Parametric Model-Order Reduction for Radiation Transport Simulations Based on an Affine Decomposition of the Operators

References

• M. P. ADAMS et al., “Provably Optimal Parallel Transport Sweeps on Semi-Structured Grids,” J. Comput. Phys., 407, 109234 (2020); https://doi.org/10.1016/j.jcp.2020.109234.
   Web of Science ®Google Scholar
• J. I. VERMAAK et al., “Massively Parallel Transport Sweeps on Meshes with Cyclic Dependencies,” J. Comput. Phys., 425, 109892 (2021); https://doi.org/10.1016/j.jcp.2020.109892.
   Web of Science ®Google Scholar
• M. HANUS and J. C. RAGUSA, “Thermal Upscattering Acceleration Schemes for Parallel Transport Sweeps,” Nucl. Sci. Eng., 194, 10, 873 (2020); https://doi.org/10.1080/00295639.2020.1767436.
   Web of Science ®Google Scholar
• J. S. HESTHAVEN, G. ROZZA, and B. STAMM, Certified Reduced Basis Methods for Parameterized Partial Differential Equations, Springer International Publishing (2016); https://doi.org/10.1007/978-3-319-22470-1.
   Google Scholar
• Reduced Order Methods for Modeling and Computational Reduction, A. QUARTERONI and G. ROZZA Eds., Springer International Publishing (2014); https://doi.org/10.1007/978-3-319-02090-7.
   Google Scholar
• Reduced Basis Methods for Partial Differential Equations, A. QUARTERONI, A. MANZONI, and F. NEGRI Eds., Springer International Publishing (2016); https://doi.org/10.1007/978-3-319-15430-2.
   Google Scholar
• Model Reduction of Parameterized Systems, P. BENNER et al. Eds., Springer International Publishing (2017); https://doi.org/10.1007/978-3-319-58785-1.
   Google Scholar
• J. J. DUDERSTADT and L. J. HAMILTON, Nuclear Reactor Analysis, John Wiley & Sons, Inc. (1976).
   Google Scholar
• G. I. BELL and S. GLASSTONE, Nuclear Reactor Theory, Van Nostrand Reinhold Company (1970).
   Google Scholar
• M. HADIGOL and A. DOOSTAN, “Least Squares Polynomial Chaos Expansion: A Review of Sampling Strategies,” Comput. Methods Appl. Mech. Eng., 332, 382 (2018); https://doi.org/10.1016/j.cma.2017.12.019.
   Web of Science ®Google Scholar
• M. M. R. WILLIAMS, “Polynomial Chaos Functions and Neutron Diffusion,” Nucl. Sci. Eng., 155, 1, 109 (2007); https://doi.org/10.13182/NSE05-73TN.
   Web of Science ®Google Scholar
• M. WILLIAMS, “Polynomial Chaos Functions and Stochastic Differential Equations,” Ann. Nucl. Energy, 33, 9, 774 (2006); https://doi.org/10.1016/j.anucene.2006.04.005.
   Web of Science ®Google Scholar
• M. EATON and M. WILLIAMS, “A Probabilistic Study of the Influence of Parameter Uncertainty on Solutions of the Neutron Transport Equation,” Prog. Nucl. Energy, 52, 6, 580 (2010); https://doi.org/10.1016/j.pnucene.2010.01.002.
   Web of Science ®Google Scholar
• L. GILLI et al., “Performing Uncertainty Analysis of a Nonlinear Point-Kinetics/Lumped Parameters Problem Using Polynomial Chaos Techniques,” Ann. Nucl. Energy, 40, 1, 35 (2012); https://doi.org/10.1016/j.anucene.2011.09.016.
   Web of Science ®Google Scholar
• L. GILLI et al., “Uncertainty Quantification for Criticality Problems Using Non-intrusive and Adaptive Polynomial Chaos Techniques,” Ann. Nucl. Energy, 56, 71 (2013); https://doi.org/10.1016/j.anucene.2013.01.009.
   Web of Science ®Google Scholar
• Z. M. PRINCE and J. C. RAGUSA, “Parametric Uncertainty Quantification Using Proper Generalized Decomposition Applied to Neutron Diffusion,” Int. J. Numer. Methods Eng., 119, 9, 899 (2019); https://doi.org/10.1002/nme.6077.
