# Steady-State Response Analysis of an Uncertain Rotor Based on Chebyshev Orthogonal Polynomials
## 2 The Proposed Method
### Sparse Grid Numerical Integration  
In Equation (8), if the number of nodes taken for each dimensional random variable is the same and equal to \(q\), then the total number of function evaluations required is \(q^{n}\). This approach for calculating \(c_{i_{1},\cdots,i_{n}}\) is referred to as full-tensor-product grid (FTG). Regarding the selection of the number of nodes \(q\) (usually \(q>3\)) in each dimension, a larger \(q\) results in higher accuracy; however, it also leads to an increase in computational workload. Consequently, when the dimension \(n\) exceeds 5, the strategy encounters a significant issue known as the \"curse of dimensionality\". To alleviate this issue, an efficient approach to computing high-dimensional integrals is to use numerical integration with sparse-grid (SG) technology. The sparse grid numerical integration, also known as the sparse tensor product grid, was first proposed by [PERSON] [24] and has been widely used by researchers to address high-dimensional integration problems, thereby improving the efficiency of integration. The SG with accuracy level \(L\) and \(n\)-dimensional integration is denoted as \(\mathcal{G}(L,n)\), and its construction formula is as follows:  
\[\mathcal{G}(L,n)=\underset{\max(n,L+1)\leq\left\|k\right\|_{1}\leq L+n=1}{ \overset{n}{\underset{i=1}{\cup}}}U_{1}^{k_{i}} \tag{9}\]  
where \(\left\|k\right\|_{1}=\sum_{i=1}^{n}k_{i}\) (\(k_{i}\geq 1\)) is the 1-norm of \(k\), and \(k_{i}\) is a positive integer that determines the number of integration nodes assigned to each dimensional random variable. The accuracy level \(L\) is defined as \(q=L+1\). \(U_{1}^{k_{i}}\) represents a univariate quadrature rule with \(k_{i}\) integral nodes. The SG decomposes the direct tensor product into different sub-tensor products, using only the sub-tensor products corresponding to multi-indices that satisfy \(\max(n,L+1)\leq\left\|k\right\|_{1}\leq L+n\). For example, if \(n=2\) and \(L=2\), the required multi-indices are \(k=\) ({1,2}, (1,3), (2,1), (2,2), (3,1)},\) and the construction of the corresponding sparse grid \(\mathcal{G}(2,2)\) is given in Figure 1.  
In Figure 1, the dots represent integration nodes; each square box represents a [\(-1\),\(1\)]\({}^{2}\) space. The Gauss-Chebyshev integration points are used for each one-dimensional integration node, and the sub-grid corresponding to the multi-exponential \(\mathbf{k}\) that satisfies the requirement is between the two dot-dash lines. Note that the tensor product results for sub-grids \(U_{1}^{1}\otimes U_{2}^{2}\) and \(U_{1}^{3}\otimes U_{2}^{1}\) in Figure 1a have one identical integration node, and the corresponding number of FTG nodes is 9. However, the number of integral nodes of  
Figure 1: Schematic diagram of the construction of a sparse grid. \(\mathcal{G}(2,2)\) in Figure 1b is 13. In lower-dimensional cases (\(n=2\)), achieving the same integration accuracy (\(L=2\)) requires more integration nodes for the SG method, compared to the FTG method (\(13>9\)). This discrepancy arises because SG techniques are specifically designed for high-dimensional problems (typically \(n>5\)). Table 1 presents the number of integration nodes required for both SG and FTG methods, corresponding to different dimensions and accuracy requirements. Table 1 demonstrates that as dimensionality increases, the number of integration nodes required for the FTG method rises rapidly, while the growth rate of required nodes in the SG method is significantly slower. This results in a substantial reduction in the number of integration nodes compared to the full-grid method, effectively alleviating the \"curse of dimensionality\". SG numerical integration is constructed from tensor products of hierarchical difference sets based on a one-dimensional product rule.