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

大规模高维非线性可靠性分析和贝叶斯反演的挑战来自于正演模型中的高维随机性、非线性和大规模空间域的耦合。为了克服这些困难，本项目旨在发展先进的随机有限元方法，以高效、准确地求解大规模高维非线性正演模型，并基于所获得的随机解进行相应的可靠性分析和贝叶斯反演。具体而言，本项目分为以下五个部分：1）开发高效的高维非线性随机有限元：由于随机空间的高维性和非线性的耦合，有效地解决高维非线性随机问题仍然是一个挑战。为此，高维非线性SFEM开发通过结合在我们以前的工作中开发的高维线性SFEM和一种新的随机牛顿方法用于处理非线性。2)开发基于区域分解的大规模随机有限元：另一个挑战是有效地解决定义在大规模空间域上的高维非线性随机问题。为此，将区域分解方法引入到上述高维非线性随机有限元中，发展了基于随机区域分解的并行随机有限元方法来求解大规模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曲面和失效概率的估计是昂贵的。基于上述SFEM得到的随机解，我们可以廉价而快速地生成随机解的大量样本实现。从而有效、准确地计算了可靠度分析中的极限状态面和失效概率。4)开发基于SFEM的贝叶斯反演：同样，大规模高维非线性贝叶斯反演的似然函数和后验分布的高精度评估需要求解大量的前向随机模型，因此成本非常昂贵。受益于大量的样本实现所提出的SFEM获得的随机解，重复求解昂贵的正演模型的计算负担是很容易避免的。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。文中着重分析了所提方法的计算精度和效率，并与参考解进行了比较，其中正演模型的参考随机解、可靠性分析中的参考失效概率和贝叶斯反演中的参考后验分布均采用标准MCS，并具有大量的样本实现.

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.