Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation
Wolfgang Dahmen, Olga Mula
TL;DR
This work addresses the principal eigenpair of a generalized Boltzmann (neutron transport) operator arising in the nuclear criticality problem, aiming for certifiable numerical accuracy without assuming excess regularity. It proposes a novelty: instead of a fixed discretization, construct outer iterations in function space (Newton or power) and realize each step approximately with computable a posteriori bounds, leveraging a stable ultraweak variational formulation. The key contributions include a rigorous two-tier framework (idealized Newton and perturbed, accuracy-controlled realizations), detailed perturbation and convergence analysis in function space, and a posteriori error mechanisms that guide adaptive numerical realization. The findings provide a principled path toward certifiable eigensolvers for radiative transfer models, with practical implications for reactor design and safety through quantifiable solution accuracy, albeit acknowledging intrinsic obstacles such as uncertain spectral gaps and nonnormal operator behavior.
Abstract
The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus.