Efficient computational framework for uncertainty quantification of large-scale high-dimensional nonlinear stochastic systems

大规模高维非线性随机系统不确定性量化的高效计算框架

• 批准号: 527222589
• 负责人: Professor Dr.-Ing. Udo Nackenhorst
• 金额: --
• 依托单位: Institut für Baumechanik und Numerische Mechanik (IBNM)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/527222589?language=en
• 关键词: Efficient; computational; framework; uncertainty; quantification

The challenges of large-scale high-dimensional nonlinear reliability analysis and Bayesian inversion come from the coupling of the high stochastic dimensions, the nonlinearities and the large-scale spatial domains in the forward models. To overcome these difficulties, this project aims to develop advanced stochastic finite element methods to efficiently and accurately solve large-scale high-dimensional nonlinear forward models, and perform corresponding reliability analysis and Bayesian inversion based on the obtained stochastic solutions. Specifically, this project is divided into the following five parts: 1) Develop efficient high-dimensional nonlinear SFEM: It remains a challenge to efficiently solve high-dimensional nonlinear stochastic problems due to the coupling of high dimensionalities of stochastic spaces and nonlinearities. To this end, high-dimensional nonlinear SFEMs are developed via combining the high-dimensional linear SFEMs developed in our previous work and a novel stochastic Newton method used for handling nonlinearities. 2) Develop domain decomposition-based large-scale SFEM: Another challenge is to efficiently solve high-dimensional nonlinear stochastic problems defined on large-scale spatial domains. To this end, introducing domain decomposition methods into the above high-dimensional nonlinear SFEM, stochastic domain decomposition-based parallel SFEMs are developed to solve large-scale high-dimensional nonlinear stochastic problems. 3) Develop SFEM-based reliability analysis: The estimation of limit state surfaces and failure probabilities for large-scale high-dimensional nonlinear reliability analysis is expensive. Based on the stochastic solutions obtained by the above SFEMs, we can cheaply and quickly generate a large number of sample realizations of the stochastic solutions. Thus, the limit state surfaces and the failure probabilities in reliability analysis are calculated efficiently and accurately. 4) Develop SFEM-based Bayesian inversion: Similarly, the high-accuracy evaluation of likelihood functions and posterior distributions for large-scale high-dimensional nonlinear Bayesian inversion requires solving a large number of forward stochastic models and is thus very expensive. Benefiting from a large number of sample realizations of the stochastic solutions obtained by the proposed SFEMs, the computational burden of repeatedly solving expensive forward models is readily avoided. 5) Benchmark problems and validation: To validate the methods proposed in the above parts, several benchmark problems are chosen in the context of solid mechanics. The computational accuracy and efficiency of the proposed methods are emphasized and compared with the reference solutions, where the reference stochastic solutions of forward models, the reference failure probabilities in reliability analysis and the reference posterior distributions in Bayesian inversion are solved using standard MCS with a large number of sample realizations.

大规模高维非线性可靠性分析和贝叶斯反演的挑战来自向前模型中高随机维度，非线性和大规模空间域的耦合。为了克服这些困难，该项目旨在开发先进的随机有限元方法，以有效，准确地求解大规模的高维非线性远期模型，并基于获得的随机解决方案执行相应的可靠性分析和贝叶斯反转。具体而言，该项目分为以下五个部分：1）开发有效的高维非线性SFEM：由于高维空间和非线性的高维度耦合，因此有效解决高维非线性随机问题仍然是一个挑战。为此，通过结合以前工作中开发的高维线性SFEM和一种用于处理非线性的新型随机牛顿方法，开发了高维的非线性SFEM。 2）开发基于域分解的大规模SFEM：另一个挑战是有效解决在大型空间域上定义的高维非线性随机问题。为此，开发了基于随机域分解的平行SFEM，将域分解方法引入上述高维非线性SFEM，以解决大型高维非线性随机问题。 3）开发基于SFEM的可靠性分析：大规模高维非线性可靠性分析的极限状态表面和故障概率的估计很昂贵。基于上述SFEM获得的随机解决方案，我们可以便宜并迅速产生大量随机溶液的样本实现。因此，有效，准确地计算了可靠性分析中极限状态表面和故障概率。 4）开发基于SFEM的贝叶斯倒置：同样，大型高维非线性贝叶斯反演的似然函数和后验分布的高临界性评估需要解决大量的正向随机模型，因此非常昂贵。受益于所提出的SFEM获得的随机解决方案的大量样品实现，很容易避免反复解决昂贵的前向模型的计算负担。 5）基准问题和验证：为了验证上述部分中提出的方法，在固体力学的背景下选择了几个基准问题。强调了所提出方法的计算准确性和效率，并将其与参考解决方案进行比较，在此，使用具有大量样品实现的标准MC，可以使用标准MCS求解前向模型的参考随机解，可靠性分析中的参考故障概率和贝叶斯反转的参考后验分布。