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：核工程中的临界性问题要求求解描述反应堆堆芯中子输运的玻尔兹曼算子的主特征对。为了可靠地设计和控制此类反应堆，需要在可量化的精度容差范围内评估这些特征量。本文提出了一种有别于常规实践的研究范式：传统方法通常采用固定且假定足够精细的离散化方案来近似求解相应的谱问题，而本方法首先构建在函数空间中表述的迭代格式，该格式被证明能在不假设任何先验超正则性条件下以定量速率收敛，且仅利用底层辐射传输模型中光学参数的性质。我们发展了分析和数值工具，用于在通过后验估计验证的、合理选取的精度容差范围内近似实现每个迭代步骤，从而仍能保证向精确特征对的可量化收敛。我们首先完整实现了牛顿法的这一过程。由于该方法仅具局部收敛性，我们还分析了函数空间中幂迭代的收敛性以生成足够精确的初始猜测值。在此过程中，我们必须处理由紧致但非对称算子带来的本质困难，这些困难阻碍了有限维情形中标准论证方法的应用。我们的核心观点在于：可以避免对初始猜测值必须位于精确解小邻域内的任何限制条件。最后，我们讨论了可验证数值实现中剩余的本质性障碍，主要涉及主特征值与模长次小特征值之间的间隙未知问题。