Adaptive numerical algorithms for PDE problems with random i nput data

该项目是在数值分析，科学计算和不确定性量化领域。它旨在开发和实施以严格的数学分析为基础的自适应算法，以有效地数值求解具有输入数据不确定性的偏微分方程（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，输入的精确知识（例如，材料特性、初始条件、外力）可能不可用，或者输入可能存在不确定性。在这些情况下，模型由随机数据的偏微分方程描述。在许多科学和工业背景下，当精确地模拟复杂过程并进行可靠的风险评估至关重要时，都会出现此类问题。随机数据偏微分方程数值解的主要挑战之一是所得离散化的高维性。这促使发展稳健和有效的数值方法，充分利用现有的计算资源。学生将开发一种新的方法，支持自适应算法的设计。这将涉及：（i）在随机Galerkin和/或随机配置有限元法的背景下的后验误差分析;（ii）所开发算法的收敛性分析。学生还将在开源软件中实现所开发的算法。

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.