Collaborative Research: A Posteriori Error Analysis for Complex Models with Applications to Efficient Numerical Solution and Uncertainty Quantification

许多对国家基础设施和国防具有重要意义的科学和工程问题都涉及多物理场系统，其中多个物理过程以复杂的方式相互作用。一个重要的例子是液体输送反应化学物质的流动，其中的反应会影响液体的流体性质。这种反应流出现在从生物系统到与能源使用相关的燃烧过程的应用中。一般来说，多物理场系统的复杂性阻碍了对关键特征的直接实验观察。因此，他们的研究主要依赖于计算描述过程及其相互作用的数学模型的近似解。然而，这样的模拟使最强大的计算机的计算能力紧张，因此，近似中的计算误差总是显著的，并且可能是压倒性的。二十多年来，项目研究人员已经开发出一种系统的方法，用于对多物理场系统模型的近似解的误差进行精确的计算估计。在这个项目中，研究人员探索使用这种方法的误差估计来指导有效利用计算资源，以最大限度地提高多物理场系统近似解的保真度。他们还应用误差估计来精确量化基于模型近似解的多物理场系统行为预测中的不确定性。该项目的结果将提高国家工程师和科学家调查和预测对国家安全和基础设施重要的复杂物理系统行为的能力。该项目解决了与使用先进的多尺度、多物理场模型的尖端多离散化数值方法相关的关键问题，以追求科学推理和工程设计。该研究基于对多物理场、多离散化问题的后验误差分析，通过使用伴随问题和可计算残差来量化各种离散化步骤的影响。该项目的主要重点是双重的：(1)开发和分析方法，使用准确的误差估计来指导离散化选择，以便在大致最小的计算成本下达到所需的精度；(2)研究如何扩展准确的误差估计方法，以解决多物理场系统的不确定性量化问题，其中“离散化”包括随机过程的抽样，总体误差是离散化和抽样误差的结合。研究人员致力于开发新的多阶段方法来构建有效的数值解，并将后验误差分析扩展到统计计算。该项目还涉及将后验误差分析理论扩展到双曲问题和感兴趣的非标准量。

Many scientific and engineering problems of importance to the nation's infrastructure and defense are concerned with multi-physics systems in which multiple physical processes interact in complex ways. An important example is the flow of a liquid transporting reacting chemicals in which the reaction affects the fluid properties of the liquid. Such reacting flows arise in applications ranging from biological systems to combustion processes associated with energy use. In general, the complexity of multi-physics systems prevents direct experimental observation of crucial features. Thus their study depends critically on computing approximate solutions of mathematical models describing the processes and their interactions. However, such simulations strain the computational capabilities of the most powerful computers and consequently, computational errors in the approximations are always significant and may be overwhelming. Over two decades, the project investigators have developed a systematic approach for producing accurate computational estimates of the error of approximate solutions of models of multi-physics systems. In this project, the investigators explore the use of error estimates from this approach to guide the efficient use of computational resources in order to maximize the fidelity of approximate solutions of multi-physics systems. They also apply the error estimates to accurately quantify the uncertainty in predictions of behavior of multi-physics systems based on the approximate solutions of models. The results of this project will enhance the ability of the nation's engineers and scientists to investigate and predict the behavior of complex physical systems important to the nation's security and infrastructure. This project tackles critical problems associated with using sophisticated cutting-edge multi-discretization numerical methods for multiscale, multiphysics models to pursue scientific inference and engineering design. The research is based on a posteriori error analysis for multi-physics, multi-discretization problems that quantifies the effects of a wide variety of discretization steps through the use of adjoint problems and computable residuals. The primary focus of the project is twofold: (1) Developing and analyzing methods for using accurate error estimates to guide discretization choices in order to achieve a desired accuracy at roughly minimal computational cost; and (2) Investigating how to extend accurate error estimation methods to address uncertainty quantification for multiphysics systems, where 'discretization' includes the sampling of a random process and the overall error is a combination of discretization and sampling errors. The investigators pursue the development of novel multi-stage approaches to the construction of efficient numerical solutions and the extension of a posteriori error analysis to statistical computations. The project also involves the extension of the theory of a posteriori error analysis to hyperbolic problems and nonstandard quantities of interest.