   Web of Science ®Google Scholar
• J. P. SENECAL and W. JI, “Characterization of the Proper Generalized Decomposition Method for Fixed-Source Diffusion Problems,” Ann. Nucl. Energy, 126, 68 (2019); https://doi.org/10.1016/j.anucene.2018.10.062.
   Web of Science ®Google Scholar
• S. GONZÁLEZ-PINTOR, D. GINESTAR, and G. VERDÚ, “Using Proper Generalized Decomposition to Compute the Dominant Mode of a Nuclear Reactor,” Math. Comput. Modell., 57, 7, 1807 (2013); https://doi.org/10.1016/j.mcm.2011.11.066.
   Google Scholar
• A. L. ALBERTI and T. S. PALMER, “Reduced-Order Modeling of Nuclear Reactor Kinetics Using Proper Generalized Decomposition,” Nucl. Sci. Eng., 194, 10, 837 (2020); https://doi.org/10.1080/00295639.2020.1758482.
   Web of Science ®Google Scholar
• K. A. DOMINESEY and W. JI, “Reduced-Order Modeling of Neutron Transport Separated in Space and Angle via Proper Generalized Decomposition,” presented at the International Conference on Mathematics Computational Methods Applied to Nuclear Science and Engineering, August 25-29 (2019).
   Google Scholar
• K. A. DOMINESEY and W. JI, “Reduced-Order Modeling of Neutron Transport Separated in Energy by Proper Generalized Decomposition with Applications to Nuclear Reactor Physics,” J. Comput. Phys., 449, 110744 (2022); https://doi.org/10.1016/j.jcp.2021.110744.
   Web of Science ®Google Scholar
• J. KOU, S. L. CLAINCHE, and W. ZHANG, “A Reduced-Order Model for Compressible Flows with Buffeting Condition Using Higher Order Dynamic Mode Decomposition with a Mode Selection Criterion,” Phys. Fluids, 30, 1, 016103 (2018); https://doi.org/10.1063/1.4999699.
   Web of Science ®Google Scholar
• H. LU and D. M. TARTAKOVSKY, “Lagrangian Dynamic Mode Decomposition for Construction of Reduced-Order Models of Advection-Dominated Phenomena,” J. Comput. Phys., 407, 109229 (2020); https://doi.org/10.1016/j.jcp.2020.109229.
   Web of Science ®Google Scholar
• D. A. BISTRIAN and I. M. NAVON, “Randomized Dynamic Mode Decomposition for Nonintrusive Reduced Order Modelling,” Int. J. Numer. Methods Eng., 112, 1, 3 (2017); https://doi.org/10.1002/nme.5499.
   Web of Science ®Google Scholar
• Z. K. HARDY, J. E. MOREL, and C. AHRENS, “Dynamic Mode Decomposition for Subcritical Metal Systems,” Nucl. Sci. Eng., 193, 11, 1173 (2019); https://doi.org/10.1080/00295639.2019.1609317.
   Web of Science ®Google Scholar
• R. G. MCCLARREN, “Calculating Time Eigenvalues of the Neutron Transport Equation with Dynamic Mode Decomposition,” Nucl. Sci. Eng., 193, 8, 854 (2019); https://doi.org/10.1080/00295639.2018.1565014.
   Web of Science ®Google Scholar
• T. SAYADI et al., “Parametrized Data-Driven Decomposition for Bifurcation Analysis, with Application to Thermo-Acoustically Unstable Systems,” Phys. Fluids, 27, 3, 037102 (2015); https://doi.org/10.1063/1.4913868.
   Web of Science ®Google Scholar
• Q. HUNH, M. TANO, and J. C. RAGUSA, “Parametric Dynamic Mode Decomposition for Reduced Order Modeling,” J. Comput. Phys. (2022, accepted for publication).
   Google Scholar
• Z. BAI, “Krylov Subspace Techniques for Reduced-Order Modeling of Large-Scale Dynamical Systems,” Appl. Numer. Math., 43, 1, 9 (2002); https://doi.org/10.1016/S0168-9274(02)00116-2.
   Web of Science ®Google Scholar
• Y. YUE and K. MEERBERGEN, “Parametric Model Order Reduction of Damped Mechanical Systems via the Block Arnoldi Process,” Appl. Math. Lett., 26, 6, 643 (2013); https://doi.org/10.1016/j.aml.2013.01.006.
