A multi-fidelity adaptive dynamical low-rank based optimization algorithm for fission criticality problems

TL;DR

This work develops a rank-adaptive dynamical low-rank (DLRA) algorithm to compute the dominant eigenpair $(k_{ ext{eff}}, \phi)$ in multigroup fission-criticality models, reducing memory and computation by representing the neutron flux as a low-rank factorization. The key novelty is a rank-adaptive inverse-power iteration that progressively increases the rank via a change-of-rank (CR) mechanism and tolerance scheduling, enabling accurate results with limited resources, and an extension to low-rank optimization over design parameters to reach target criticality. The method is demonstrated on plutonium and uranium-steel sphere benchmarks and a light-water reactor, showing significant reductions in average rank and computational cost while preserving high accuracy. These results support efficient, multi-query criticality analyses and reactor-design optimization in nuclear engineering.

Abstract

Computing the dominant eigenvalue is important in nuclear systems as it determines the stability of the system (i.e. whether the system is sub or supercritical). Recently, the work of Kusch, Whewell, McClarren and Frank \cite{KWMF} showed that performing a low-rank approximation can be very effective in reducing the high memory requirement and computational cost of such problems. In this work, we propose a rank adaptive approach that changes the rank during the inverse power iteration. This allows us to progressively increase the rank (i.e. changing the fidelity of the model) as we get closer to convergence, thereby further reducing computational cost. We then exploit this multi-fidelity approach to optimize a simplified nuclear reactor. In this case the system is parameterized and the values of the parameters that give criticality are sought.

Figures (7)

• Figure 1: Numerical results for a plutonium sphere, with fixed increase of rank $\kappa=1$. The left figure shows the behaviors of $\Delta_n$ and of the errors in both the eigenvalue and the eigenmatrix. In the right plot, the adaptation of the rank is presented.
• Figure 2: Numerical results of stainless-steel reflected uranium sphere with the rank adaptive algorithm for different values of $\rho$.
• Figure 3: Numerical results of stainless-steel reflected uranium sphere, with adaptation with some values of $\beta$ and $\rho= 0.5$.
• Figure 4: Numerical results of light water ractor.
• Figure 5: Computational results for the fixed rank gradient optimization. The computational cost of any gradient step is computed as the sum of all the previous low-rank power method iterations.

Theorems & Definitions (2)

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