Efficient Bayesian Multi-fidelity Schemes for Analysis and Design of Complex Multiphysics Systems

在工程系统的设计过程中，考虑多个物理场是至关重要的。用于分析、设计和优化结构的计算工具已经发展到能够解释这些影响并使研究这种复杂系统成为可能。然而，如果所用模型的任何组成部分不能被精确确定或表现出随机变异性，纯粹的确定性分析可能会导致不满意和不可靠的结果。这种认识性和任意性的不确定性在绝大多数真实世界的系统中都会遇到。如果采用概率观点并在模型中考虑输入参数中的不确定性，则可以显著提高计算模型的预测能力以及所设计系统的弹性。尽管近年来提出了许多不确定量化策略，但现有方法对随机维度的可扩展性较差，并且需要对昂贵的、非线性的正向模型进行大量的评估。我们在本项目中提出的挑战是开发一个新的、高效的UQ框架，用于分析和设计具有高随机维度的复杂、非线性、多物理模型。应用将涉及到强耦合问题，如流体-结构-相互作用。建议的一套UQ方法将能够处理具有高维参数不确定性的复杂的真实世界系统。所开发的方法将为感兴趣的输出量的统计数据提供可证明的估计，并为不确定的模型输入参数提供灵敏度测量。在如此复杂和具有挑战性的环境中，与UQ相关的迄今令人望而却步的计算成本将通过严格纳入来自廉价、低保真模型的信息而得到缓解。这些与对昂贵的高保真模型的几个智能选择的评估相结合，以便以当前UQ方法的一小部分成本获得准确的估计。此外，通过采用贝叶斯方法，可以计算可靠的区间，量化估计中的置信度，并指导自适应改进。此外，UQ方法还将作为开发一种新的随机优化框架的基础，用于在存在不确定性的情况下设计复杂系统。所提出的方法是非常普遍的，将适用于广泛的问题。例如，心血管生物力学中出现的大规模、非线性问题将被用来展示方法的能力和效率。该提议直接解决了优先计划提出的几个最基本的问题。将制定一套基准问题，以便对不同观点进行比较和交叉授粉。

The consideration of multiple physical fields is of paramount importance in the design process of engineering systems. Computational tools for analysis, design, and optimization of structures have evolved to account for these effects and enable the study of such complex systems. Nevertheless, a purely deterministic analysis can lead to unsatisfactory and unreliable results if any component of the models employed cannot be precisely determined or exhibits random variability. Such epistemic and aleatoric uncertainties are encountered in the overwhelming majority of real-world systems. The predictive capabilities of computational models, as well as the resilience of the systems designed, can be significantly improved if a probabilistic point of view is adopted and the uncertainties in the input parameters are accounted for in the model. Although many strategies for uncertainty quantification (UQ) have been proposed in recent years, current approaches exhibit poor scalability with the stochastic dimension and require an exuberant number of evaluations of the expensive, nonlinear forward model.The challenge we propose to undertake in this project is the development of a novel and efficient UQ framework that can be used for analysis and design of complex, nonlinear, multiphysics models with high stochastic dimension. Applications will involve strongly coupled problems such as fluid-structure-interaction. The proposed set of UQ methods will be able to handle complex, real-world systems, characterized by high-dimensional parametric uncertainties. The methods developed will provide certifiable estimates for the statistics of the output quantities of interest, as well as sensitivity measures for the uncertain model input parameters. The hitherto prohibitive computational costs associated with UQ in such complex and challenging settings will be mitigated by rigorously incorporating information from inexpensive, lower-fidelity models. These are combined with a few, intelligently selected evaluations of the expensive, high-fidelity model, in order to obtain accurate estimates at a fraction of the cost compared to current UQ methods. Moreover, by adopting a Bayesian approach, credible intervals can be computed which quantify the confidence in the estimates as well as guide adaptive refinements. In addition, the UQ approach will also serve as the basis for the development of a novel stochastic optimization framework for the design of complex systems in the presence of uncertainties. The proposed methods are very general and will be applicable to a wide range of problems. As example, large-scale, nonlinear problems arising in cardiovascular biomechanics will be used to demonstrate capabilities and efficiency of the methods.The proposal directly addresses several of the most fundamental questions posed by the priority program. A set of benchmark problems will be developed that will enable comparison as well as cross-pollination of different perspectives.