gardner, R.P. and Mattingly, J. and Smith, R. and Worrall, L.G. and Dewji, S.}, year={2017}, month={Nov}, pages={1–15} }
 @article{barichello_tres_picoloto_azmy_2016, title={Recent Studies on the Asymptotic Convergence of the Spatial Discretization for Two-Dimensional Discrete Ordinates Solutions}, volume={45}, ISSN={["2332-4325"]}, DOI={10.1080/23324309.2016.1171242}, abstractNote={ABSTRACT In this work, four types of quadrature schemes are used to define discrete directions in the solution of a two-dimensional fixed-source discrete ordinates problem in Cartesian geometry. Such schemes enable generating numerical results for averaged scalar fluxes over specified regions of the domain with high number (up to 105) of directions per octant. Two different nodal approaches, the ADO and AHOT-N0 methods, are utilized to obtain the numerical results of interest. The AHOT-N0 solutions on a sequence of refined meshes are then used to develop an asymptotic analysis of the spatial discretization error in order to derive a reference solution. It was more clearly observed that the spatial discretization error converges asymptotically with second order for the source region with all four quadratures employed, while for the other regions refined meshes along with tighter convergence criterion must be applied to evidence the same behavior. However, in that case, some differences among the four quadrature schemes results were found.}, number={4}, journal={JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT}, author={Barichello, L. B. and Tres, A. and Picoloto, C. B. and Azmy, Y. Y.}, year={2016}, pages={299–313} }
 @article{schunert_azmy_2015, title={Comparison of spatial discretization methods for solving the S-N equations using a three-dimensional method of manufactured solutions benchmark suite with escalating order of nonsmoothness}, volume={180}, DOI={10.13182/nse14-77}, abstractNote={Abstract A comparison of the accuracy and computational efficiency of spatial discretization methods of the three-dimensional SN equations is conducted, including discontinuous Galerkin finite element methods, the arbitrarily high-order transport method of nodal type (AHOTN), the linear-linear method, the linear-nodal (LN) method, and the higher-order diamond difference method. For this purpose, we have developed a suite of method of manufactured solutions benchmarks that provides an exact solution of the SN equations even in the presence of scattering. Most importantly, our benchmark suite permits the user to set an arbitrary level of smoothness of the exact solution across the singular characteristics. Our study focuses on the computational efficiency of the considered spatial discretization methods. Numerical results indicate that the best-performing method depends on the norm used to measure the discretization error. We employ discrete Lp norms and integral error norms in this work. For configurations with continuous exact angular flux, high-order AHOTNs perform best under Lp error norms, while the LN method performs best when measured by integral error norms. When the angular flux is discontinuous, a new singular-characteristic tracking method for three-dimensional geometries performs best among the considered methods.}, number={1}, journal={Nuclear Science and Engineering}, author={Schunert, S. and Azmy, Y.}, year={2015}, pages={1–29} }
 @article{anistratov_azmy_2015, title={Iterative stability analysis of spatial domain decomposition based on block Jacobi algorithm for the diamond-difference scheme}, volume={297}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2015.05.033}, abstractNote={We study convergence of the integral transport matrix method (ITMM) based on a parallel block Jacobi (PBJ) iterative strategy for solving particle transport problems. The ITMM is a spatial domain decomposition method proposed for massively parallel computations. A Fourier analysis of the PBJ-based iterations applied to SN diamond-difference equations in 1D slab and 2D Cartesian geometries is performed. It is carried out for infinite-medium problems with homogeneous material properties. To analyze the performance of the ITMM with the PBJ algorithm and evaluate its potential in scalability we consider a limiting case of one spatial cell per subdomain. The analysis shows that in such limit the spectral radius of the iteration method is one without regard to values of the scattering ratio and optical thickness of the spatial cells. This implies lack of convergence in infinite medium. Numerical results of finite-medium problems are presented. They demonstrate effects of finite size of spatial domain on the performance of the iteration algorithm as well as its asymptotic behavior when the extent of the spatial domain increases. These numerical experiments also show that for finite domains iterative convergence to a finite criterion is achievable in a multiple of the sum of number of cells in each dimension.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anistratov, Dmitriy Y. and Azmy, Yousry Y.}, year={2015}, month={Sep}, pages={462–479} }
