Numerical analysis of adaptive UQ algorithms for PDEs with random inputs

偏微分方程（PDE）是科学和工程中物理过程数学建模的关键工具。传统的基于偏微分方程的确定性模型假设所有输入（材料属性，初始条件，外力等）的精确知识。存在大量的数值方法，可用于计算此类模型的解，以达到任何所需的精度。然而，在实际应用中，PDE模型的所有输入的完整特征可能不可用。例子包括应力体的弹性模量（在线性弹性模型中）和非均匀介质的波动特性（在波传播模型中）。在这些情况下，基于确定性模型的模拟无法估计不期望事件的概率（例如，应力板的断裂），并因此执行可靠的风险评估。不确定性量化（UQ）的新兴领域涉及不同层次的数学建模。它涉及使用概率技术，以便（i）确定和量化基于PDE的模型的输入中的不确定性，以及（ii）分析这些不确定性如何传播到输出（PDE的解决方案，或从解决方案导出的感兴趣的量）。然后用随机数据的偏微分方程来描述模型，其中输入和输出都采用随机场的形式。这种偏微分方程模型的数值解比确定性类似物的解更具挑战性。该项目的中心重点是开发用于求解相关参数依赖PDE模型的鲁棒、准确和实用的数值方法。在20世纪90年代的工程文献中出现了基于参数化的PDE问题的数值方法，在随机空间的维数适中（10个随机参数的数量级）的情况下，这些方法是蒙特-卡罗采样的更有效和快速收敛的替代方案。最近的研究表明，这些方法的有利的近似性能，可以最好地实现通过使用自适应细化策略，当空间和随机组件的近似解是明智地选择在数值计算的过程中。然而，最优自适应算法的设计仍然是一个悬而未决的问题。拟议的研究计划旨在设计，理论分析和有效实施的国家的最先进的自适应算法适用于一系列随机输入的偏微分方程问题。通过提高不确定性量化数值方法的效率和可靠性，该研究项目与英国管理核废料和最大限度地减少地下水污染风险的社会挑战直接相关。

Partial differential equations (PDEs) are key tools in the mathematical modelling of physical processes in science and engineering. Traditional deterministic PDE-based models assume precise knowledge of all inputs (material properties, initial conditions, external forces, etc.). There exist an abundance of numerical methods that can be used to compute a solution to such models to any required accuracy. In practical applications, however, a complete characterisation of all the inputs to a PDE model may not be available. Examples include the modulus of elasticity of a stressed body (in linear elasticity models)and wave characteristics of inhomogeneous media (in wave propagation models). In these cases, simulations based on deterministic models are unable to estimate probabilities of undesirable events (e.g., the fracture of a stressed plate) and, hence, to perform a reliable risk assessment. The emergent area of uncertainty quantification (UQ) deals with mathematical modelling at a different level. It involves the use of probabilistic techniques in order to(i) determine and quantify uncertainties in the inputs to PDE-based models, and(ii) analyse how these uncertainties propagate to the outputs(either the solution to the PDE, or a quantity of interest derived from the solution). The models are then described by PDEs with random data, where both inputs and outputs take the form of random fields. Numerical solution of such a PDE model is significantly more challenging than the solution of the deterministic analogues. The development of robust, accurate, and practical numerical methods for solving associated parameter-dependent PDE models is the central focus of the project. Numerical methods based on a parametric reformulation of such PDE problems emerged in the engineering literature in the 1990s as more efficient and rapidly convergent alternatives to Monte-Carlo sampling in cases where the dimension of the stochastic space is moderate (of the order of 10 random parameters). Recent research into these methods suggests that their advantageous approximation properties can best be achieved by using an adaptive refinement strategy, when spatial and stochastic components of the approximate solution are judiciously chosen in the course of numerical computation. The design of optimal adaptive algorithms remains an open question however. The proposed research programme aims at the design, theoretical analysis and efficient implementation of the state-of-the-art adaptive algorithms applicable to a range of PDE problems with random inputs. By improving the efficiency and reliability of numerical methods for uncertainty quantification, the research project is directly relevant to the UK societal challenge of managing nuclear waste and minimising the risks of contamination of groundwater.