Multilevel Monte Carlo Methods for Elliptic Problems with Applications to Radioactive Waste Disposal

我们建议对不确定参数问题的有效方法进行基础数学研究，并将其应用于放射性废物处理。英国政府的核电政策指出，核电是一种经过验证的低碳发电技术，应该成为英国未来能源供应的一部分。如果在放射性废物问题上取得足够进展，能源公司将被允许建造新的核电站。与其他国家一样，地质处置是英国长期处理放射性废物的首选方案。地质处置的安全论证是一项重大的科学工作。国家和国际研究方案已使人们对放射性核素可能返回人类环境的机制及其一旦返回的后果有了很好的了解。其中一个突出的挑战是如何处理地质系统固有的不确定性，以及长期以来储存库的演变，这是拟议研究的核心。如果放射性核素从储存库中逸出，它们可能返回环境的主要机制是由地下岩石中的地下水输送。模拟这种流动的数学方程是很容易理解的，但是为了求解它们并预测放射性核素的运输，必须指定储存库周围各处岩石的渗透率和孔隙度。只有在相对较少的地点才能测量这些数量。其他地方的值必须推断出来，这不可避免地会产生不确定性。在早期的性能评估中，主要由于计算成本的原因，使用了相对初级的方法来处理这些不确定性。从那时起，在计算机硬件和不确定性量化的数学领域有了相当大的进步。量化不确定性最常用的方法之一是使用概率技术。这意味着流动方程中的系数将被建模为随机场，从而导致具有随机系数的偏微分方程（stochastic PDEs），并且与它们的确定性等价物相比，求解这些方程要困难得多，计算量也要大得多。近年来出现了许多快速收敛的随机偏微分方程求解技术，这些技术适用于不确定性可以用少量随机参数很好地逼近的情况。然而，现场数据的证据表明，在储存库安全案例中，需要大量的随机参数来捕捉系统中的不确定性。在这种情况下，目前只有蒙特卡罗（MC）采样和平均方法是可行的，这些方法的收敛速度相对较慢是一个主要问题。在这里提出的工作中，我们将开发和分析一种新的令人兴奋的方法来加速随机偏微分方程MC模拟的收敛。多层MC方法将确定性偏微分方程的多网格思想与经典MC方法相结合。我们预测这种方法在计算成本上的巨大节省源于这样一个事实，即大多数工作可以在计算成本低廉的粗糙空间网格上完成。只有很少的样本需要在更细的网格上进行计算，以获得必要的空间精度。这种方法已经（由一位pi）成功地应用于数学金融中的随机常微分方程。在这个项目中，我们将把这项技术扩展到pde，发展所需方法的分析，并将这项技术应用于与放射性废物储存库评估有关的地下水流动的现实模型。对放射性废物处理的未来工作以及对不确定性量化起主要作用的其他领域（例如碳捕获和储存）的潜在影响是相当大的。

We propose to carry out fundamental mathematical research into efficient methods for problems with uncertain parameters and apply them to radioactive waste disposal.The UK Government's policy on nuclear power states that it is a proven low-carbon technology for generating electricity and should form part of the UK's future energy supply. Energy companies will be allowed to build new nuclear power stations provided sufficient progress is made on the radioactive waste issue. In common with other nations, geological disposal is the UK's preferred option for dealing with radioactive waste in the long term. Making a safety case for geological disposal is a major scientific undertaking. National and international research programmes have produced a good understanding of the mechanisms by which radionuclides might return to the human environment and of their consequences once there. One of the outstanding challenges is how to deal with the uncertainties inherent in geological systems and in the evolution of a repository over long time periods and this is at the heart of the proposed research.The main mechanism whereby radionuclides might return to the environment, in the event that they escape from the repository, is transport by groundwater flowing in rocks underground. The mathematical equations that model this flow are well understood, but in order to solve them and to predict the transport of radionuclides the permeability and porosity of the rocks must be specified everywhere around the repository. It is only feasible to measure these quantities at relatively few locations. The values elsewhere have to be inferred and this, inevitably, gives rise to uncertainty. In early performance assessments, relatively rudimentary approaches to treating these uncertainties were used, primarily due to the computational cost. Since then, there have been considerable advances in computer hardware and in the mathematical field of uncertainty quantification. One of the most common approaches to quantify uncertainty is to use probabilistic techniques. This means that the coefficients within the flow equations will be modelled as random fields, leading to partial differential equations with random coefficients (stochastic PDEs), and solving these is much harder and more computationally demanding than their deterministic equivalents. Many fast converging techniques for stochastic PDEs have recently emerged, which are applicable when the uncertainty can be approximated well with a small number of stochastic parameters. However, evidence from field data is such that in repository safety cases much larger numbers of stochastic parameters will be required to capture the uncertainty in the system. Only Monte Carlo (MC) sampling and averaging methods are currently feasible in this case, and the relatively slow rate of convergence of these methods is a major issue.In the work proposed here we will develop and analyse a new and exciting approach to accelerate the convergence of MC simulations for stochastic PDEs. The multilevel MC approach combines multigrid ideas for deterministic PDEs with the classical MC method. The dramatic savings in computational cost which we predict for this approach stem from the fact that most of the work can be done on computationally cheap coarse spatial grids. Only very few samples have to be computed on finer grids to obtain the necessary spatial accuracy. This method has already been applied (by one of the PIs), with great success, to stochastic ordinary differential equations in mathematical finance. In this project we will extend the technique to PDEs, developing the analysis of the method required, and apply the technique to realistic models of groundwater flow relevant to radioactive waste repository assessments. The potential impact for future work on radioactive waste disposal and also for other areas where uncertainty quantification plays a major role (e.g. carbon capture and storage) is considerable.