Numerical analysis of adaptive UQ algorithms for PDEs with random inputs

具有随机输入的 PDE 自适应 UQ 算法的数值分析

基本信息

批准号：
EP/P013317/1
负责人：
David Silvester
金额：
$ 48.56万
依托单位：
University of Manchester
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP013317%2F1
关键词：
Numerical analysis adaptive UQ algorithms

项目摘要

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.

部分微分方程（PDE）是科学和工程学物理过程数学建模的关键工具。基于传统的确定性PDE模型假设所有输入（材料属性，初始条件，外力等）的精确知识。存在丰富的数值方法，可用于将这些模型的解决方案计算为任何必需的准确性。但是，在实际应用中，可能没有对PDE模型的所有输入的完整表征。示例包括压力体的弹性模量（在线性弹性模型中）和不均匀培养基的波特征（在波传播模型中）。在这些情况下，基于确定性模型的仿真无法估计不良事件的概率（例如，压力板的断裂），因此无法执行可靠的风险评估。不确定性定量（UQ）的新兴领域涉及不同级别的数学建模。它涉及概率技术的使用，以（i）确定和量化基于PDE的模型的输入中的不确定性，（ii）分析这些不确定性如何传播到输出（PDE的解决方案，或者是从解决方案中得出的一定兴趣））。然后，模型由PDE描述，其中包括随机数据，其中输入和输出都采用随机字段的形式。与确定性类似物的解决方案相比，这种PDE模型的数值解决方案更具挑战性。求解相关参数依赖性PDE模型的强大，准确和实用的数值方法的开发是项目的中心重点。基于对此类PDE问题的参数重新制定的数值方法在1990年代在工程文献中出现了，因为在随机空间的维度中等的情况下（10个随机参数的顺序），在蒙特卡罗取样的情况下，对蒙特卡罗取样的效率更高且快速收敛。对这些方法的最新研究表明，当在数值计算过程中明智地选择了近似解决方案的空间和随机组件时，最好使用自适应策略来最佳地实现它们的优势近似特性。但是，最佳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适用于随机输入的一系列PDE问题的最先进的自适应算法。通过提高数值方法对不确定性量化的效率和可靠性，该研究项目与英国的社会挑战直接相关，即管理核废料和最大程度地减少地下水污染的风险。