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随机维度、非线性和大尺度空间域的耦合。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来高效、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。具体地说，本课题分为以下五个部分：1)发展高效的高维非线性随机有限元：由于随机空间的高维和非线性的耦合，有效地求解高维非线性随机问题仍然是一个挑战。为此，本文将我们前期工作中发展的高维线性SFEM与一种用于处理非线性的新型随机牛顿方法相结合，发展了高维非线性SFEM。2)发展基于区域分解的大规模随机有限元方法：另一个挑战是如何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求解大规模高维非线性随机问题。3)开展基于有限元的可靠性分析：大型高维非线性可靠性分析的极限状态面估计和失效概率估计代价较高。基于上述SFEM获得的随机解，我们可以廉价而快速地产生大量随机解的样本实现。从而有效、准确地计算了可靠性分析中的极限状态面和失效概率。4)发展基于SFEM的贝叶斯逆问题：同样，大规模高维非线性贝叶斯逆问题的似然函数和后验分布的高精度估计需要求解大量的正向随机模型，因此代价很高。得益于所提出的SFEM获得的随机解的大量样本实现，很容易避免重复求解昂贵的正演模型的计算负担。5)基准问题及验证：为验证上述方法的有效性，在固体力学的背景下选取了几个基准问题。重点讨论了该方法的计算精度和计算效率，并与参考解进行了比较。在参考解中，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都是用大量样本实现的标准MCS求解的。

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.