Stochastic analysis of the neutron transport equation and applications to nuclear safety

该提议的技术基础涉及中子输运方程（NTE），该方程用于描述发生核裂变的物理环境（例如反应堆堆芯）中的中子密度。该方程在核工业中至关重要，因为它用于构建反应堆堆芯、核医疗设备（例如质子治疗）和发生辐照的其他工业场景的模型。这些模型主要用于评估安全性，并在处理放射性材料时告知监管程序。虽然NTE可以通过物质输运的物理考虑推导出来，但它也可以使用完全概率的方法推导出来。更准确地说，NTE可以从空间分支过程的随机分析中推导出来。后者模拟了中子粒子在现实中的演化，结合了随机散射和随机裂变的特征，随着时间的推移，粒子的数量不断增加。自20世纪60/70年代以来，使用空间分支过程的推导就已经为人所知，然而，从那时起，通过概率分析在文献中出现了很少的创新。这反映了20世纪80年代后基础数学研究对核裂变建模的普遍停滞。然而，近年来，核电和核监管行业对深入了解NTE的光谱特性有了更大的需求。这样的分析量帮助例如工程师对反应堆堆芯内的核裂变活动的临界性和密度进行建模。反过来，这从几个不同的角度（安全，能源生产，效率等）通知最佳反应堆设计。以及解决监管限制。随着旧核电站的退役和新的、更高效、更环保的核电站的建设，对使用NTE进行数学建模的需求从未如此之大。NTE在实际应用中的不均匀性使得工业界转向基于40-50年前的潜在概率处理的蒙特-卡罗技术。许多相关的算法只能在超级计算机上运行，因为它们归结为整个裂变过程的昂贵的蒙特-卡罗循环，本质上是在计算机中复制虚拟的物理现实。这有一个巨大的缺点，即计算并行化是不可能的。在几十年来，新的概率发展已经缺席的治疗NTE，有一个显着的演变，在数学理论的空间分支过程和相关的随机过程。本提案中的研究旨在重新调整对NTE的理解与空间分支过程的现代理论。这主要是出于这样的含义，即可以开发出一整套全新的蒙特-卡罗技术，如工业所期望的，这些技术从根本上说比现有算法的复杂度低。该项目的总体目标是为这一全新的方法开发“概念验证”，提供理论基础和量化相对效率的随机数值分析。特别是，将出现的新算法的最重要特征是并行计算的能力。该项目将与工业合作伙伴Amec-Foster-惠勒密切科学合作，一个主要的英国-为核能和核医疗行业提供安全和监管模拟软件的全球领导者之一。所有研究成果都将公开来源于一个专门用于该项目的网页。

The technical basis of this proposal pertains to the Neutron Transport Equation (NTE), which is used to describe neutron density in a physical environment where nuclear fission is taking place, such as a reactor core. This equation is of prime importance in the nuclear industry as it is used to construct models of reactor cores, nuclear medical equipment (e.g. for proton therapy) and other industrial scenarios where irradiation occurs. Primarily these models are used to assess safety and inform regulatory procedure when handling radioactive materials. Although the NTE can be derived through physical considerations of mass transport, it can also be derived using entirely probabilistic means. To be more precise, the NTE can be derived from the stochastic analysis of a spatial branching process. The latter models the evolution of neutron particles as they behave in reality, incorporating the features of random scattering and random fission, with increasing numbers of particles as time evolves. The derivation using spatial branching processes has been known since the 1960/70s, however, since then, very little innovation in the literature has emerged through probabilistic analysis. This mirrors a general lull in fundamental mathematical research contributing to modelling of nuclear fission after the 1980s. In recent years, however, the nuclear power and nuclear regulatory industries have a greater need for a deep understanding the spectral properties of the NTE. Such analytical quantities help e.g. engineers model the criticality and density of nuclear fission activity within a reactor core. In turn this informs optimal reactor design from several different view points (safety, energy production, efficiency etc.) as well as address regulatory constraints. With the decommissioning of old and the construction of new, more efficient and environmentally friendly nuclear power stations the demand for mathematical modelling using the NTE was never greater. The inhomogeneous nature of the NTE as it is used in practice has seen industry turn to Monte-Carlo techniques based on the underlying probabilistic treatment from 40-50 years ago. Many of the associated algorithms can only be run on supercomputers as they boil down to costly Monte-Carlo cycles of the entire fission processes, in essence replicating a virtual physical reality in a computer. This has the huge drawback that computational parallelization is not possible. In the decades that new probabilistic developments have been absent from the treatment of the NTE, there has been a significant evolution in the mathematical theory of spatial branching processes and related stochastic processes. The research in this proposal aims to re-align the understanding of the NTE with the modern theory of spatial branching processes. This is principally motivated by the implication that a whole suite of completely new Monte-Carlo techniques can be developed, as desired by industry, which are, fundamentally, of a lower order of complexity than existing algorithms. The overall aim of this project is to develop a `proof of concept' for this completely new approach, providing the theoretical basis and a stochastic numerical analysis that quantifies relative efficiency. In particular, the most important feature of the new algorithms that will emerge is the ability to parallelize computations.The project will be carried out in close scientific collaboration with industrial partner Amec-Foster-Wheeler, a major UK-based energy consultancies and one of the global leaders in servicing the nuclear energy and nuclear medical industries with simulation software for safety and regulatory purposes.All research output will be made open source on a webpage dedicated to the project.