Inverse problems aim to recover hidden features of a physical system — coefficients, sources, or geometry — from measurements that can only be performed indirectly. In practice, this means reversing a forward model (typically a PDE) using data that are incomplete and contaminated by noise. The purpose of these lecture notes is to build intuition and a robust analytical toolbox by focusing on a small set of prototypical models that already contain the main difficulties of the field. 

We use these prototypes to highlight three recurring themes. First, identifiability: when do the available measurements determine the unknown uniquely, and when do they only determine an “effective” parameter? Second, instability (or weak stability): even when uniqueness holds, inversion can amplify noise dramatically, leading to severe ill-posedness. Third, regularization: stability can be recovered only by injecting additional information — through restricted admissible classes, spectral filtering, or carefully designed measurement setups.

The discussion moves from a one-dimensional elliptic ODE (contrasting boundary versus interior data) to a paradigmatic PDE example, the backward heat equation, where diffusion acts as a low-pass filter and makes high frequencies essentially invisible in final-time data. We then introduce an operator viewpoint that unifies these phenomena and leads naturally to compactness, spectral theory, and singular value decompositions: ill-posedness becomes a quantitative statement about singular value decay and noise amplification. Finally, we connect these abstract mechanisms to the Calderon inverse conductivity problem, briefly commenting on what changes beyond smooth coefficients, on the intrinsic (typically logarithmic) limits of stability, and on what can be achieved with structured priors and finite measurements.