The criticality problem in nuclear engineering asks for the principal eigen-pair 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 withing judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigen-pair. 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.

翻译：核工程中的临界问题要求计算描述反应堆堆芯中子输运的玻尔兹曼算子的主特征对。能够可靠地设计并控制此类反应堆，需要在可量化的精度容差内评估这些量。本文提出一种新范式，偏离了传统做法——即采用固定且假定足够精细的离散化来近似求解相应的谱问题。相反，本方法基于首先在函数空间中构造迭代格式，这些格式被证明在无需任何先验超正则性假设的情况下以量化速率收敛，且仅利用底层辐射传输模型中光学参数的性质。我们开发了分析和数值工具，通过在每次迭代步骤中审慎选择由后验估计验证的精度容差来近似实现该迭代步骤，从而仍能保证可量化收敛至精确特征对。这一过程首先针对牛顿格式完整展开。由于该格式仅具有局部收敛性，我们额外分析了函数空间中的幂迭代收敛性，以产生足够精确的初始猜测。在此过程中，我们必须处理紧致但非对称算子带来的固有困难——这些困难使得有限维情形中的标准论证失效。本文的核心论点在于：我们能够避免对初始猜测需位于精确解小邻域内的任何条件。最后，我们讨论了可认证数值实现中尚存的固有障碍，主要涉及主特征值与按模长次大特征值之间间隙未知的问题。