Mixed Tikhonov regularization in Banach spaces based on domain decomposition Ivan Cimrák b , Valdemar Melicher a,⇑,1 a Department of Mathematical Analysis, NaM 2 Research Group, Ghent University, Galglaan 2, 9000 Ghent, Belgium b Department of Software Technologies, Faculty of Management Science and Informatics, University of Z ˇ ilina, Univerzitná 8215/1, 01026 Z ˇ ilina, Slovakia article info Keywords: Tikhonov regularization Domain decomposition Banach spaces Bounded variation Total variation abstract We analyze Tikhonov regularization where the forward operator is defined on a direct sum U of several Banach spaces U i ; i ¼ 1; ... ; m. The regularization term on U is a sum of differ- ent regularizations on each U i . The theoretical framework, known for the case m ¼ 1, can be easily reformulated to the case m > 1. Under certain assumptions on weak topologies and on the forward operator it is possible to extend the results on the well-posedness of the minimization process, on its stability with respect to the data, and on the correspond- ing existence and convergence results, to the general case m P 1. We consider two particular regularizations, the total or the so-called bounded variation (BV) regularization and the smooth (H 1 ) regularization. Assume that the domain X is decomposed into two non-overlapping subdomains X 1 ; X 2 . The direct sum of Banach spaces for m ¼ 2 will be U ¼ H 1 ðX 1 Þ BV ðX 2 Þ and the underlying regularization will be mixed H 1 –BV regularization. We define proper weak topology on U and prove that the assumptions for the above mentioned framework are fulfilled. In such a way we get the theoretical results for a broad range of H 1 –BV regularizations including the image denois- ing for the identity forward operator, the deconvolution problems for the convolution for- ward operator, or any other inverse problem for the continuous nonlinear forward operators. We demonstrate how the mixed regularization can be applied in practice. We suggest a denoising algorithm and compare its behavior with the BV regularization and the H 1 reg- ularization. We discuss the results of the denoising on several real world images justifying the usefulness of the mixed H 1 –BV regularization. Ó 2012 Published by Elsevier Inc. 1. Introduction In many areas of inverse problems, the function to be reconstructed is expected to have different properties on different parts of the computational domain. As examples we mention magnetic resonance imaging or electrical impedance tomog- raphy where the image depicts different tissues with different physical and structural properties. Another example is image processing where the image contains smooth regions as well as sharp edges or textures. We use a domain decomposition technique that splits the domain into several parts and on each part the most suitable regularization is applied, chosen according to the expected level of smoothness of the solution. 0096-3003/$ - see front matter Ó 2012 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.amc.2012.05.042 ⇑ Corresponding author. E-mail addresses: Ivan.Cimrak@fri.uniza.sk (I. Cimrák), Valdemar.Melicher@UGent.be (V. Melicher). 1 Valdemar Melicher would like to acknowledge the support of GOA project BOF 07/GOA/006 of Ghent University, Ghent, Belgium and of BOF doctor- assistant research mandate 01P09209T of Ghent University, Ghent, Belgium. Applied Mathematics and Computation 218 (2012) 11583–11596 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc