Inverse problems require data to make inference about unknown parameters (typically spatial or spatiotemporal functions) of interest. However, only a limited amount of data can be collected, so it is imperative to collect data optimally to minimize the uncertainty associated with the unknown parameters. This talk will make new connections between to column subset selection, which is a well-studied problem in numerical linear algebra. This connection leads to several efficient algorithms, which are further accelerated by using randomization. The resulting algorithms are robust, computationally efficient, amenable to parallelization, require virtually no parameter tuning, and come with strong theoretical guarantees. Some of the proposed algorithms are also adjoint-free which is beneficial in situations, where the adjoint is expensive to evaluate or is not available. We demonstrate the performance of the algorithms on model problems involving partial differential equations and tomography.