# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 1 Introduction
[PERSON] and [PERSON] [19] introduced the SFE approach for predicting the properties of filler-modified polymers, with a focus on electrical properties. [PERSON] et al. [20] proposed an invasive formula for the dynamic SFE method, which propagates cognitive uncertainty in material properties over time into the finite element system. [PERSON] et al. [21] proposed a global SFM method to simulate the impact of concrete aging uncertainties on the serviceability and durability of large reinforced concrete and prestressed structures. [PERSON] et al. [22] proposed an interval finite element method based on the Neumann series. By representing the inverse of the stiffness matrix as a Neumann series expansion, they obtained explicit expressions for structural responses with interval variables, enabling the efficient computation of the upper and lower bounds of structural responses. [PERSON] and [PERSON] [23] incorporated Neumann expansion into the Monte Carlo method for deriving the finite element solution under random variations in material properties. They compared the results from the Neumann expansion method with those from the first-order and second-order perturbation approximation methods, as well as the direct Monte Carlo simulation method, in terms of accuracy, convergence, and computational efficiency. [PERSON] et al. [24] proposed an SFE method grounded in the Gaussian Process, enabling the direct utilization of experimental data and the quantification of uncertainties inherent in simulation-based prediction outcomes. [PERSON] and [PERSON] [25] studied how to map the continuous random field of material properties onto the finite element mesh in the stochastic finite element method. [PERSON] et al. [26] employed the stochastic spectral finite element method to investigate the effects of material randomness and heterogeneity on the seismic response of gravity dam-foundation systems. The results indicate that material randomness and heterogeneity may reduce the seismic response of gravity dams, while deterministic models may overestimate dynamic responses. Although remarkable progress has been made in SFE methods, the existing calculation algorithms still have the problem of low calculation efficiency when applied to large-scale structures. To enhance the computational efficiency, an SFE approach leveraging Krylov subspace techniques is introduced for assessing the static reliability of structures. This methodology initiates by preprocessing the static response equation, incorporatingrandomness, to diminish the condition number of the coefficient matrix, thereby facilitating the computation. Subsequently, a Krylov subspace is constructed based on this refined static response equation. By expressing the static displacement under random sampling as a linear superposition of subspace basis vectors, rapid solutions are achieved. Ultimately, statistics and failure probabilities are derived from the static responses acquired through extensive random sampling. Three numerical case studies are presented, contrasting this new method with a stochastic finite element approach grounded in the Neumann series. The outcomes demonstrate that the Krylov subspace-based stochastic finite element method surpasses its Neumann series counterpart in both accuracy and efficiency.