Multilevel Intrusive UQ Methods

物理过程，如传热和流体流动，通常使用偏微分方程（PDE）建模。如果所有的输入（系数，边界条件等）是已知的，那么标准的数值方案，如有限元方法可以用来执行模拟和预测与模型解决方案相关的感兴趣的数量。然而，在工程问题中，我们经常会遇到对一个或多个模型输入不确定的情况。处理这个问题最常见的方法是诉诸概率论，将不确定输入表示为随机变量的函数。估计与具有指定概率分布的随机输入的模型的解相关的感兴趣的量被称为前向不确定性量化（UQ）。尽管存在许多用于执行前向UQ的算法，但是对于复杂PDE模型有效且准确地估计感兴趣的统计量仍然是一个重要的科学挑战。该项目将在理论和计算方面取得进展，开发所谓的多级侵入式（MINT）算法，用于计算效率高且可证明准确的前向UQ。与采样方法不同，侵入式方案寻求作为随机输入的多项式的近似。标准的侵入式方法不受欢迎，因为它们需要求解庞大的线性方程组，这会迅速耗尽可用的计算资源。主要问题是它们使用了大的张量积近似空间，这导致了计算的浪费。通过构建具有灵活多层结构的低维近似空间，并由自动和准确的误差评估驱动，将取得进展。

Physical processes such as heat transfer and fluid flows are typically modelled using partial differential equations (PDEs). If all the inputs (coefficients, boundary conditions etc) are known then standard numerical schemes such as finite element methods can be used to perform simulations and predict quantities of interest related to the model solution. In engineering problems, however, we frequently encounter scenarios where we are uncertain about one or more model inputs. The most common way to deal with this is to appeal to probability theory and represent uncertain inputs as functions of random variables. Estimating quantities of interest related to solutions of models with random inputs with a prescribed probability distribution is called forward uncertainty quantification (UQ). Although many algorithms for performing forward UQ exist, estimating statistical quantities of interest efficiently and accurately for complex PDE models remains an important scientific challenge. This project will make theoretical and computational advances in the development of so-called multilevel intrusive (MINT) algorithms for forward UQ that are computationally efficient and also provably accurate. Unlike sampling methods, intrusive schemes seek approximations which are polynomials of the random inputs. Standard intrusive methods are unpopular because they require the solution of huge linear systems of equations which quickly exhausts available computational resources. The main issue is that they use large tensor product approximation spaces which leads to wasted computations. Advances will be made by constructing lower-dimensional approximation spaces with flexible multilevel structure driven by an automated and accurate assessment of error.