TY - JOUR
TI - Various minimization problems involving the total variation in one dimension
DO - https://doi.org/doi:10.7282/T3TX3JHB
PY - 2017
AB - We consider certain minimization problems in one dimension. The first one is the ROF filter, which was originally introduced in the context of image processing. For the one-dimensional case, we show that the problem can be reformulated as a variational inequality, and use this to extend existing regularity results. In addition, we look at the jump set of solutions and investigate its behavior as certain parameters are changed. The second functional to be considered arises in the context of regularized interpolation. The second problem is in the context of regularized interpolation, and the functional to be minimized uses the total variation as a penalty term. This problem is shown to be ill-posed with multiple solutions, and the set of solutions is described. Next, we introduce further regularization methods that lead to unique solutions, and use these regularized solutions to determine special solutions of the original problem. Finally, we consider the functional in the space L^2. To investigate it, the lower semicontinuous envelope is constructed. We then characterize the minimizers of the LSC envelope.
KW - Mathematics
KW - Variational inequalities (Mathematics)
LA - eng
ER -