Computational Methods for Parameter-Dependent Partial Differential Equations

基于偏微分方程模型的现代数值模拟具有高维性和参数依赖性的特点。 为了实现离散近似的精度，需要高维数，特别是对于三维或多组分模型。参数依赖性来源于多种因素。 参数可以对应于时间或特定的术语，如流体中的雷诺数，在各种参数选择下需要解决方案的属性。或者，它们可以用于指定模型的组件，例如材料属性、几何形状或边界条件，需要对其执行多个模拟。 这些相同的参数集可能是不确定的，并被视为随机变量，模拟用于识别解的统计特性。 所有这些方案都需要计算许多离散的解决方案，这可能是昂贵的，当离散模型是大规模的。 我们在这个项目中的目标是探索，开发和改进计算算法，以降低这些成本。 我们的重点将是有效地使用减少的基础方法，该项目的高维模型到子空间的显着更小的维度，目的是快速，有效地构建精确的近似解在广泛的参数范围内。这种方法的潜在影响在于它在各种工程和科学模拟的使用。其中包括地下水流和其他环境现象的模型，将边界条件或存在流体的介质的渗透性等不确定特性作为参数处理;空气动力学模拟，将结构的材料特性或可燃材料的质量作为参数，必须分析其对效率和安全的影响;以及生物过程的模型，例如血液流动或细胞中的化学反应，其取决于诸如流体粘度或反应速率的参数。 基于减少问题规模的高效算法策略的开发将显著提高快速执行此类模拟的前景，使工程师和科学家能够在现场使用模拟结果并进行实时决策。

Modern numerical simulation using models based on partial differential equations is characterized by high dimensionality together with parameter dependence. High dimensionality is required to achieve accuracy in discrete approximations, especially for three-dimensional or multi-component models. Parameter dependence stems from a variety of sources. Parameters may correspond to time or to specific terms such as Reynolds numbers in fluids, for which the properties of solutions are wanted at a variety of parameter choices. Alternatively, they may be used to specify components of a model such as material properties, geometry, or boundary conditions, for which it is desired to perform multiple simulations. These same sets of parameters may instead be uncertain and treated as random variables, for which simulations are used to identify statistical properties of solutions. All these scenarios require the computation of many discrete solutions, which may be prohibitively expensive when the discrete models are large in scale. Our aim in this project is to explore, develop and refine computational algorithms to reduce these costs. Our emphasis will be on effective use of reduced basis methods, which project high-dimensional models into subspaces of significantly smaller dimension with the aim of quickly and efficiently constructing accurate approximate solutions over a wide range of parameters.The potential impact of this approach lies in its use in wide varieties of engineering and scientific simulations. These include models of groundwater flows and other environmental phenomena, where uncertain properties such as boundary conditions or permeabilities of media in which fluids are found are treated as parameters; aerodynamic simulations, where material properties of structures or qualities of combustible material are parameters that must be analyzed for their effects on efficiency and safety; and in models of biological processes, for example blood flows or chemical reactions in cells, which depend on parameters such as fluid viscosity or reaction rates. The development of efficient algorithmic strategies based on reduction of problem size will significantly enhance the prospects of performing such simulations quickly, enabling engineers and scientists to use the results of simulations in the field and for real-time decision making.