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

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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.