The Kaczmarz method is a numerical method to solve systems of linear equations and compute minimum-norm solutions of underdetermined systems. Because the method has very low memory requirements it has gained new attention in recent years.

In this talk we propose a flexible algorithmic framework that extends the Kaczmarz method such that it also can handle, among other things, sparse solutions of underdetermined systems. The framework also includes other known methods such as the linearized Bregman iteration and can be extended to compute solutions of minimal total variation methods. The analysis is build on the notion of split feasibility problems. Furthermore, we show that randomization of the method allows to prove linear convergence of the method.

The algorithmic framework has a variety of applications and is especially useful for problems in which the linear measurements are slow and expensive to obtain. We present examples for online compressed sensing, TV tomographic reconstruction and radio interferometry.