 @article{hykes_azmy_2015, title={Radiation Source Mapping with Bayesian Inverse Methods}, volume={179}, ISSN={["1943-748X"]}, DOI={10.13182/nse13-91}, abstractNote={Abstract We present a method to map the spectral and spatial distributions of radioactive sources using a limited number of detectors. Locating and identifying radioactive materials is important for border monitoring, in accounting for special nuclear material in processing facilities, and in cleanup operations following a radioactive material spill. Most methods to analyze these types of problems make restrictive assumptions about the distribution of the source. In contrast, the source mapping method presented here allows an arbitrary three-dimensional distribution in space and a gamma peak distribution in energy. To apply the method, the problem is cast as an inverse problem where the system’s geometry and material composition are known and fixed, while the radiation source distribution is sought. A probabilistic Bayesian approach is used to solve the resulting inverse problem since the system of equations is ill-posed. The posterior is maximized with a Newton optimization method. The probabilistic approach also provides estimates of the confidence in the final source map prediction. A set of adjoint, discrete ordinates flux solutions, obtained in this work by the Denovo code, is required to efficiently compute detector responses from a candidate source distribution. These adjoint fluxes form the linear mapping from the state space to the response space. The test of the method’s success is simultaneously locating a set of 137Cs and 60Co gamma sources in a room. This test problem is solved using experimental measurements that we collected for this purpose. Because of the weak sources available for use in the experiment, some of the expected photopeaks were not distinguishable from the Compton continuum. However, by supplanting 14 flawed measurements (out of a total of 69) with synthetic responses computed by MCNP, the proof-of-principle source mapping was successful. The locations of the sources were predicted within 25 cm for two of the sources and 90 cm for the third, in a room with an ˜4-× 4-m floor plan. The predicted source intensities were within a factor of ten of their true value.}, number={4}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hykes, Joshua M. and Azmy, Yousry Y.}, year={2015}, month={Apr}, pages={364–380} }
 @article{schunert_azmy_2013, title={Using the Cartesian Discrete Ordinates Code DORT for Assembly-Level Calculations}, volume={173}, ISSN={["1943-748X"]}, DOI={10.13182/nse11-17}, abstractNote={Abstract For the sake of a high-fidelity representation of the curved surfaces characteristic of fuel pins, the standard reactor design process employs the method of collision probabilities (CP), the method of characteristics (MOC), or unstructured-grid discrete ordinates (SN) transport solvers for assembly-level calculations. In this work we provide a proof of principle using highly simplified assembly configurations that an approximate staircased representation of the fuel pin’s circumference via an orthogonal mesh is accurate enough for reactor physics computations. For the purpose of comparing the performance of these approaches, we employ the orthogonal-grid SN code DORT and the lattice code DRAGON (CP and MOC) to perform k-eigenvalue-type computations for both a boiling water reactor (BWR) and pressurized water reactor (PWR) test assembly. In the framework of a computational model refinement study, the multiplication factor and the fission source distribution are computed and compared to a high-fidelity multigroup MCNP reference solution. The accuracy of the considered methods at each considered model refinement level (fidelity of curved surface representation in DORT, number of tracks in MOC, etc.) is quantified via the difference of the multiplication factor from its reference value and via the root-mean-square and maximum norm of the error in the fission source distribution. We find that for the BWR assembly DORT outperforms MOC and CP in both accuracy and computational efficiency, while for the PWR test case, MOC computes the most accurate fission source distribution but fails to compute the multiplication factor accurately.}, number={3}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Schunert, Sebastian and Azmy, Yousry}, year={2013}, month={Mar}, pages={233–258} }
