# Greedy method for the model order reduction in neutronics : application of the reduced basis method

We are interested in a methodology that perform a computation in a very short amount of time while preserving the accuracy. A reduced basis approach could meet this requirement.

In the framework of the reduced basis methods [1,3], we devise an approximation space associated to a parameter-dependent partial differential equation. The construction of this approximation space includes a phase of exploration of the space of parameters where it is important to quantify the error between the solution obtained from the approximation space (in construction) and the solution obtained from a standard (fine) discretization.

This crucial step allows to certify the construction of the reduced basis.

Recently, some research work in the laboratory have provided a posteriori error estimator in the context of neutronics [4].

In this context [2], we are interested in generalized non-symmetric eigenvalue problems. Typically, we consider a linear Boltzmann operator of the form:

Find (u, v) such that Lu = Hu + v Fu,

where Lu is the advection operator, Hu is the scattering operator that modelize the collisions of the neutrons, Fu is the fission operator and the unknown u represents the neutron flux. The equation is also called the neutron transport equation. The fact the operator is not symetric comes form the scattering operator.

A first implementation of the reduced basis method based on the Proper Orthogonal Decomposition has been made for the neutron diffusion model in the APOLLO3® code [5]. Reduced basis methods have been studied for the neutron diffusion model [6-8] and the neutron transport model [9-14] with a varying degree of intrusivity.

The objective of this thesis is to contribute to the construction of greedy reduced basis methods for a model of neutronics, especially on assembling the reduced problem and the computation of an a posteriori estimator based on an affine decomposition of the operator. In a second step, many possibilities may be investigated :

- The extension of the reduced basis method to the simplified transport;

- The extension of the reduced basis method to the transport model;

- The application to the loading pattern of a research reactor.

[1] Y. Maday, O. Mula, A generalized empirical interpolation method: application of reduced basis techniques to data assimilation. Analysis and Numerics of Partial Differential Equations, XIII:221-231,2013.

[2] O. Mula, Some contributions towards the parallel simulation of time dependent neutron transport and the integration of observed data in real time, Chapter 1, 2014.

[3] G. Rozza, D. Huynh, and A. Patera, “Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations,” Archives of Computational Methods in Engineering, vol. 15, no. 3, pp. 1–47, 2008.

[4] Y. Conjungo Taumhas, G. Dusson, V. Ehrlacher, T. Lelièvre, F. Madiot. Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations. 2023. ?HAL cea-04156959?

[5] Y. Conjungo Taumhas, F. Madiot, T. Lelièvre, V. Ehrlacher, and G. Dusson. An Application of Reduced Basis Methods to Core Computation in APOLLO3®. M&C 2023

[6] Sartori, A. Cammi, L. Luzzi, M. E. Ricotti, and G. Rozza. Reduced order methods: applications to nuclear reactor core spatial dynamics.15566, in ICAPP 2015 Proceedings, 2015.

[7] S. Lorenzi, An adjoint proper orthogonal decomposition method for a neutronics reduced order model, Annals of Nuclear Energy, 114 (2018), pp. 245–

258.

[8] P. German and J. C. Ragusa, Reduced-order modeling of parameterized multi-group diffusion k-eigenvalue problems, Annals of Nuclear Energy, 134

(2019), pp. 144–157

[9] I Halvic, JC Ragusa. Non-intrusive model order reduction for parametric radiation transport simulations. Journal of Computational Physics 492 (2023), 112385

[10] P Behne, J Vermaak, J Ragusa. Parametric Model-Order Reduction for Radiation Transport Simulations Based on an Affine Decomposition of the Operators. Nuclear Science and Engineering 197 (2), 233-261 (2023)

[11] P Behne, J Vermaak, JC Ragusa. Minimally-invasive parametric model-order reduction for sweep-based radiation transport. Journal of Computational Physics 469, 111525

[12] Z Peng, Y Chen, Y Cheng, F Li. A reduced basis method for radiative transfer equation. Arxiv preprint (2021).

[13] Sun, Y., Yang, J., Wang, Y., Li, Z., & Ma, Y. (2020). A POD reduced-order model for resolving the neutron transport problems of nuclear reactor. Annals of Nuclear Energy, 149, 107799.

[14] Wei, C., Di, Y., Junjie, Z., Chunyu, Z., Helin, G., Bangyang, X., ... & Lianjie, W. (2021). Study of non-intrusive model order reduction of neutron transport problems. Annals of Nuclear Energy, 162, 108495.