   Web of Science ®Google Scholar
• K. LEE and K. T. CARLBERG, “Model Reduction of Dynamical Systems on Nonlinear Manifolds Using Deep Convolutional Autoencoders,” J. Comput. Phys., 404, 108973 (2020); https://doi.org/10.1016/j.jcp.2019.108973.
   Web of Science ®Google Scholar
• B. A. KHUWAILEH and W. A. METWALLY, “Gaussian Processa Aproach for Dose Mapping in Radiation Fields,” Nucl. Eng. Technol., 52, 8, 1807 (2020); https://doi.org/10.1016/j.net.2020.01.013.
   Web of Science ®Google Scholar
• K. PEARSON, “On Lines and Planes of Closest Fit to Systems of Points in Space,” London, Edinburgh Dublin Philosophical Magazine J. Sci., 2, 11, 559 (1901); https://doi.org/10.1080/14786440109462720.
   Google Scholar
• D. T. CROMMELIN and A. J. MAJDA, “Strategies for Model Reduction: Comparing Different Optimal Bases,” J. Atmos. Sci., 61, 17, 2206 (Sep. 1, 2004); https://journals.ametsoc.org/view/journals/atsc/61/17/1520-0469_2004_061_2206_sfmrcd_2.0.co_2.xml (current as of Feb. 23,2022).
   Web of Science ®Google Scholar
• I. T. JOLLIFFE and J. CADIMA, “Principal Component Analysis: A Review and Recent Developments,” Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 374, 2065, 20150202 (2016); https://doi.org/10.1098/rsta.2015.0202.
   PubMed Web of Science ®Google Scholar
• C. W. ROWLEY and J. E. MARSDEN, “Reconstruction Equations and the Karhunen–Lo`eve Expansion for Systems with Symmetry,” Physica D, 142, 1, 1 (2000); https://doi.org/10.1016/S0167-2789(00)00042-7.
   Web of Science ®Google Scholar
• P. V. BAYLY et al., “Predicting Patterns of Epicardial Potentials During Ventricular Fibrillation,” IEEE Trans. Biomed. Eng., 42, 9, 898 (1995); https://doi.org/10.1109/10.412656.
   PubMed Web of Science ®Google Scholar
• E. KRUEZER and O. KUST, “Analysis of Long Torsional Strings by Proper Orthogonal Decomposition,” Arch. Appl. Math., 67, 68 (1996); https://doi.org/10.1007/BF00787141.
   Google Scholar
• M. AZEEZ and A. VAKAKIS, “Proper Orthogonal Decomposition (POD) of a Class of Vibroimpact Oscillations,” J. Sound Vib., 240, 5, 859 (2001); https://doi.org/10.1006/jsvi.2000.3264.
   Web of Science ®Google Scholar
• B. EPUREANU, K. HALL, and E. DOWELL, “Reduced-Order Models of Unsteady Viscous Flows in Turbomachinery Using Viscous-Inviscid Coupling,” J. Fluids Struct., 15, 2, 255 (2001); https://doi.org/10.1006/jfls.2000.0334.
   Web of Science ®Google Scholar
• M. RAJAEE, S. K. F. KARLSSON, and L. SIROVICH, “Low-Dimensional Description of Free-Shear-Flow Coherent Structures and Their Dynamical Behavior,” J. Fluid Mech., 258, 1 (1994); https://doi.org/10.1017/S0022112094003228.
   Web of Science ®Google Scholar
• J. BURKARDT, M. GUNZBURGER, and H.-C. LEE, “POD and CVT-based Reduced-Order Modeling of Navier–Stokes Flows,” Comput. Methods Appl. Mech. Eng., 196, 1, 337 (2006); https://doi.org/10.1016/j.cma.2006.04.004.
   Web of Science ®Google Scholar
• F. FANG et al., “A POD Reduced Order Unstructured Mesh Ocean Modelling Method for Moderate Reynolds Number Flows,” Ocean Modell., 28, 1, 127 (2009); https://doi.org/10.1016/j.ocemod.2008.12.006.
   Web of Science ®Google Scholar
• A. BAKHSHINEJAD et al., “Merging Computational Fluid Dynamics and 4D Flow MRI Using Proper Orthogonal Decomposition and Ridge Regression,” J. Biomech., 58, 162 (2017); https://doi.org/10.1016/j.jbiomech.2017.05.004.
