# Structural Reliability Analysis Using Stochastic Finite Element Method Based on Krylov Subspace
## 2 Theoretical Development
For the complete analysis, the number of flops required for each sampling to compute the displacement using Equation (5) is the sum of the flops needed for inverting matrix \(K_{d}\) and for multiplying matrices \(K_{d}^{-1}\) and \(l\). For a system with \(n\)-DOFs, the number of flops for Equation (5) is about \(n^{3}+n^{2}\). For the NSFE model, the number of flops required for each sampling to calculate the displacement mainly consists of the following parts: (1) calculating \(\Delta K\) using Equation (6) requires \(n^{2}\) times; (2) calculating the second-order and higher-order terms in Equation (9) requires \(2(m-1)n^{2}\) times; and (3) calculating \(x_{d}\) using Equation (9) requires \((m-1)n^{2}\) times. Therefore, the total number of flops for the NSFE model is about \(3 mn^{2}-2n^{2}\) times. For the SFE-KS method, the number of flops required for each sampling to calculate the displacement mainly consists of the following parts: (1) calculating \(\Pi\) using Equations (12) and (14) requires \(2(m-1)n^{2}\) times; (2) calculating \(D\) using Equation (18) requires \(2 mn^{2}\) times; (3) calculating \(z\) using Equation (19) requires \(mn\) times; (4) calculating \(\dot{y}\) using Equation (20) requires \(m^{3}+m^{2}\) times; and (5) calculating \(x_{d}\) using Equation (21) requires \(mn\) times. Therefore, the total number of flops for the SFE-KS model is about \(4 mn^{2}-2n^{2}+2 mn+m^{3}+m^{2}\) times. In the subsequent numerical examples, the number of flops for each method will be estimated according to these formulas, and the computation time for each method will also be provided. These results, taken together, will validate that the proposed method exhibits a high level of computational efficiency. In terms of the computational accuracy, under the condition that the number of terms \(m\) is the same, the proposed method is expected to achieve a higher accuracy compared to the NSFE method due to the additional least-squares fitting performed using the basis vectors of the Krylov subspace, as shown in Equations (17)-(20). The key advantage lies in the utilization of the Krylov subspace, which captures important information about the solution space through a sequence of matrix-vector multiplications. By projecting the solution onto this subspace and performing a least-squares fit, the proposed method can better approximate the true solution. It is worth noting that the actual improvement in the accuracy will depend on various factors, including the specific problem being analyzed, the choice of basic functions, the number of samples used, and the computational resources available. However, in general, the proposed method's use of the Krylov subspace and least-squares fitting provides a solid theoretical foundation for achieving higher accuracy in the computation of failure probabilities and other stochastic quantities of interest. In the subsequent case studies, the computational accuracy will primarily be judged by comparing the errors between the statistical metrics (mean and standard deviation) obtained using various methods and the exact solution derived from a complete analysis.