# Hierarchical Tensor Decomposition of Module Partition for the Mechanical Simulation
## 2 Proposed Methodology
### Optimization Algorithms for HTD  
A convex loss function can be solved using the Alternating Direction Method of Multipliers (ADMM) [48], as shown in Equation (1). HTD is defined as a linear combination of several convex regularization terms that satisfy the ADMM assumption. As part of its goal, the ADMM algorithm aims to minimize the sum of a group of convex functions that are constrained. The augmented Lagrangian for Equation (1) is defined as  
\[ \begin{split}\mathrm{L}=&\|\mathbf{E}\|_{F}^{2}+ \sum_{n=1}^{N}\lambda_{n}\|\mathbf{G}_{v}^{n}\mathbf{B}_{n}\mathbf{G}_{h}^{n} \|_{F}^{2}+\lambda_{\mathbf{S}}\|\mathbf{S}\|_{1}+\frac{1}{2\mu}\|\mathbf{X}- \sum_{n=1}^{N}\mathbf{B}_{n}-\mathbf{S}-\mathbf{E}\|^{2}\\ &+\langle\mathbf{Y},\mathbf{X}-\sum_{n=1}^{N}\mathbf{B}_{n}- \mathbf{S}-\mathbf{E}\rangle\end{split} , \tag{2}\]  
where \(\mathbf{Y}\in\mathbb{R}^{I\times I}\) is the Lagrange multiplier matrix, \(\langle\cdot,\cdot\rangle\) is the outer product operator, and \(\mu\) is a positive penalty scalar. With the increasing number of iteration steps, \(\mu\) becomes smaller, which leads to a suppression of residual \(\|\sum_{n=1}^{N}\mathbf{B}_{n}+\mathbf{S}+\mathbf{E}-\mathbf{X}\|^{2}\) and improves the performance of the ADMM algorithm. ADMM initializes \(\mathbf{B}_{n}^{0}=\mathbf{S}^{0}=\mathbf{E}^{0}\), where \(n=\{1,2,\cdots,N\}\). Then, with \(\mathbf{B}_{n}^{k},\mathbf{S}^{k},\mathbf{E}^{k}\) solved at iteration \(k\), the solution \(\mathbf{B}_{n}^{k+1},\mathbf{S}^{k+1},\mathbf{E}^{k+1}\) at iterations \(k+1\) can be split into \(N+2\) separate problems, each of which can be solved in parallel. Explicitly, \(\mathbf{B}_{n}^{k+1}\), in Equation (2), is updated by \(\frac{\partial\mathrm{L}}{\partial\mathbf{B}_{n}^{k+1}}=\mathbf{0}\) with the other terms fixed, which results in  
\[ \begin{split}\mathbf{B}_{n}^{k+1}:&=\arg\min_{ \mathbf{B}_{n}}\lambda_{n}\|\mathbf{G}_{v}^{n}\mathbf{B}_{n}\mathbf{G}_{h}^{ n}\|_{F}^{2}+\frac{1}{2\mu^{k}}\|\mathbf{X}-\mathbf{B}_{n}-\sum_{j=1}^{n-1} \mathbf{B}_{j}^{k+1}-\sum_{j=n+1}^{N}\mathbf{B}_{j}^{k}-\mathbf{S}^{k}- \mathbf{E}^{k}\|^{2}\\ &+\langle\mathbf{Y}^{k},\mathbf{X}-\mathbf{B}_{n}-\sum_{j=1}^{n-1 }\mathbf{B}_{j}^{k+1}-\sum_{j=i+1}^{N}\mathbf{B}_{j}^{k}-\mathbf{S}^{k}- \mathbf{E}^{k}\rangle=\mathbf{0}\end{split} \]  
where \(k=\{0,1,2,\cdots\}\). Then, \(\mathbf{B}_{n}^{k+1}:=2\lambda_{n}\mu^{k}(\mathbf{G}_{v}^{n})^{T}\mathbf{G}_{v }^{n}\mathbf{B}_{n}^{k+1}\mathbf{G}_{h}^{n}(\mathbf{G}_{h}^{n})^{T}+\mathbf{B} _{n}^{k+1}=\mathbf{X}+\mu^{k}\mathbf{Y}^{k}-\sum_{j=1}^{n-1}\mathbf{B}_{j}^{k+ 1}-\sum_{j=n+1}^{N}\mathbf{B}_{j}^{k}-\mathbf{S}^{k}-\mathbf{E}^{k}\), where \(\mathbf{I}\) is an identity matrix. It can be written as Sylvester equation [49] : \(\mathbf{QD}+\mathbf{DP}=\mathbf{U}\), where \(\mathbf{Q}=2\lambda_{n}(\mathbf{G}_{v}^{n})^{T}\mathbf{G}_{v}^{n}\), \(\mathbf{D}=\mathbf{B}_{n}^{k+1}\), \(\mathbf{P}=(\mathbf{G}_{h}^{n}(\mathbf{G}_{h}^{n})^{T})^{-1}\), and \(\mathbf{U}=(\mathbf{X}+\mu^{k}\mathbf{Y}^{k}-\sum_{j=1}^{n-1}\mathbf{B}_{j}^{k+ 1}-\sum_{j=n+1}^{N}\mathbf{B}_{j}^{k}-\mathbf{S}^{k}-\mathbf{E}^{k})(\mathbf{ G}_{h}^{n}(\mathbf{G}_{h}^{n})^{T})^{-1}\). Then, it can be solved. \(\mathbf{S}^{k+1}\), in Equation (2), is updated by \(\frac{\partial\mathrm{L}}{\partial\mathbf{S}^{k+1}}=\mathbf{0}\) with the other terms fixed, which results in\[\mathbf{S}^{k+1} :=\arg\min_{\mathbf{S}}\lambda_{\mathbf{S}}\|\mathbf{S}\|_{1}+\frac{ 1}{2\mu^{k}}\|\mathbf{X}-\sum_{n=1}^{N}\mathbf{B}_{n}^{k+1}-\mathbf{S}- \mathbf{E}^{k}\|_{F}^{2}\] \[\qquad+(\mathbf{Y}^{k},\mathbf{X}-\sum_{n=1}^{N}\mathbf{B}_{n}^{k +1}-\mathbf{S}-\mathbf{E}^{k})=\mathbf{0},\] \[\mathbf{S}^{k+1} =\mathfrak{S}_{\mu^{k}\lambda_{\mathbf{S}}}(\mathbf{X}+\mu^{k} \mathbf{Y}^{k}-\sum_{n=1}^{N}\mathbf{B}_{n}^{k+1}-\mathbf{E}^{k}),\]  
where \(\mathfrak{S}_{\mu}(\mathbf{x}):=\operatorname{sign}(\mathbf{x})\max(| \mathbf{x}|-\mu,0)\). Then, the solution of the proposed HTD is given in Algorithm 1.