Adaptive multilevel stochastic collocation methods for uncertainty quantification

科学和工程中的计算机模拟依赖于基本现象和过程的数学模型。这些数学模型通常根据与物理量（例如，固体中的温度或流动流体的速度）在空间和时间上。复杂现象和过程的现实模型必须考虑到始终存在的不确定性，例如对基于PDE的模型的所有输入（例如材料特性，初始条件，外力等）的不精确或不完整的知识。这种现象的例子包括波在不均匀介质中的传播与不确定的波的特性和流体流动通过多孔介质的渗透率不知道精确地在计算域中的每一个点。在这些情况下，代替标准确定性模型，模拟必须依赖于概率技术，以便对输入中的潜在不确定性进行建模（使用随机变量或随机场），分析不确定性如何传播到模型输出，估计不期望事件的概率（例如，核废料储存库泄漏造成的地下水污染），并进行可靠的风险评估。然后用随机数据的偏微分方程来表示模型，其中输入和输出都采用随机场的形式，发展有效的近似技术和数值算法来求解随机输入偏微分方程是不确定性量化的一项重要任务，因为它打开了现实模拟的大门，并确保在存在不确定性的情况下可靠和准确的预测。在这一研究领域的关键数学挑战关注（一）近似方法的设计与保证和可靠的误差控制，以及（ii）可证明准确的自适应算法的发展，使可用的计算资源的最佳利用。该项目将通过在具有随机输入的偏微分方程的强大逼近技术（称为多层随机配置有限元法）的框架内开发、分析、实施和测试一种可靠的误差估计和自适应误差控制的新型方法来解决上述挑战。该项目与工程和制造业中的许多应用相关（例如，在核电工业中），提高不确定性量化的数值方法的效率和可靠性将加快决策速度，并对公共安全产生直接影响。

Computer simulations in science and engineering rely on mathematical models of the underlying phenomena and processes. These mathematical models are typically written in terms of partial differential equations (PDEs) relating rates of changes of physical quantities (e.g., temperature in a solid or velocity of a flowing fluid) in space and time. Realistic models of complex phenomena and processes must account for the ever-present uncertainties resulting e.g. from imprecise or incomplete knowledge of all inputs to a PDE-based model (such as material properties, initial conditions, external forces, etc.). Examples of such phenomena include wave propagation in inhomogeneous media with uncertain wave characteristics and fluid flow through a porous media with permeability not known precisely at every point in the computational domain. In these cases, instead of standard deterministic models, simulations must rely on probabilistic techniques in order to model the underlying uncertainties in the inputs (using random variables or random fields), analyse how the uncertainties propagate to the model outputs, estimate probabilities of undesirable events (e.g., the contamination of groundwater resulting from a leakage from nuclear waste repository), and perform reliable risk assessments. The models are then represented by PDEs with random data, where both inputs and outputs take the form of random fields.The development of effective approximation techniques and numerical algorithms for solving PDEs with random inputs is an important task in uncertainty quantification, because it opens the door to realistic simulations and ensures reliable and accurate predictions in the presence of uncertainties. Key mathematical challenges in this research area concern (i) the design of approximation methods with guaranteed and reliable error control, and (ii) the development of provably accurate adaptive algorithms that make the best use of available computational resources. This project will address both aforementioned challenges by developing, analysing, implementing and testing a novel methodology for reliable error estimation and adaptive error control in the framework of a powerful approximation technique for PDEs with random inputs known as the multilevel stochastic collocation finite element method. The project is relevant to many applications in engineering and manufacturing (e.g., in nuclear power industry) where improvements in the efficiency and reliability of numerical methods for uncertainty quantification would speed up decision making and have a direct impact on public safety.