Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines
Abstract
In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadershipclass computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates illconditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadershipclass machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. Thesemore »
 Authors:

 Univ. of California, Berkeley, CA (United States). Nuclear Engineering Dept.
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Radiation Transport and Criticality Group
 Publication Date:
 Research Org.:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF); Univ. of California, Berkeley, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC); USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1427595
 Grant/Contract Number:
 AC0500OR22725
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Nuclear Science and Engineering
 Additional Journal Information:
 Journal Volume: 190; Journal Issue: 1; Journal ID: ISSN 00295639
 Publisher:
 American Nuclear Society  Taylor & Francis
 Country of Publication:
 United States
 Language:
 English
 Subject:
 22 GENERAL STUDIES OF NUCLEAR REACTORS; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 42 ENGINEERING; eigenvalue; Rayleigh quotient; preconditioning
Citation Formats
Slaybaugh, R. N., RamirezZweiger, M., Pandya, Tara, Hamilton, Steven, and Evans, T. M. Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines. United States: N. p., 2018.
Web. doi:10.1080/00295639.2017.1413875.
Slaybaugh, R. N., RamirezZweiger, M., Pandya, Tara, Hamilton, Steven, & Evans, T. M. Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines. United States. doi:10.1080/00295639.2017.1413875.
Slaybaugh, R. N., RamirezZweiger, M., Pandya, Tara, Hamilton, Steven, and Evans, T. M. Tue .
"Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines". United States. doi:10.1080/00295639.2017.1413875. https://www.osti.gov/servlets/purl/1427595.
@article{osti_1427595,
title = {Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines},
author = {Slaybaugh, R. N. and RamirezZweiger, M. and Pandya, Tara and Hamilton, Steven and Evans, T. M.},
abstractNote = {In this paper, three complementary methods have been implemented in the code Denovo that accelerate neutral particle transport calculations with methods that use leadershipclass computers fully and effectively: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid in energy (MGE) preconditioner. The MG Krylov solver converges more quickly than Gauss Seidel and enables energy decomposition such that Denovo can scale to hundreds of thousands of cores. RQI should converge in fewer iterations than power iteration (PI) for large and challenging problems. RQI creates shifted systems that would not be tractable without the MG Krylov solver. It also creates illconditioned matrices. The MGE preconditioner reduces iteration count significantly when used with RQI and takes advantage of the new energy decomposition such that it can scale efficiently. Each individual method has been described before, but this is the first time they have been demonstrated to work together effectively. The combination of solvers enables the RQI eigenvalue solver to work better than the other available solvers for large reactors problems on leadershipclass machines. Using these methods together, RQI converged in fewer iterations and in less time than PI for a full pressurized water reactor core. These solvers also performed better than an Arnoldi eigenvalue solver for a reactor benchmark problem when energy decomposition is needed. The MG Krylov, MGE preconditioner, and RQI solver combination also scales well in energy. Finally, this solver set is a strong choice for very large and challenging problems.},
doi = {10.1080/00295639.2017.1413875},
journal = {Nuclear Science and Engineering},
number = 1,
volume = 190,
place = {United States},
year = {2018},
month = {2}
}
Figures / Tables:
Works referenced in this record:
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
journal, July 1986
 Saad, Youcef; Schultz, Martin H.
 SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3
Inverse Iteration, IllConditioned Equations and Newton’s Method
journal, July 1979
 Peters, G.; Wilkinson, J. H.
 SIAM Review, Vol. 21, Issue 3
Application of Quadruple Range Quadratures to ThreeDimensional Model Shielding Problems
journal, November 2009
 Jarrell, Joshua J.; Adams, Marvin L.; Risner, Joel M.
 Nuclear Technology, Vol. 168, Issue 2
The Rayleigh quotient iteration and some generalizations for nonnormal matrices
journal, September 1974
 Parlett, B. N.
 Mathematics of Computation, Vol. 28, Issue 127
Fast iterative methods for discreteordinates particle transport calculations
journal, January 2002
 Adams, Marvin L.; Larsen, Edward W.
 Progress in Nuclear Energy, Vol. 40, Issue 1
Denovo: A New ThreeDimensional Parallel Discrete Ordinates Code in SCALE
journal, August 2010
 Evans, Thomas M.; Stafford, Alissa S.; Slaybaugh, Rachel N.
 Nuclear Technology, Vol. 171, Issue 2
An overview of the Trilinos project
journal, September 2005
 Heroux, Michael A.; Phipps, Eric T.; Salinger, Andrew G.
 ACM Transactions on Mathematical Software, Vol. 31, Issue 3
Modified GramSchmidt (MGS), Least Squares, and Backward Stability of MGSGMRES
journal, January 2006
 Paige, Christopher C.; Rozlozník, Miroslav; Strakos, Zdenvek
 SIAM Journal on Matrix Analysis and Applications, Vol. 28, Issue 1
The inverse power method for calculation of multiplication factors
journal, May 2002
 Allen, E. J.; Berry, R. M.
 Annals of Nuclear Energy, Vol. 29, Issue 8
Preconditioning Techniques for Large Linear Systems: A Survey
journal, November 2002
 Benzi, Michele
 Journal of Computational Physics, Vol. 182, Issue 2
Multigrid in energy preconditioner for Krylov solvers
journal, June 2013
 Slaybaugh, R. N.; Evans, T. M.; Davidson, G. G.
 Journal of Computational Physics, Vol. 242
An S _{n} Algorithm for the Massively Parallel CM200 Computer
journal, March 1998
 Baker, Randal S.; Koch, Kenneth R.
 Nuclear Science and Engineering, Vol. 128, Issue 3
Massively Parallel, ThreeDimensional Transport Solutions for the k Eigenvalue Problem
journal, June 2014
 Davidson, Gregory G.; Evans, Thomas M.; Jarrell, Joshua J.
 Nuclear Science and Engineering, Vol. 177, Issue 2
Figures / Tables found in this record: