On Tikhonov-type regularization with approximated penalty terms

Abstract

<p style='text-indent:20px;'>In this paper, we deal with (nonlinear) ill-posed problems that are regularized by minimizing Tikhonov-type functionals. If the minimization is tedious for some penalty term <inline-formula><tex-math id="M1">\begin{document}$ P_0 $\end{document}</tex-math></inline-formula>, we approximate it by a family of penalty terms <inline-formula><tex-math id="M2">\begin{document}$ ({P_\beta}) $\end{document}</tex-math></inline-formula> having nicer properties and analyze what happens as <inline-formula><tex-math id="M3">\begin{document}$ \beta\to 0 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We investigate the discrepancy principle for the choice of the regularization parameter and apply all results to linear problems with sparsity constraints. Numerical results show that the proposed method yields good results.</p>