A Novel Fractional Tikhonov Regularization Coupled with an Improved Super-Memory Gradient Method and Application to Dynamic Force Identification Problems

Abstract

This paper presents a novel inverse technique to provide a stable optimal solution for the ill-posed dynamic force identification problems. Due to ill-posedness of the inverse problems, conventional numerical approach for solutions results in arbitrarily large errors in solution. However, in the field of engineering mathematics, there are famous mathematical algorithms to tackle the ill-posed problem, which are known as regularization techniques. In the current study, a novel fractional Tikhonov regularization (NFTR) method is proposed to perform an effective inverse identification, then the smoothing functional of the ill-posed problem processed by the proposed method is regarded as an optimization problem, and finally a stable optimal solution is obtained by using an improved super-memory gradient (ISMG) method. The result of the present method is compared with that of the standard TR method and FTR method; new findings can be obtained; that is, the present method can improve accuracy and stability of the inverse identification problem, robustness is stronger, and the rate of convergence is faster. The applicability and efficiency of the present method in this paper are demonstrated through a mathematical example and an engineering example.