Editing Inverse problem (section)

==General statement of the inverse problem==
The inverse problem is the "inverse" of the forward problem: instead of determining the data produced by particular model parameters, we want to determine the model parameters that produce the data <math>d_\text{obs}</math> that is the observation we have recorded (the subscript '''obs''' stands for observed).
Our goal, in other words, is to determine the model parameters <math>p</math> such that (at least approximately)
<math display="block"> d_\text{obs} = F(p)</math>
where <math>F</math> is the forward map. We denote by <math>M</math> the (possibly infinite) number of model parameters, and by <math>N</math> the number of recorded data.
We introduce some useful concepts and the associated notations that will be used below:
* The '''space of models''' denoted by <math>P</math>: the [[vector space]] spanned by model parameters; it has <math>M</math> dimensions;
* The '''space of data''' denoted by <math>D</math>: <math>D = \R^N</math> if we organize the measured samples in a vector with <math>N</math> components (if our measurements consist of functions, <math>D</math> is a vector space with infinite dimensions);
* <math>F(p)</math>: the '''response of model''' <math>p</math>; it consists of the '''data predicted by model''' <math>p</math>;
* <math>F(P)</math>: the image of <math>P</math> by the forward map, it is a subset of <math>D</math> (but not a subspace unless <math>F</math> is linear) made of responses of all models;
* <math>d_\text{obs} - F(p)</math>: the '''data misfits (or residuals)''' associated with model  <math>p</math>: they can be arranged as a vector, an element of <math>D</math>.
The concept of residuals is very important: in the scope of finding a model that matches the data, '''their analysis reveals if the considered model can be considered as realistic or not'''. Systematic unrealistic discrepancies between the data and the model responses also reveals that the forward map is inadequate and may give insights about an improved forward map.

When operator <math>F</math> is linear, the inverse problem is linear. Otherwise, that is most often, the inverse problem is nonlinear.
Also, models cannot always be described by a finite number of parameters. It is the case when we look for [[Distributed parameter system|distributed parameters]] (a distribution of wave-speeds for instance): in such cases the goal of the inverse problem is to retrieve one or several functions. Such inverse problems are inverse problems with infinite dimension.