## 2 Green Kernel Function  
For a multi-degrees-of-freedom system, the general vibration differential equation is often written as  
\[\left\{\begin{array}{l}\dot{\mathbf{M}}\dot{\mathbf{x}}(t)+\dot{\mathbf{C}}\dot{\mathbf{x}}( t)+\mathbf{K}\mathbf{x}(t)=\mathbf{f}(t)\\ \mathbf{x}(0)=\mathbf{x}_{0},\dot{\mathbf{x}}(0)=\dot{\mathbf{x}}_{0}\end{array}\right.. \tag{1}\]  
Here, \(\mathbf{M},\mathbf{K},\mathbf{C}\) are the mass, stiffness and damping matrix of the system, respectively, while \(\mathbf{x},\dot{\mathbf{x}},\ddot{\mathbf{x}}\) are the displacement, the velocity, and the acceleration vectors of the system response. The variable \(f\) represents external forces acting on the system. Assuming a zero initial state, i.e., \(x_{0}\) and \(\dot{x}_{0}\) are both zero, the displacement response of the \(i\)-th degree of freedom is  
\[x_{i}(t)=\sum\limits_{j=1}^{n}x_{ij}(t)=\sum\limits_{j=1}^{n}\int_{0}^{t}h_{ij} (t-\tau)f_{j}(\tau)d\tau, \tag{2}\]  
where \(x_{ij}(t)\) is the displacement response of \(i\)-th degree of freedom caused by the external force acting on the \(j\)-th degree, and \(h_{ij}\) is the unit impulse response function, while \(f_{j}\) represents the external force acting on the \(j\)-th degree of freedom.  
Assuming the sampling interval of the response is \(\Delta t\), \(x_{ij}\) can be simplified using integral discretization  
\[x_{ij}(n\Delta t)\approx\left[\sum\limits_{i=0}^{n-1}h(t-i\Delta t)f(i\Delta t )\right]\Delta t. \tag{3}\]  
Rewrite Equation (3) into matrix form as  
\[\left[\begin{array}{c}x_{ij}(1)\\ x_{ij}(2)\\ \vdots\\ x_{ij}(m)\end{array}\right]=\left[\begin{array}{cccc}h_{ij}(1)\\ h_{ij}(2)&h_{ij}(1)\\ \vdots&\vdots&\ddots\\ h_{ij}(m)&h_{ij}(m-1)&\cdots&h_{ij}(1)\end{array}\right]\left[\begin{array}[] {c}f_{j}(0)\\ f_{j}(1)\\ \vdots\\ f_{j}(m-1)\end{array}\right]\Delta t. \tag{4}\]  
Equation (4) can be abbreviated as  
\[\mathbf{Y}_{ij}\mathbf{=}H_{ij}\mathbf{F}_{j}. \tag{5}\]  
Here, \(Y_{ij}\) is a discrete vector form of \(x_{ij}\), while \(F_{j}\) is a discrete vector form of \(f_{j}\). When only one load is acting on the system, i.e., on the \(j\)-th degree of freedom, the response of the \(i\)-th degree of freedom can be expressed as  
\[\mathbf{Y}_{i}=\mathbf{Y}_{ij}=\mathbf{H}_{ij}\mathbf{F}_{j}. \tag{6}\]  
Equation (6) is the linear equations in the case of a single input and a single output. The unknown force \(F_{j}\) can be obtained by solving Equation (6). Here, \(H_{ij}\) is the kernel function matrix. When there are \(M\) loads acting on different degrees of freedom, the number of the responses to be collected \(N\) needs to meet the following formula \(M\leq N\). Thus, the relationship between \(M\) loads and \(N\) responses can be obtained:  
\[\mathbf{Y}_{i}=\sum\limits_{j=1}^{M}\mathbf{Y}_{ij}=\sum\limits_{j=1}^{M} \mathbf{H}_{ij}\mathbf{F}_{j}\qquad(i=1,2,\cdots,N)\enspace. \tag{7}\]  
Rewrite Equation (7) in matrix form as  
\[\left[\begin{array}{c}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{N}\end{array}\right]=\left[\begin{array}{cccc}\mathbf{H}_{11}&\mathbf{H }_{12}&\cdots&\mathbf{H}_{1M}\\ \mathbf{H}_{21}&\mathbf{H}_{22}&\cdots&\mathbf{H}_{2M}\\ \vdots&\vdots&&\vdots\\ \mathbf{H}_{N1}&\mathbf{H}_{N2}&\cdots&\mathbf{H}_{NM}\end{array}\right] \left[\begin{array}{c}F_{1}\\ F_{2}\\ \vdots\\ F_{M}\end{array}\right]. \tag{8}\]  
Equation (8) is the linear equations in the case of multiple inputs and outputs, and the unknown force \(F_{1},F_{2},\cdots,F_{M}\) can be obtained by solving this equation. Clearly, Equations (6) and (8) both conform to the form of \(\boldsymbol{y}=\boldsymbol{H}\!f\). The inversion of equations in this form usually introduces a significant error into the solution due to the ill-posedness of the problem that is caused by the ill-posedness of the kernel function matrix \(\boldsymbol{H}\), especially for large systems, i.e., hundreds of degrees of freedom. Next, we discuss reducing the ill-posedness problem using Augmented Tikhonov regularization method.