 @article{ferrer_azmy_2012, title={A Robust Arbitrarily High-Order Transport Method of the Characteristic Type for Unstructured Grids}, volume={172}, ISSN={["0029-5639"]}, DOI={10.13182/nse10-106}, abstractNote={Abstract A reformulation of the arbitrarily high-order transport method of the characteristic type (AHOT-C) for unstructured grids (AHOT-C-UG) is presented in this work, which resolves the previous difficulties encountered in the original formalism. A general equivalence between the arbitrary-order neutron balance and arbitrary-order characteristic equations is derived, which improves the numerical computation of the spatial moments of the angular flux and allows a series expansion of the characteristic integral kernel in cases where the medium is optically thin. Numerical results are presented, which verify the convergence behavior of AHOT-C-UG for various expansion orders.}, number={1}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Ferrer, Rodolfo M. and Azmy, Yousry Y.}, year={2012}, month={Sep}, pages={33–51} }
 @article{gill_azmy_2011, title={Newton's Method for Solving k-Eigenvalue Problems in Neutron Diffusion Theory}, volume={167}, ISSN={["0029-5639"]}, DOI={10.13182/nse09-98}, abstractNote={Abstract We present an approach to the k-eigenvalue problem in multigroup diffusion theory based on a nonlinear treatment of the generalized eigenvalue problem. A nonlinear function is posed whose roots are equal to solutions of the k-eigenvalue problem; a Newton-Krylov method is used to find these roots. The Jacobian-vector product is found exactly or by using the Jacobian-free Newton-Krylov (JFNK) approximation. Several preconditioners for the Krylov iteration are developed. These preconditioners are based on simple approximations to the Jacobian, with one special instance being the use of power iteration as a preconditioner. Using power iteration as a preconditioner allows for the Newton-Krylov approach to heavily leverage existing power method implementations in production codes. When applied as a left preconditioner, any existing power iteration can be used to form the kernel of a JFNK solution to the k-eigenvalue problem. Numerical results generated for a suite of two-dimensional reactor benchmarks show the feasibility and computational benefits of the Newton formulation as well as examine some of the numerical difficulties potentially encountered with Newton-Krylov methods. The performance of the method is also seen to be relatively insensitive to the dominance ratio for a one-dimensional slab problem.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Gill, Daniel F. and Azmy, Yousry Y.}, year={2011}, month={Feb}, pages={141–153} }
 @article{gill_azmy_warsa_densmore_2011, title={Newton's Method for the Computation of k-Eigenvalues in S-N Transport Applications}, volume={168}, ISSN={["0029-5639"]}, DOI={10.13182/nse10-01}, abstractNote={Abstract Recently, Jacobian-Free Newton-Krylov (JFNK) methods have been used to solve the k-eigenvalue problem in diffusion and transport theories. We propose an improvement to Newton’s method (NM) for solving the k-eigenvalue problem in transport theory that avoids costly within-group iterations or iterations over energy groups. We present a formulation of the k-eigenvalue problem where a nonlinear function, whose roots are solutions of the k-eigenvalue problem, is written in terms of a generic fixed-point iteration (FPI). In this way any FPI that solves the k-eigenvalue problem can be accelerated using the Newton approach, including our improved formulation. Calculations with a one-dimensional multigroup SN transport implementation in MATLAB provide a proof of principle and show that convergence to the fundamental mode is feasible. Results generated using a three-dimensional Fortran implementation of several formulations of NM for the well-known Takeda and C5G7-MOX benchmark problems confirm the efficiency of NM for realistic k-eigenvalue calculations and highlight numerous advantages over traditional FPI.}, number={1}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Gill, Daniel F. and Azmy, Yousry Y. and Warsa, James S. and Densmore, Jeffery D.}, year={2011}, month={May}, pages={37–58} }
 @article{rosa_azmy_morel_2010, title={On the Degradation of Cell-Centered Diffusive Preconditioners for Accelerating S-N Transport Calculations in the Periodic Horizontal Interface Configuration}, volume={166}, ISSN={["0029-5639"]}, DOI={10.13182/nse09-69}, abstractNote={Abstract We investigate the degraded effectiveness of diffusion-based acceleration schemes in terms of the adjacent-cell preconditioner (AP) in a periodically heterogeneous limit devised for the two-dimensional (2-D) periodic horizontal interface (PHI) configuration. Specifically, we demonstrate that the diffusive low-order operator employed in the AP scheme lacks the structure of the integral transport operator in the above asymptotic limit since it (1) ignores cross-derivative coupling and (2) incorrectly estimates the strength of intra-layer coupling in the optically thin layers. In order to prove propositions 1 and 2, we derive expressions for the elements of the matrix representing a certain angular (SN) and spatially discretized form of the 2-D neutron transport integral operator. This is the transport operator that produces the full scalar flux solution if it is directly inverted on the once-collided particle source. The properties of this operator’s elements are then investigated in the asymptotic limit for PHI. The results of the asymptotic analysis point to a sparse but nonlocal matrix structure due to long-range coupling of a cell’s average flux with its neighboring cells, independent of the distance between the cells in the spatial mesh. In particular, for a cell in a thin layer, cross-derivative coupling of the cell’s flux to its diagonal neighbors is of the same asymptotic order as self-coupling and coupling with its north/south Cartesian neighbors. Similarly, its coupling with the fluxes in the same thin layer is of the same order, independent of the distance between the cells in the layer, as the coupling with the east/west Cartesian neighbors. We also show that modifying the standard diffusion-based AP can lead to effective acceleration in PHI. Specifically, we devise three novel acceleration schemes, named APB, Optimized-AP (OAP), and Hybrid-AP (HAP), obtained by modifying the original AP formalism in 2-D. In the APB the five-point AP operator is extended to a nine-point stencil that accounts for cross-derivative coupling by including the matrix elements of the integral transport operator B, which couple a cell-averaged scalar flux to its first diagonal neighbors. In the OAP the five-point stencil of the original AP operator is retained while optimizing the value of the elements in the preconditioner that affect the coupling of a cell with its east/west Cartesian neighbors. Specifically, the optimum elements are obtained by minimizing the iteration’s spectral radius and offer a more correct estimate of the strength of intra-layer coupling in a thin layer. Finally, the nine-point HAP operator represents a “hybrid” of the APB and OAP approaches, in the sense that the spectral properties of the optimized five-point OAP are further improved via the inclusion of cross-derivative terms. Fourier analysis of the novel acceleration schemes indicates that robustness of the accelerated iterations can be recovered, in spite of sharp material discontinuities, by accounting for cross-derivative coupling and by optimizing the preconditioner elements. The new acceleration schemes have also been implemented in a 2-D transport code, and numerical tests successfully verify the predictions of the Fourier analysis. However, it is important to emphasize that the modifications attempted in this work are specific to the selected asymptotic limit for PHI and do not translate into new low-order operators for the general heterogeneous-material case. Rather, the above modified operators suggest that it may be possible to eventually derive such a general low-order unconditionally robust operator.}, number={3}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Rosa, Massimiliano and Azmy, Yousry Y. and Morel, Jim E.}, year={2010}, month={Nov}, pages={218–238} }