   PubMed Web of Science ®Google Scholar
• K. K. NAGARAJAN et al., “Open-Loop Control of Cavity Noise Using Proper Orthogonal Decomposition Reduced-Order Model,” Comput. Fluids, 160, 1 (2018); https://doi.org/10.1016/j.compfluid.2017.10.019.
   Web of Science ®Google Scholar
• D. LI et al., “An Efficient Implementation of Aeroelastic Tailoring Based on Efficient Computational Fluid Dynamics-based Reduced Order Model,” J. Fluids Struct., 84, 182 (2019); https://doi.org/10.1016/j.jfluidstructs.2018.10.011.
   Web of Science ®Google Scholar
• T. BUI-THANH et al., “Goal-Oriented, Model-Constrained Optimization for Reduction of Large-Scale Systems,” J. Comput. Phys., 224, 2, 880 (2007); https://doi.org/10.1016/j.jcp.2006.10.026.
   Web of Science ®Google Scholar
• P. GERMAN and J. C. RAGUSA, “Reduced-Order Modeling of Parameterized Multi-Group Diffusion k-Eigenvalue Problems,” Ann. Nucl. Energy, 134, 144 (2019); https://doi.org/10.1016/j.anucene.2019.05.049.
   Web of Science ®Google Scholar
• A. G. BUCHAN et al., “A POD Reduced-Order Model for Eigenvalue Problems with Application to Reactor Physics,” Int. J. Numer. Methods Eng., 95, 12, 1011 (2013); https://doi.org/10.1002/nme.4533.
   Web of Science ®Google Scholar
• A. BUCHAN et al., “A POD Reduced Order Model for Resolving Angular Direction in Neutron/Photon Transport Problems,” J. Comput. Phys., 296, 138 (2015); https://doi.org/10.1016/j.jcp.2015.04.043.
   Web of Science ®Google Scholar
• A. C. HUGHES and A. G. BUCHAN, “A Discontinuous and Adaptive Reduced Order Model for the Angular Discretization of the Boltzmann Transport Equation,” Int. J. Numer. Methods Eng., 121, 24, 5647 (2020); https://doi.org/10.1002/nme.6516.
   Web of Science ®Google Scholar
• J. TENCER et al., “Reduced Order Modeling Applied to the Discrete Ordinates Method for Radiation Heat Transfer in Participating Media,” Proc. Heat Transfer Summer Conf. (Nov. 11, 2016); https://doi.org/10.1115/HT2016-7010.
   Google Scholar
• J. TENCER et al., “Accelerated Solution of Discrete Ordinates Approximation to the Boltzmann Transport Equation for a Gray Absorbing-Emitting Medium via Model Reduction,” J. Heat Transfer, 139, 12, 122701 (2017); https://doi.org/10.1115/1.4037098.
   Web of Science ®Google Scholar
• I. UDAGEDARA et al., “Reduced Order Modeling for Accelerated Monte Carlo Simulations in Radiation Transport,” Appl. Math. Comput., 267, 237 (2015); https://doi.org/10.1016/j.amc.2015.03.113.
   Web of Science ®Google Scholar
• Y. SUN et al., “A POD Reduced-Order Model for Resolving the Neutron Transport Problems of Nuclear Reactor,” Ann. Nucl. Energy, 149, 107799 (2020); https://doi.org/10.1016/j.anucene.2020.107799.
   Web of Science ®Google Scholar
• A. SARTORI et al., “Comparison of a Modal Method and a Proper Orthogonal Decomposition Approach for Multi-group Time-Dependent Reactor Spatial Kinetics,” Ann. Nucl. Energy, 71, 217 (2014); https://doi.org/10.1016/j.anucene.2014.03.043.
   Web of Science ®Google Scholar
• K. TSUJITA, T. ENDO, and A. YAMAMOTO, “Fast Reproduction of Time-Dependent Diffusion Calculations Using the Reduced Order Model Based on the Proper Orthogonal and Singular Value Decompositions,” J. Nucl. Sci. Technol., 58, 2, 173 (2021); https://doi.org/10.1080/00223131.2020.1814891.
   Web of Science ®Google Scholar
• J. M. COALE, “Reduced Order Models for Thermal Radiative Transfer Problems Based on Low-Order Transport Equations and the Proper Orthogonal Decomposition,” North Carolina State University ProQuest Dissertations Publishing. (2019).
