1. Well-posed and ill-posed inverse problems and diagnostic tools

1.1 Hadamard's conditions

Let x be the vector of unknown parameters to be identified ( x ΠX and y _mo the modeled vector ( y _mo ΠY.

We're talking about finite dimensions here, and X and Y are two normed vector spaces.

Consider an operator H;;, linear or not, representing the model and applying X in Y. We then have y _mo = H x and both the model and the direct problem (finding y _mo knowing ;;x) are shown in figure 1 a.

Inversion consists in...