 @article{duo_azmy_zikatanov_2009, title={A posteriori error estimator and AMR for discrete ordinates nodal transport methods}, volume={36}, ISSN={["0306-4549"]}, DOI={10.1016/j.anucene.2008.12.008}, abstractNote={In the development of high fidelity transport solvers, optimization of the use of available computational resources and access to a tool for assessing quality of the solution are key to the success of large-scale nuclear systems’ simulation. In this regard, error control provides the analyst with a confidence level in the numerical solution and enables for optimization of resources through Adaptive Mesh Refinement (AMR). In this paper, we derive an a posteriori error estimator based on the nodal solution of the Arbitrarily High Order Transport Method of the Nodal type (AHOT-N). Furthermore, by making assumptions on the regularity of the solution, we represent the error estimator as a function of computable volume and element-edges residuals. The global L2 error norm is proved to be bound by the estimator. To lighten the computational load, we present a numerical approximation to the aforementioned residuals and split the global norm error estimator into local error indicators. These indicators are used to drive an AMR strategy for the spatial discretization. However, the indicators based on forward solution residuals alone do not bound the cell-wise error. The estimator and AMR strategy are tested in two problems featuring strong heterogeneity and highly transport streaming regime with strong flux gradients. The results show that the error estimator indeed bounds the global error norms and that the error indicator follows the cell-error’s spatial distribution pattern closely. The AMR strategy proves beneficial to optimize resources, primarily by reducing the number of unknowns solved for to achieve prescribed solution accuracy in global L2 error norm. Likewise, AMR achieves higher accuracy compared to uniform refinement when resolving sharp flux gradients, for the same number of unknowns.}, number={3}, journal={ANNALS OF NUCLEAR ENERGY}, author={Duo, Jose I. and Azmy, Yousry Y. and Zikatanov, Ludmil T.}, year={2009}, month={Apr}, pages={268–273} }
 @article{rosa_azmy_morel_2009, title={Properties of the S-N-equivalent integral transport operator in slab geometry and the iterative acceleration of neutral particle transport methods}, volume={162}, DOI={10.13182/NSE162-234}, abstractNote={Abstract General expressions for the matrix elements of the discrete SN-equivalent integral transport operator are derived in slab geometry. Their asymptotic behavior versus cell optical thickness is investigated both for a homogeneous slab and for a heterogeneous slab characterized by a periodic material discontinuity wherein each optically thick cell is surrounded by two optically thin cells in a repeating pattern. In the case of a homogeneous slab, the asymptotic analysis conducted in the thick-cell limit for a highly scattering medium shows that the discretized integral transport operator approaches a tridiagonal matrix possessing a diffusion-like coupling stencil. It is further shown that this structure is approached at a fast exponential rate with increasing cell thickness when the arbitrarily high order transport method of the nodal type and zero-order spatial approximation (AHOT-N0) formalism is employed to effect the spatial discretization of the discrete ordinates transport operator. In the case of periodically heterogeneous slab configurations, the asymptotic behavior is realized by pushing apart the cells’ optical thicknesses; i.e., the thick cells are made thicker while the thin cells are made thinner at a prescribed rate. We show that in this limit the discretized integral transport operator is approximated by a pentadiagonal structure. Notwithstanding, the discrete operator is amenable to algebraic transformations leading to a matrix representation still asymptotically approaching a tridiagonal structure at a fast exponential rate bearing close resemblance to the diffusive operator. The results of the asymptotic analysis of the integral transport matrix are then used to gain insight into the excellent convergence properties of the adjacent-cell preconditioner (AP) acceleration scheme. Specifically, the AP operator exactly captures the asymptotic structure acquired by the integral transport matrix in the thick-cell limit for homogeneous slabs of pure-scatterer or partial-scatterer material, and for periodically heterogeneous slabs hosting purely scattering materials. In the above limits the integral transport matrix reduces to a diffusive structure consistent with the diffusive matrix template used to construct the AP. In the case of periodically heterogeneous slabs containing absorbing materials, the AP operator partially captures the asymptotic structure acquired by the integral transport matrix. The inexact agreement is due either to discrepancies in the equations for the boundary cells or to the nondiffusive structure acquired by the integral transport matrix. These findings shed light on the immediate convergence, i.e., convergence in two iterations, displayed by the AP acceleration scheme in the asymptotic limit for slabs hosting purely scattering materials, both in the homogeneous and periodically heterogeneous cases. For periodically heterogeneous slabs containing absorbing materials, immediate convergence is achieved by modifying the original recipe for constructing the AP so that the correct asymptotic structure of the integral transport matrix coincides with the AP operator in the asymptotic limit.