On generalized iterated Tikhonov regularization with operator-dependent seminorms

Abstract

We investigate the recently introduced Tikhonov regularization filters with penalty terms having seminorms that depend on the operator itself. Exploiting the singular value decomposition of the operator, we provide optimal order conditions, smoothing properties, and a general condition (with a minor condition of the seminorm) for the saturation level. Moreover, we introduce and analyze both stationary and nonstationary iterative counterparts of the generalized Tikhonov method with operator-dependent seminorms. We establish their convergence rate under conditions affecting only the iteration parameters, proving that they overcome the saturation result. Finally, some selected numerical results confirm the effectiveness of the proposed regularization filters.

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

“…Moreover, Table 1 also shows that when ∈ [20,30,40,50], RE and iterative steps of the identified result with application of the above methods are seen to increase with the noise level; then CC is the opposite.…”

“…First of all, when ∈ [20,30,40,50] and the noise level is set to 10%, 5%, and 1%, then RE and CC and iterative steps of the identified result are listed in Table 1. Table 1 shows that whether the noise level is set to 10%, 5%, or 1%, RE and iterative steps of the identified result with application of the above methods are seen to increase with ; then CC is the opposite.…”

“…Furthermore, the FTR method has been applied to many fields and obtained some remarkable achievements [18][19][20]. M. E. Hochstenbach et al [21] used the FTR method to obtain approximate solution of higher quality which is better than the standard TR method in solving linear system equation with a severe ill-posedness.…”

A Posteriori Fractional Tikhonov Regularization Method for the Problem of Analytic Continuation

“…Tikhonov et al [21]. Related research on fractional regularization methods can refer to the literature [22][23][24][25][26][27][28][29]. The fractional Tikhonov method with the a priori parameter for the same analytic continuation problem has been researched by [30].…”

Graph approximation and generalized Tikhonov regularization for signal deblurring

“…For a reference, see [1]. Clearly, in the real applications it is not usually granted that K and A commute, and in some works it was proposed to use A = F (KK * ), where F is a suitable function on the spectrum of KK * that mimic the spectral distribution of the Laplace operator, see [4], [15], [18].…”