In this talk we discuss the numerical solution of minimisation problems promoting higher-order sparsity properties. In particular, we are interested in total variation minimisation, which enforces sparsity on the gradient of the solution. There are several methods presented in the literature for performing very efficiently total variation minimisation, e.g., for image processing problems of small or medium size. Because of their iterative-sequential formulation, none of them is able to address in real-time extremely large problems, such as 4D imaging (spatial plus temporal dimensions) for functional magnetic-resonance in nuclear medical imaging, astronomical imaging or global terrestrial seismic tomography. For these cases, we propose subspace splitting techniques, which accelerate the numerics by dimension reduction and preconditioning. A careful analysis of these algorithms is furnished with a presentation of their application to some imaging tasks.