}, number={3}, journal={Nuclear Science and Engineering}, author={Rosa, M. and Azmy, Y. Y. and Morel, J. E.}, year={2009}, pages={234–252} }
 @article{duo_azmy_2009, title={Spatial Convergence Study of Discrete Ordinates Methods Via the Singular Characteristic Tracking Algorithm}, volume={162}, ISSN={["1943-748X"]}, DOI={10.13182/NSE08-28}, abstractNote={Abstract This paper analyzes the spatial discretization of the discrete ordinates (DO) approximation of the transport equation. A new method, the singular characteristics tracking algorithm, is developed to account for potential nonsmoothness across the singular characteristics in the exact solution of the DO approximation to the transport equation. Numerical results in two-dimensional problems show improved rate of convergence of the exact solution of the DO equations in nonscattering and isotropic scattering media. Unlike the standard weighted diamond difference scheme, the new algorithm achieves local convergence in the case of discontinuous angular flux across the singular characteristics. The method also significantly reduces the error for problems where the angular flux presents discontinuous spatial derivatives across these lines. For purposes of testing the performance of the new method, the method of manufactured solutions is used to generate analytical reference solutions that permit accurate estimation of the local error in case of discontinuous flux.}, number={1}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Duo, J. I. and Azmy, Y. Y.}, year={2009}, month={May}, pages={41–55} }
 @article{bekar_azmy_2009, title={TORT solutions to the NEA suite of benchmarks for 3D transport methods and codes over a range in parameter space}, volume={36}, ISSN={["0306-4549"]}, DOI={10.1016/j.anucene.2008.11.036}, abstractNote={We present the TORT solutions to the 3D transport codes’ suite of benchmarks exercise. An overview of benchmark configurations is provided, followed by a description of the TORT computational model we developed to solve the cases comprising the benchmark suite. In the numerical experiments reported in this paper, we chose to refine the spatial and angular discretizations simultaneously, from the coarsest model (40 × 40 × 40, 200 angles) to the finest model (160 × 160 × 160, 800 angles). The MCNP reference solution is used for evaluating the effect of model-refinement on the accuracy of the TORT solutions. The presented results show that the majority of benchmark quantities are computed with good accuracy by TORT, and that the accuracy improves with model refinement. However, this deliberately severe test has exposed some deficiencies in both deterministic and stochastic solution approaches. Specifically, TORT fails to converge the inner iterations in some benchmark configurations while MCNP produces zero tallies, or drastically poor statistics for some benchmark quantities. We conjecture that TORT’s failure to converge is driven by ray effects in configurations with low scattering ratio and/or highly skewed computational cells, i.e. aspect ratio far from unity. The failure of MCNP occurs in quantities tallied over a very small area or volume in physical space, or quantities tallied many (∼25) mean free paths away from the source. Hence automated, robust, and reliable variance reduction techniques are essential for obtaining high quality reference values of the benchmark quantities. Preliminary results of the benchmark exercise indicate that the occasionally poor performance of TORT is shared with other deterministic codes. Armed with this information, method developers can now direct their attention to regions in parameter space where such failures occur and design alternative solution approaches for such instances.}, number={3}, journal={ANNALS OF NUCLEAR ENERGY}, author={Bekar, Kursat B. and Azmy, Yousry Y.}, year={2009}, month={Apr}, pages={368–374} }
