All inverse problems rely on data to recover unknown parameters, yet not all data are equally informative. This raises the central question of data selection. A distinctive challenge in PDE-based inverse problems is their inherently infinite-dimensional nature: both the parameter space and the design space are infinite, which greatly complicates the selection process. Somewhat unexpectedly, randomized numerical linear algebra (RNLA), originally developed in very different contexts, has provided powerful tools for addressing this challenge. These methods are inherently probabilistic, with guarantees typically stating that information is preserved with probability at least 1-p when using N randomly selected, weighted samples. Here, the notion of information can take different mathematical forms depending on the setting. In this review, we survey the problem of data selection in PDE-based inverse problems, emphasize its unique infinite-dimensional aspects, and highlight how RNLA strategies have been adapted and applied in this context.

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