Adaptive numerical algorithms for PDE problems with random i nput data

这个项目是在数值分析、科学计算和不确定性量化领域。它的目标是开发和实现基于严格数学分析的自适应算法，用于输入数据不确定的偏微分方程组(PDE)的有效数值解。PDE是科学和工程过程数学建模的关键工具。在基于偏微分方程的实际模型中，可能无法获得关于输入的精确知识(例如，材料属性、初始条件、外力)，或者可能存在关于输入的不确定性。在这些情况下，模型由带有随机数据的偏微分方程来描述。这类问题出现在许多科学和工业环境中，因为必须对复杂的过程进行准确的建模并进行可靠的风险评估。随机数据偏微分方程组数值求解的主要挑战之一是离散结果的高维性。这推动了健壮和有效的数值方法的发展，充分利用可用的计算资源。学生将开发一种新的方法，以支持自适应算法的设计。这将涉及：(I)在随机Galerkin和/或随机配置有限元方法的背景下的后验误差分析；(Ii)所开发算法的收敛分析。学生还将在开放源代码软件中实施所开发的算法。

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.