Well-posed and ill-posed inverse problems and diagnostic tools
Inverse techniques and parameter estimation Part 2

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 1a.

Inversion consists in determining the vector x from the data y_mo;;, the model being represented by the operator H. The ideal inversion, shown in figure 1b, can be...