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

该提议的技术基础与中子传输方程（NTE）有关，该方程用于描述核裂变发生的物理环境中的中子密度，例如反应堆堆芯。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情景的模型。这些模型主要用于评估安全性，并为处理放射性物质时的监管程序提供信息。虽然NTE可以通过对质量输运的物理考虑推导出来，但它也可以完全使用概率方法推导出来。更精确地说，NTE可以从空间分支过程的随机分析中得到。后者模拟了中子粒子在现实中的演化行为，结合了随机散射和随机裂变的特征，随着时间的推移粒子数量增加。使用空间分支过程的推导自20世纪60/70年代以来就已经知道，然而，从那时起，通过概率分析出现的文献创新很少。这反映了20世纪80年代以后，基础数学研究在核裂变建模方面的普遍停滞。然而，近年来，核电和核监管行业更需要深入了解NTE的光谱特性。这样的分析量可以帮助工程师模拟反应堆堆芯内核裂变活动的临界和密度。反过来，这从几个不同的角度（安全、能源生产、效率等）为优化反应堆设计提供了信息，同时也解决了监管限制问题。随着旧核电站的退役和更高效、更环保的新核电站的建设，对使用NTE进行数学建模的需求空前高涨。NTE在实践中使用的非同质性使得行业转向基于40-50年前潜在概率处理的蒙特卡洛技术。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。近几十年来，在NTE的研究中没有出现新的概率发展，但空间分支过程和相关随机过程的数学理论有了重大发展。本研究旨在将对NTE的理解与空间分支过程的现代理论重新结合起来。这主要是由于一种暗示，即一整套全新的蒙特卡罗技术可以开发出来，正如工业界所希望的那样，从根本上说，这些技术比现有算法的复杂性要低。该项目的总体目标是为这种全新的方法开发“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，即将出现的新算法最重要的特征是并行计算的能力。该项目将与工业伙伴Amec-Foster-Wheeler进行密切的科学合作，Amec-Foster-Wheeler是一家总部位于英国的主要能源咨询公司，也是为核能和核医学工业提供安全和监管模拟软件的全球领导者之一。所有的研究成果都将在一个专门用于该项目的网页上公开。

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.