Nuclear theories span a broad range of refinement from the liquid drop model to highly involved ab-initio theories. In the middle of that scale lie self-consistent models, also known as nuclear Density Functional Theory (DFT). They constitute presently the best compromise between expense and benefit. The talk will give an overview of thepresent status of this class of models.

Nuclear DFT aims at a universal description of nuclei and nuclear matter.  Its structure is deduced from general considerations.  Its parameters require calibration to a well selected set of empirical data.  The goal is to cover as many observables as possible. Nuclear radii and binding energies are the most prominent observables for which the present DFT models work very well.  More demanding are excitation properties and isotopic radius differences. These observables probe the models and stimulate actual developments. The talk will report on both, successes and still unsolved problems.

The ab initio approach to nuclear structure allows us to describe atomic nuclei with controlled and systematically improvable approximations. Applying it to nuclei that are at the same time both heavy and open-shell is largely impossible with current many-body techniques. This is due to the computational cost of handling huge dense tensors. 

I will show how some surprisingly simple tricks may help us to tackle this hurdle. These tricks are driven by the observation that different contributions to ab initio calculations describe different physics. We leverage this by using adapted model spaces. 

In addition, we use modern linear algebra methods to develop dimensionality reduction techniques based on the singular value decomposition. By avoiding the construction of large many-body tensors in the first place, we are able to extend the reach of ab initio calculations to nuclei where standard approaches would be too expensive to run.