 @article{azmy_gupta_pugh_2008, title={Computational Modelling of Genome-Side Transcription Assembly Networks Using a Fluidics Analogy}, volume={3}, ISSN={["1932-6203"]}, DOI={10.1371/journal.pone.0003095}, abstractNote={Understanding how a myriad of transcription regulators work to modulate mRNA output at thousands of genes remains a fundamental challenge in molecular biology. Here we develop a computational tool to aid in assessing the plausibility of gene regulatory models derived from genome-wide expression profiling of cells mutant for transcription regulators. mRNA output is modelled as fluid flow in a pipe lattice, with assembly of the transcription machinery represented by the effect of valves. Transcriptional regulators are represented as external pressure heads that determine flow rate. Modelling mutations in regulatory proteins is achieved by adjusting valves' on/off settings. The topology of the lattice is designed by the experimentalist to resemble the expected interconnection between the modelled agents and their influence on mRNA expression. Users can compare multiple lattice configurations so as to find the one that minimizes the error with experimental data. This computational model provides a means to test the plausibility of transcription regulation models derived from large genomic data sets.}, number={8}, journal={PLOS ONE}, author={Azmy, Yousry Y. and Gupta, Anshuman and Pugh, Franklin}, year={2008}, month={Aug} }
 @article{fischer_azmy_2007, title={Comparison via parallel performance models of angular and spatial domain decompositions for solving neutral particle transport problems}, volume={49}, ISSN={0149-1970}, url={http://dx.doi.org/10.1016/j.pnucene.2006.08.003}, DOI={10.1016/j.pnucene.2006.08.003}, abstractNote={A previously reported parallel performance model for angular domain decomposition (ADD) of the discrete ordinates approximation for solving multidimensional neutral particle transport problems is revisited for stronger validation. Three communication schemes, native MPI, the bucket algorithm, and the distributed bucket algorithm, are included in the validation exercise that is successfully conducted on a Beowulf cluster. The parallel component of the parallel performance model is largely independent of the communication scheme, in contrast with the communication component that is strongly dependent on the global reduce algorithm. Correct trends for each component and each communication scheme are measured for the Arbitrarily High Order Transport (AHOT) code, thus validating the performance models. Furthermore, extensive experiments illustrate the superiority of the bucket algorithm, in the sense that it incurs a smaller communication penalty compared to the native MPI and distributed bucket algorithms. The primary question addressed in this work is for a given problem size, which domain decomposition scheme, angular or spatial, is best suited to parallelize discrete ordinates methods on a specific computational platform? We address this question for three-dimensional applications via parallel performance models for the abovementioned ADD, and a previously constructed and validated spatial domain decomposition (SDD) model. The constructed parallel performance models include parameters specifying the problem size and system performance. We conclude that for large problems the parallel component dwarfs the communication component even on moderately large numbers of processors. The main advantages of SDD are (a) scalability to higher numbers of processors of the order of the number of computational cells; (b) smaller memory requirement; (c) better performance than ADD on high-end platforms and large number of processors. On the other hand, the main advantages of ADD are (a) perfect load balance; (b) simple implementation, even on unstructured grids; (c) better performance than SDD on medium- and low-end platforms and large number of discrete ordinates. It follows that programmers and users of discrete ordinates codes must carefully select the appropriate domain decomposition method for the class of problems and multiprocessor platforms they wish to target.}, number={1}, journal={Progress in Nuclear Energy}, publisher={Elsevier BV}, author={Fischer, James W. and Azmy, Y.Y.}, year={2007}, month={Jan}, pages={37–60} }
