## 3 Regularization Method
### ADMM Algorithm  
Now, we consider the augmented Lagrangian function  
\[\min l_{\mu}(\alpha,\sigma): =||(\alpha,\sigma)||_{*}^{2}+\lambda\sum_{i=1}^{N}\sum_{j=1}^{M_{i} }[V(\tau_{0},r_{0},S_{0},T_{i},K_{ij},\alpha,\sigma(\tau))-\dot{V}_{ij}]^{2}\] \[+\sum_{i=1}^{N}\sum_{j=1}^{M_{i}}\mu_{ij}[V(\tau_{0},r_{0},S_{0},T _{i},K_{ij},\alpha,\sigma(\tau))-\dot{V}_{ij}],\]  
where \(\mu_{ij}>0\) is the Lagrange multiplier. The ADMM algorithm consists of staring from \((\sigma^{0},\alpha^{0},\mu_{ij}^{0})\) to generate inductively a sequence \((\sigma^{k},\alpha^{k},\mu_{ij}^{k})\) as follows:  
* **Step 1:** minimization with repect to \(\sigma\): \[\sigma^{k+1}:=\operatorname*{arg\,min}_{\sigma}l_{\mu^{k}}(\alpha^{k},\sigma),\]
* **Step 2:** minimization with repect to \(\alpha\): \[\alpha^{k+1}:=\operatorname*{arg\,min}_{\alpha}l_{\mu^{k}}(\alpha,\sigma^{k+1}),\]
* **Step 3:** update the Lagrange multiplier: \[\mu_{ij}^{k+1}:=\mu_{ij}^{k}+\beta[V(\tau_{0},r_{0},S_{0},T_{i},K_{ij},\alpha ^{k+1},\sigma^{k+1}(\tau))-\dot{V}_{ij}],\] where \(\beta\) is the step size. For step 1 Similar to Reference [31], introduce a 'false' time parameter \(\theta\) and a function \(\hat{\sigma}(t,\theta)\). Starting with initial guess \(\hat{\sigma}(t,0)\), solve the following parabolic equation \[ \begin{split}\frac{\partial\hat{\sigma}}{\partial\theta}& =\frac{\partial^{2}\hat{\sigma}}{\partial t^{2}}-\lambda\sum_{i=1}^{N }\sum_{j=1}^{M_{i}}\frac{\partial V}{\partial\sigma}(\tau,r_{0},S_{0},T_{i},K _{ij},\alpha^{k},\sigma(\tau))[V(\tau,r_{0},S_{0},T_{i},K_{ij},\alpha^{k}, \sigma(\tau))-\dot{V}_{ij}]\\ &-\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{M_{i}}\mu_{ij}\frac{ \partial V}{\partial\sigma}(\tau,r_{0},S_{0},T_{i},K_{ij},\alpha^{k},\sigma( \tau)).\end{split} \] (10) From the necessary conditions for the existence of extreme values, we can obtain that if \(\hat{\sigma}(t,\theta)\) tends to a steady-state solution as \(\theta\to\infty\), then this solution will satisfy the Euler Lagrange equation for the functional \(l_{\mu}\). In order to solve the above equation, we need to calculate \(\frac{\partial V}{\partial\hat{\sigma}}(\tau,r_{0},S_{0},T_{i},K_{ij},\alpha^ {k},\sigma(\tau))\) by its definition, define \(\delta(\cdot)\) as the Dirac delta function, then, calculate the following expression  
\[\frac{\partial V}{\partial\sigma}(\tau,r_{0},S_{0},T_{i},K_{ij},\alpha^{k}, \sigma(\tau))=[\frac{d}{d\epsilon}V(\tau,r_{0},S_{0},T_{i},K_{ij},\alpha^{k}, \sigma(\tau)+\epsilon\delta)]_{\epsilon=0}.\]  
Solving the variational derivative is similar to the solution approach for Equation (7). First, we define an operator as follows  
\[G_{\sigma}=-_{0}^{c}D_{\tau}^{a}+\frac{1}{2}\sigma(\tau)S^{2}\frac{\partial^{2 }}{\partial S^{2}}+\rho\sigma(\tau)\omega S\frac{\partial^{2}}{\partial S \partial\tau}+\frac{1}{2}\omega^{2}\frac{\partial^{2}}{\partial\tau^{2}}+(a+ br)\frac{\partial}{\partial\tau}+rS\frac{\partial}{\partial S}-rI,\]  
where \(I\) is the identity operator. Therefore, when the volatility function is disturbed, the expected pricing function satisfies the following equation  
\[G_{\sigma+\epsilon\delta}V(\tau,r_{0},S_{0},T_{i},K_{ij},\alpha^{k},\sigma+ \epsilon\delta)=0. \tag{11}\]  
Then, differential (11) with \(\epsilon\) and evaluating for \(\epsilon=0\), we have  
\[G_{\sigma}\frac{\partial V}{\partial\sigma}(\tau,r_{0},S_{0},T_ {i},K_{ij},\alpha^{k},\sigma) =-\sigma(\tau)S^{2}\frac{\partial^{2}V}{\partial S^{2}}(\tau,r_{ 0},S_{0},T_{i},K_{ij},\alpha^{k},\sigma)\] \[-\rho\omega S\frac{\partial^{2}V}{\partial S\partial r}(\tau,r_{0},S_{0},T_{i},K_{ij},\alpha^{k},\sigma).\]  
The variational derivative adheres to both the homogeneous boundary condition and the initial condition.