Inverse problems governed by PDEs are inherently infinite-dimensional: the data correspond to operators acting on function spaces, and the goal is to recover an unknown function. In practice, however, reconstruction algorithms operate in finite dimensions: only finitely many data pairs are available, and the solution is parameterized. Bridging the infinite-dimensional theory with finite-dimensional algorithms often requires problem-specific assumptions on the PDE under consideration.

In this talk, I will explore this transition and highlight its unexpected connections with randomized numerical linear algebra (RNLA). Although RNLA solvers were originally developed for other purposes, they turn out to offer valuable tools for analyzing data selection in PDE-based inverse problems. In particular, they provide theoretical guarantees on data efficiency, shedding light on how to design reconstruction procedures that balance rigor with computational feasibility.