 @article{alim_bekar_ivanov_unlu_brenizer_azmy_2006, title={Modeling and optimization of existing beam port facility of PSBR}, volume={33}, ISSN={0306-4549}, url={http://dx.doi.org/10.1016/j.anucene.2006.10.007}, DOI={10.1016/j.anucene.2006.10.007}, abstractNote={Due to inherited design issues with the current arrangement of beam ports (BPs) and reactor core-moderator assembly in The Perm State Breazeale Reactor (PSBR), the development of innovative experimental facilities utilizing neutron beams is extremely limited. Therefore, a study has started to examine the existing BPs for neutron and gamma outputs and develop a new core-moderator location and BP geometry in PSBR. Although 7 BPs are placed in PSBR, 2 of them are using currently. In this study BP 4, one of the currently being used BP, is examined. With changing the location of the BP 4 and structure of the core assembly, some artificial models are developed and compared with the original model.}, number={17-18}, journal={Annals of Nuclear Energy}, publisher={Elsevier BV}, author={Alim, Fatih and Bekar, Kursat and Ivanov, Kostadin and Unlu, Kenan and Brenizer, Jack and Azmy, Yousry}, year={2006}, month={Nov}, pages={1391–1395} }
 @article{klingensmith_azmy_gehin_orsi_2006, title={Tort solutions to the three-dimensional MOX benchmark, 3-D Extension C5G7MOX}, volume={48}, ISSN={0149-1970}, url={http://dx.doi.org/10.1016/j.pnucene.2006.01.011}, DOI={10.1016/j.pnucene.2006.01.011}, abstractNote={Solutions to the OECD/NEA three-dimensional fuel assembly benchmark problem, C5G7MOX, obtained using the discrete-ordinates neutron transport code TORT are presented. The accuracy and convergence of the effective multiplication factor and maximum pin power are examined while varying the fidelity of the geometric model and the order of the angular quadrature, specifically the Square Legendre–Chebychev quadrature. The calculated value of the effective multiplication factor is shown to converge asymptotically as the order of the quadrature is increased. The behavior of that asymptotic limit of keff with decreasing computational cell size and increasing staircase resolution of the curved rod-moderator is less converged. Moreover, rapidly increasing computational cost with geometric configuration refinement prevented us from achieving asymptotic convergence with the spatial approximation. The maximum pin power results are generally shown to asymptotically converge to within the statistical error of the reference solution for the problem slices where control rods are not present. For the zones that contain control rods, the solution does not appear to be tightly converged for the meshes and quadrature orders employed.}, number={5}, journal={Progress in Nuclear Energy}, publisher={Elsevier BV}, author={Klingensmith, Jesse J. and Azmy, Yousry Y. and Gehin, Jess C. and Orsi, Roberto}, year={2006}, month={Jul}, pages={445–455} }
 @article{azmy_gehin_orsi_2004, title={Dort solutions to the two-dimensional C5G7MOXbenchmark problem}, volume={45}, ISSN={0149-1970}, url={http://dx.doi.org/10.1016/j.pnueene.2004.09.011}, DOI={10.1016/j.pnueene.2004.09.011}, abstractNote={Abstract We present a comprehensive study of the solution error for the C5G7MOXBenchmark problem using the two-dimensional transport code DORT. We set a stringent criterion that a successful solution to the benchmark exercise must satisfy, namely that at least for some of the benchmark quantities convergence to the reference solution with computational model refinement must be demonstrated. In the present exercise this amounts to examining the evolution of the DORT solution, e.g. the multiplication factor, with increasing angular quadrature order, decreasing computational cell size, and tighter representation of the curved rod-moderator interface for all circular rods in the core model. In addition we explored the effect of angular quadrature type, comparing solution accuracy for the fully symmetric and the Square Legendre-Chebychev quadratures establishing superiority of the latter. We establish the high quality of DORT's solution to this benchmark exercise by demonstrating that the multiplication factor asymptotically approaches the reference solution as stated in our self-imposed success criterion. High accuracy of the pin power distribution is also attained, however, convergence to the reference values is not realized. We conjecture that this is due to the lack of error information in the reference values. Our results also illustrate DORT's high accuracy on reasonable meshes and quadrature orders, as well as the sufficiency of a crude, square, geometric approximation of the rod-moderator interface to achieve high accuracy.}, number={2-4}, journal={Progress in Nuclear Energy}, publisher={Elsevier BV}, author={Azmy, Yousry Y. and Gehin, Jess C. and Orsi, Roberto}, year={2004}, month={Jan}, pages={215–231} }