Adaptive Numerical Algorithms for Forward UQ in Time-Dependent PDEs

不确定性量化（UQ）是一个快速发展的领域，融合了几个传统的数学学科。该项目将开发新的自适应数值算法，用于时变CFD（计算流体动力学）模型中不确定性的前向传播。当我们使用数学模型来模拟真实世界的过程（如流体流动）时，我们经常会遇到一个或多个输入（粘度，材料参数，初始条件，几何形状等）不确定的情况。在远期UQ中，主要目的是评估模型输入中的不确定性对与模型输出相关的感兴趣数量的影响。为此，我们需要计算效率高的数值方法，可以考虑模型输入的概率分布，并提供与模型输出相关的感兴趣统计量的准确近似值。对于时间相关的问题，特别是那些具有非光滑解的问题，近似空间通常需要及时调整以保持精度。如何设计具有保误差控制的自适应数值算法是一个极具挑战性的问题。该项目是一个数值分析项目，将开发新的自适应数值格式的前向UQ驱动的严格的误差估计。

Uncertainty quantification (UQ) is a rapidly-evolving field, incorporating several traditional mathematical disciplines. This project will develop new adaptive numerical algorithms for the forward propagation of uncertainty in time-dependent CFD (computational fluid dynamics) models.When we use mathematical models to simulate real-world processes (such as fluid flows) we frequently encounter situations where we are uncertain about one or more of the inputs (viscosity, material parameters, initial conditions, geometry etc). In forward UQ, the main aim is to assess the impact of uncertainty in the model inputs on quantities of interest associated with the model's outputs. For this, we require computationally efficient numerical methods that can take in a probability distribution for the model's inputs and deliver accurate approximations of statistical quantities of interest related to the model's outputs. For time-dependent problems, and especially those with non-smooth solutions, the approximation space often needs to be adapted in time to maintain accuracy. How to design adaptive numerical algorithms with guaranteed error control is highly challenging. This project is a numerical analysis project that will develop new adaptive numerical schemes for forward UQ driven by rigorous error estimation.