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Hyper‐Reduced Model Based on the Proper Orthogonal Decomposition and the LU Factorization Applied to the Neutron Diffusion Eigenvalue Problem.

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Title: Hyper‐Reduced Model Based on the Proper Orthogonal Decomposition and the LU Factorization Applied to the Neutron Diffusion Eigenvalue Problem.
Authors: Vidal‐Ferràndiz, A.1 (AUTHOR) anvifer2@upv.es, Carreño, A.2 (AUTHOR), Javaloyas, E.2 (AUTHOR), Ginestar, D.1 (AUTHOR), Verdú, G.2 (AUTHOR)
Source: International Journal for Numerical Methods in Engineering. 2/15/2026, Vol. 127 Issue 3, p1-20. 20p.
Subjects: Proper orthogonal decomposition, Neutron diffusion, Reduced-order models, Matrix decomposition, Eigenvalues, Optimization algorithms
Abstract: An efficient method for solving large eigenvalue problems efficiently can be developed using hyper‐reduced order models, such as those arising from the LU Proper Orthogonal Decomposition (LUPOD). The LUPOD employs dominant orthogonal modes along with a flexible number of collocation points to establish a reduced scalar product, thereby enhancing computational efficiency to construct the reduced order model. This strategy produces accurate results when a similar number of collocation points and orthogonal modes are used. For problems where the number of available modes is small due to the high computational cost of its computation or because sufficient model accuracy can be achieved with less data, LUPOD may not yield enough precise results. This work proposes an extension of the LUPOD method to increase the number of collocation points in the reduced scalar product and consequently, the accuracy of the LUPOD. The performance of this method is illustrated with a diffusion‐reaction problem with analytical solution. Then, this method is applied to solve the neutron diffusion eigenvalue problem. Numerical results for these methodologies are tested for obtaining the k‐effective and the steady‐state neutron flux distribution of three‐dimensional nuclear reactor benchmark problems. They show that using 20 snapshots and 40% of the mesh points of the problem as collocation points speeds up the simulations by about 100 times with respect to the resolution of the full order eigenvalue problem while maintaining accurate results that differ by less than 100 pcm in the eigenvalue and less than 1.2% in the neutron flux determination. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:An efficient method for solving large eigenvalue problems efficiently can be developed using hyper‐reduced order models, such as those arising from the LU Proper Orthogonal Decomposition (LUPOD). The LUPOD employs dominant orthogonal modes along with a flexible number of collocation points to establish a reduced scalar product, thereby enhancing computational efficiency to construct the reduced order model. This strategy produces accurate results when a similar number of collocation points and orthogonal modes are used. For problems where the number of available modes is small due to the high computational cost of its computation or because sufficient model accuracy can be achieved with less data, LUPOD may not yield enough precise results. This work proposes an extension of the LUPOD method to increase the number of collocation points in the reduced scalar product and consequently, the accuracy of the LUPOD. The performance of this method is illustrated with a diffusion‐reaction problem with analytical solution. Then, this method is applied to solve the neutron diffusion eigenvalue problem. Numerical results for these methodologies are tested for obtaining the k‐effective and the steady‐state neutron flux distribution of three‐dimensional nuclear reactor benchmark problems. They show that using 20 snapshots and 40% of the mesh points of the problem as collocation points speeds up the simulations by about 100 times with respect to the resolution of the full order eigenvalue problem while maintaining accurate results that differ by less than 100 pcm in the eigenvalue and less than 1.2% in the neutron flux determination. [ABSTRACT FROM AUTHOR]
ISSN:00295981
DOI:10.1002/nme.70272