The regularization technique aimed at finding stable approximate solutions of inverse and ill-posed problems by incorporating additional a priori information in a sophisticated manner has been developed comprehensively since A.N. Tikhonov's work. Today such an approach is essential and widely applied to various inverse problems, and it is admitted as a main technique against the intrinsic unavoidable instability in form of ill-posedness or ill-conditioning. Although the fundamental setting of the regularization has been well established, there is recent remarkable progress in the theory itself and the expansion of the applicability. One reason is the tremendous improvement of computer ability providing us with substantially greater possibilities of gaining numerical solutions for more relevant regularization techniques with possibly higher accuracy techniques. Thus one needs to sharpen regularization techniques and deepen the theory.
In such a trend, we have edited the current special issue to collect the latest six papers. The first four papers among the six are concerned with new developments of the regularization techniques. The first paper by D. Lorenz and A. Rösch exploits regularization ideas and concepts developed originally for inverse problems to constrained control problems, and the second paper by S. Lu, S. V. Pereverzev and U. Tautenhahn establishes a new method for the regularized total least-sqaures approach. New results of the extension of Tikhonov type regularization methods to the Banach space with focus on convergence rates for nonlinear ill-posed operator equations are presented in the third and fourth papers. B. Hofmann and M. Yamamoto discuss in the third paper the chances of variational inequalities for low rate results, whereas in the fourth paper A. Neubauer et al. consider improved results for enhanced convergence rates. The remaning two papers are concerned with novel applications of regularization techniques. In the fifth paper by L. Bourgeois and J. Dardé, the regularization is applied to ill-posed elliptic Cauchy problems with sharp conditional stability estimates. In the sixth paper by J. Liu and M. Yamamoto, the regularization is applied to a backward problem in time for the time-fractional diffusion equation which appears for modelling the diffusion in porus media.
The editors hope that the special issue may be helpful in order to grasp several updated research directions in regularization techniques and create further developments.