Editing Inverse problem (section)

==== Tools to overcome the second difficulty ====
As mentioned above, noise may be such that our measurements are not the image of any model, so that we cannot look for a model that produces the data but rather look for [[model selection|the best (or optimal) model]]: that is, the one that best matches the data. This leads us to minimize an [[objective function]], namely a [[functional (mathematics)|functional]]  that quantifies how big the residuals are or how far the predicted data are from the observed data. Of course, when we have perfect data (i.e. no noise) then the recovered model should fit the observed data perfectly. A standard objective function, <math>\varphi</math>, is of the form:
<math>\varphi(p) = \|F p-d_\text{obs} \|^2 </math>
where <math>\| \cdot \| </math> is the Euclidean norm (it will be the [[Lp space|<math>L^2</math> norm]] when the measurements are functions instead of samples) of the residuals. This approach amounts to making use of [[Least squares|ordinary least squares]], an approach widely used in statistics. However, the Euclidean norm is known to be very sensitive to outliers: to avoid this difficulty we may think of using other distances, for instance the <math>L^1</math> norm, in replacement of the <math>L^2</math> norm.