Adaptive multilevel stochastic collocation methods for uncertainty quantification

科学和工程中的计算机模拟依赖于底层现象和过程的数学模型。这些数学模型通常用偏微分方程（PDEs）来表示物理量（例如，固体的温度或流动流体的速度）在空间和时间中的变化率。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，由于对基于pde的模型的所有输入（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具有不确定波特性的非均匀介质中的传播，以及流体在多孔介质中的流动，其渗透率在计算域中的每一点都不是精确已知的。在这些情况下，模拟必须依靠概率技术，而不是标准的确定性模型，以便模拟输入中的潜在不确定性（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，估计不良事件的概率（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表示模型，其中输入和输出都采用随机场的形式。求解随机输入偏微分方程的有效逼近技术和数值算法的发展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。该研究领域的关键数学挑战涉及(i)设计具有保证和可靠误差控制的近似方法，以及（ii）开发可证明准确的自适应算法，以充分利用可用的计算资源。该项目将通过开发、分析、实施和测试一种可靠误差估计和自适应误差控制的新方法来解决上述两个挑战，该方法是一种强大的近似技术，用于随机输入的偏微分方程，称为多层随机搭配有限元法。该项目与工程和制造业（例如核动力工业）中的许多应用有关，其中改进不确定性量化的数值方法的效率和可靠性将加快决策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.