Prescribed-time neurodynamic ADMM for Tikhonov regularization: Algorithms, circuits and application

Among the various continuous neurodynamic algorithms obtained through conventional discrete algorithms, two important methods are to utilize the continuous limit technique to transform the alternating direction method of multipliers (ADMM) and the linearized alternating direction method of multipliers (LADMM) into their neurodynamic forms which are named neurodynamic alternating direction method of multipliers (NADMM) and neurodynamic linearized alternating direction method of multipliers (NLADMM), respectively. However, existing NADMM and NLADMM exhibit only asymptotic or exponential stability, which limits their applicability in scenarios requiring guaranteed convergence at precise settling time. To address this challenge, this paper proposes prescribed-time NADMM (PNADMM) and prescribed-time NLADMM (PNLADMM) for solving Tikhonov regularization. Through rigorous theoretical analysis, it can be demonstrated that both PNADMM and PNLADMM can obtain the correct real-time solution within the prescribed time and are independent of the initial conditions. In addition, the PNADMM and PNLADMM have superior robustness under any bounded noise. Furthermore, analog circuits using operational amplifiers, resistors, and capacitors are designed to realize the proposed neurodynamic models, and simulation experiments verify the feasibility of the proposed analog circuits in solving Tikhonov regularization in a high-speed and hardware-based way. Finally, the effectiveness of the proposed models is verified through numerical simulations and image deblurring.