Adaptive numerical algorithms for PDE problems with random i nput data

具有随机输入数据的偏微分方程问题的自适应数值算法

• 批准号: 2281603
• 金额: --
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2281603
• 关键词: Adaptive; numerical; algorithms; PDE; problems

This project is in the areas of Numerical Analysis, Scientific Computing, and Uncertainty Quantification. It aims at developing and implementing adaptive algorithms underpinned by rigorous mathematical analysis for efficient numerical solution of partial differential equations (PDEs) with uncertainty in input data.PDEs are key tools in the mathematical modelling of processes in science and engineering. In practical PDE-based models, precise knowledge of inputs (e.g., material properties, initial conditions, external forces) may not be available, or there might be uncertainty about the inputs. In these cases the models are described by PDEs with random data. Such problems arise in many scientific and industrial contexts when it is essential to accurately model complex processes and perform a reliable risk assessment. One of the major challenges in numerical solution of PDEs with random data is the high dimensionality of the resulting discretisations. This motivates the development of robust and effective numerical methods which make best use of available computational resources.The student will be developing a novel methodology that underpins the design of adaptive algorithms. This will involve:(i) a posteriori error analysis in the context of stochastic Galerkin and/or stochastic collocation finite element methods;(ii) convergence analysis of the developed algorithms.The student will be also implementing the developed algorithms in an open-source software.

该项目属于数值分析、科学计算和不确定性量化领域。它旨在开发和实施以严格数学分析为基础的自适应算法，以有效数值求解输入数据不确定的偏微分方程 (PDE)。PDE 是科学和工程过程数学建模的关键工具。在实际的基于偏微分方程的模型中，可能无法获得输入的精确知识（例如材料属性、初始条件、外力），或者输入可能存在不确定性。在这些情况下，模型由具有随机数据的偏微分方程描述。当必须对复杂过程进行准确建模并执行可靠的风险评估时，在许多科学和工业环境中都会出现此类问题。使用随机数据对偏微分方程进行数值求解的主要挑战之一是所得离散化的高维性。这促进了稳健有效的数值方法的开发，充分利用了可用的计算资源。学生将开发一种支持自适应算法设计的新颖方法。这将涉及：（i）随机伽辽金和/或随机搭配有限元方法背景下的后验误差分析；（ii）所开发算法的收敛性分析。学生还将在开源中实现所开发的算法软件。