Mathematical Resolution of Ill-Posedness in Padé-Based Subgroup Parameter Calculation via Regularization and Pole–Residue Constraints
- 1Naval University of Engineering, Wuhan 430033, China
- 2China Nuclear Data Center, China Institute of Atomic Energy, Beijing 102413, China
Abstract
The Padé-based method for subgroup parameter calculation in resonance self-shielding analysis is ill-posed: the truncated overdetermined system, the Vandermonde-type coefficient matrix, and the unenforced pole–residue structure jointly produce unstable poles and subgroup parameters that violate non-negativity or the probability normalization. This paper proposes the regularized overdetermined Padé approximation (ROPA), a two-stage, structure-preserving procedure that resolves this ill-posedness at the level of the mathematical problem rather than the numerical workflow. In stage one, the overdetermined least-squares Padé system is built on a normalized background cross-section variable with column equilibration and Tikhonov regularization, the poles are computed as eigenvalues of a companion matrix, and a reflection gate maps them onto the positive real axis. In stage two, the subgroup probabilities are recovered by a convex quadratic program enforcing non-negativity and unit sum, with non-negative least squares for partial probability channels. The well-posedness of the regularized problem is established theoretically, including an explicit stability bound with respect to perturbations of the resonance integral data. Validation against OpenMC Monte Carlo references on ENDF/B-VIII.0 data for 235U, 238U, and 240Pu over 16 resonance groups, together with VERA pin-cell and lattice benchmarks, demonstrates that ROPA achieves accuracy comparable to or higher than existing methods while guaranteeing admissible parameters in all tested cases.
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BibTeX
@article{li2026mathematical,
title = {{Mathematical Resolution of Ill-Posedness in Padé-Based Subgroup Parameter Calculation via Regularization and Pole–Residue Constraints}},
author = {Song Li and Lei Liu and Yongfa Zhang and Jianli Hao and Zouzhe Li and Ying Chen},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193636},
url = {https://www.mdpi.com/2227-7390/14/19/3636},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Mathematical Resolution of Ill-Posedness in Padé-Based Subgroup Parameter Calculation via Regularization and Pole–Residue Constraints
AU - Song Li
AU - Lei Liu
AU - Yongfa Zhang
AU - Jianli Hao
AU - Zouzhe Li
AU - Ying Chen
PY - 2026
DA - 2026/10/08
JO - Mathematics
DO - 10.3390/math14193636
UR - https://www.mdpi.com/2227-7390/14/19/3636
AB - The Padé-based method for subgroup parameter calculation in resonance self-shielding analysis is ill-posed: the truncated overdetermined system, the Vandermonde-type coefficient matrix, and the unenforced pole–residue structure jointly produce unstable poles and subgroup parameters that violate non-negativity or the probability normalization. This paper proposes the regularized overdetermined Padé approximation (ROPA), a two-stage, structure-preserving procedure that resolves this ill-posedness at the level of the mathematical problem rather than the numerical workflow. In stage one, the overdetermined least-squares Padé system is built on a normalized background cross-section variable with column equilibration and Tikhonov regularization, the poles are computed as eigenvalues of a companion matrix, and a reflection gate maps them onto the positive real axis. In stage two, the subgroup probabilities are recovered by a convex quadratic program enforcing non-negativity and unit sum, with non-negative least squares for partial probability channels. The well-posedness of the regularized problem is established theoretically, including an explicit stability bound with respect to perturbations of the resonance integral data. Validation against OpenMC Monte Carlo references on ENDF/B-VIII.0 data for 235U, 238U, and 240Pu over 16 resonance groups, together with VERA pin-cell and lattice benchmarks, demonstrates that ROPA achieves accuracy comparable to or higher than existing methods while guaranteeing admissible parameters in all tested cases.
SN - 2227-7390
ER -