## 1 Introduction
[PERSON] et al. [20] proposed a linearized fast high-order time-stepping scheme to solve the spatiotemporal fractional Schrodinger equation. A fast L2-1\({}_{\sigma}\) formula was used to approximate the time of non-uniform mesh. The Fourier spectrum method is used to discretize the space. The result of unconditional convergence is given. [PERSON] and [PERSON] [21] considered the two-dimensional nonlinear time-space fractional Schrodinger equation and applied the second-order fractional backward difference formula in the time direction and the Fourier spectrum method in the space direction to solve the model numerically. By using the generalized discrete Gronwall inequality and the space-time error splitting argument, the convergence of the fast time-step numerical method is simply proved without applying the Courant-Friedrichs-Lewy (CFL) condition. Parameter calibration is a challenging problem in the financial field. Different from the forward problem of solving option prices with known model parameters, the parameter calibration problem refers to solving the model parameters based on the price data observed in the market. The volatility is one of the most important parameters in the option pricing, which is an indicator to measure the uncertainty of the return rate of assets and is used to reflect the risk level of financial assets. However, the volatility cannot be derived directly from market data and needs to be calibrated. This is a typical inverse problem, and it is usually ill-posed because the calibration problem may have no solution or have infinite solutions, or there may be solutions that are discontinuously dependent on observation data; many scholars have conducted research in this area. [PERSON] [22] proposed a local volatility model for option pricing in 1994, calibrated the volatility, and obtained an expression for volatility, which is called the Dupire formula. [PERSON] and [PERSON] [23] studied the Tikhonov regularization method for the volatility calibration problem under the jump diffusion model and used an iterative algorithm to solve the problem to obtain the volatility. [PERSON] et al. [24] reconstructed the implicit local volatility function using the optimal control framework and presented numerical experiments. Based on the Legendre pseudo-spectral method of space discretization and the finite difference method of time discretization, [PERSON] and [PERSON] [25] gave numerical solutions for the optimal control problem of time-fractional diffusion equations. [PERSON] and [PERSON] [26] jointly recovered the mean-reversion parameter \(\gamma\) and the volatility function under the fractional Chan-Karolyi-Longstaff-Sanders (CKLS) stochastic interest rate model. At the same time, numerical experiments are presented that show the fractional CKLS model (\(\gamma=0.8451\)) is more suitable for SSE 50 ETF than the fractional Vasicek model. [PERSON] and [PERSON] [27] investigated the inverse problem of identifying volatility in option pricing. They accomplished this problem by transforming the parabolic equation defined on an unbounded region into a degenerate parabolic equation on a bounded region through a variable substitution. The study establishes that the optimal approximate solution converges effectively to the true solution of the original problem. In our paper, we use the alternating direction multiplier method to simultaneously reconstruct fractional order \(\alpha\) and time-dependent volatility function \(\sigma(t)\). Numerical examples and empirical analysis demonstrate the effectiveness of this method. Our research aims to reconstruct the volatility \(\sigma\) and the fractional order \(\alpha\) in the time-fractional Vasicek model. This unique perspective, to the best of our knowledge, has not been thoroughly explored in previous studies.