# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 2 The Bayesian Formulation of the Distributed Dynamic Load Identification
### Distributed Dynamic Load Identification Algorithm Based on the Bayesian Method
By the logarithm and the negative of Equation (26), we have Equation (28):  
\[\hat{\theta}\big{\{}\hat{A},\hat{\tau},\hat{\lambda}\big{\}}=\text{argmax}\{p(A,\lambda,\tau|\mathbf{y})\,\} \tag{27}\]  
\[J(A,\lambda,\tau)=\frac{\tau}{2}\|QA-\mathbf{y}\|_{2}^{2}+\frac{\lambda}{2}\|A\|_{2 }^{2}-(\frac{n}{2}+\alpha_{2}-1)\ln\tau-(\frac{m}{2}+\alpha_{1}-1)\ln\lambda+ \beta_{1}\lambda+\beta_{2}\tau \tag{28}\]  
Taking the partial derivative of \(J(A,\lambda,\tau)\) and setting as 0, we have  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]0  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]1  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]2  
That is,  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]3  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]4  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]5  
Thus, the orthogonal polynomial coefficient vectors and the unknown parameters are given by:  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]6  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]7  
\[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]8  
Therefore, the unknown distributed load is identified based on the coefficient vectors \(A\) and the orthogonal polynomials. Moreover, the common difficulty to solve the maximum posterior estimator of the hierarchical model is to obtain the minimum value of the function. Some approaches have been developed, such as the gradient iterative, the simulated annealing method, and the genetic algorithm. In this paper, we mainly employ the alternating direction iterative algorithm to solve the optimization problem. The following steps present the numerical iteration of the unknown parameters, and the schematic diagram of the distributed dynamic load identification is shown in \[p(A\,|\mathbf{y})\propto(\sigma^{2})^{-\frac{\eta}{2}}\exp(-\frac{\|QA-\mathbf{y}\|_{2 }^{2}}{2\sigma^{2}})\lambda^{\frac{\eta}{2}}\exp(-\frac{1}{2}\lambda A^{ \mathsf{T}}\mathbf{W}A) \tag{20}\]9  
(1) Select the appropriate initial value \(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2},\lambda,\tau\), and set \(k=1\). (2) Determine the regularization parameters \(\eta_{k}=\lambda_{k}/\tau_{k}\). (3) Determine the orthogonal polynomial coefficient vectors \(A_{k}=(Q^{\text{T}}Q+\eta_{k}\mathbf{I})^{-1}Q^{\text{T}}\mathbf{y}\).