   Google Scholar
• R. PINNAU, Model Reduction via Proper Orthogonal Decomposition, pp. 95–109, Springer Berlin Heidelberg, Berlin, Heidelberg (2008); https://doi.org/10.1007/978-3-540-78841-65.
   Google Scholar
• P. A. BEHNE, J. C. RAGUSA, and J. E. MOREL, “Model-Order Reduction Approaches for SN Radiation Transport,” Proceedings, International Conference on Mathematics Computational Methods Applied to Nuclear Science and Engineering, Portland, OR, August 25-29, 2019, (2019).
   Google Scholar
• P. BEHNE, J. VERMAAK, and J. C. RAGUSA, “Minimally-Invasive Parametric Model-Order Reduction for Sweep-Based Radiation Transport,” J. Comput. Phys., 469 (2022).
   Web of Science ®Google Scholar
• P. BENNER, S. GUGERCIN, and K. WILLCOX, “A Survey of Projection-based Model Reduction Methods for Parametric Dynamical Systems,” SIAM Rev., 57, 4, 483 (2015); https://doi.org/10.1137/130932715.
   Web of Science ®Google Scholar
• C. E. HEANEY et al., “Reduced-Order Modelling Applied to the Multigroup Neutron Diffusion Equation Using a Nonlinear Interpolation Method for Control-Rod Movement,” Energies, 14, 5, 1350 (2021); https://doi.org/10.3390/en14051350.
   Web of Science ®Google Scholar
• P. A. BEHNE, J. C. RAGUSA, and M. TANO, “Projection-based Model Order Reduction Based on Affine Decomposition of the Operators,” Proceedings, International Conference on Mathematics Computational Methods Applied to Nuclear Science and Engineering, October 3-7, 2021 (2021).
   Google Scholar
• M. TANO et al., “Affine Reduced-Order Model for Radiation Transport Problems in Cylindrical Coordinates,” Ann. Nucl. Energy, 158, 108214 (2021); https://doi.org/10.1016/j.anucene.2021.108214.
   Web of Science ®Google Scholar
• T. BUI-THANH, K. WILLCOX, and O. GHATTAS, “Parametric Reduced-Order Models for Probabilistic Analysis of Unsteady Aerodynamic Applications,” AIAA J., 46, 10, 2520 (2008); https://doi.org/10.2514/1.35850.
   Google Scholar
• W. R. D. BOYD III, “Reactor Agnostic Multi-Group Cross Section Generation for Fine-Mesh Deterministic Neutron Transport Simulations,” PhD Thesis, Massachusetts Institute of Technology, Department of Nuclear Science and Engineering (2017).
   Google Scholar
• T. R. HILL, “ONETRAN: A Discrete Ordinates Finite Element Code for the Solution of the One-Dimensional Multigroup Transport Equation,” Los Alamos National Laboratory (1975).
   Google Scholar
• W. H. REED and T. R. HILL, “Triangular Mesh Methods for the Neutron Transport Equation,” Los Alamos Scientific Laboratory (1973).
   Google Scholar
• T. A. WAREING et al., “Discontinuous Finite Element SN Methods on Three-Dimensional Unstructured Grids,” Nucl. Sci. Eng., 138, 3, 256 (2001); https://doi.org/10.13182/NSE138-256.
   Web of Science ®Google Scholar
• T. BUI-THANH, K. WILLCOX, and O. GHATTAS, “Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space,” SIAM J. Sci. Comput., 30, 6, 3270 (2008); https://doi.org/10.1137/070694855.
   Web of Science ®Google Scholar
• B. PEHERSTORFER and K. WILLCOX, “Data-driven Operator Inference for Nonintrusive Projection-based Model Reduction,” Comput. Methods Appl. Mech. Eng., 306, 196 (2016); https://doi.org/10.1016/j.cma.2016.03.025.
   Web of Science ®Google Scholar
• K. KUNISCH and S. VOLKWEIN, “Optimal Snapshot Location for Computing POD Basis Functions,” ESAIM: Math. Model. Numer. Anal., 44, 3, 509 (2010); https://doi.org/10.1051/m2an/2010011.
   Web of Science ®Google Scholar
• T. A. BRUNNER, “Forms of Approximate Radiation Transport,” Technical report, Sandia National Laboratories (2002).
   Google Scholar