Editing Inverse problem (section)

=== Mathematical aspects: Hadamard's questions ===

The questions concern well-posedness: Does the least-squares problem have a unique solution which depends continuously on the data (stability problem)? It is the first question, but it is also a difficult one because of the non-linearity of <math>F</math>. 
In order to see where the difficulties arise from, Chavent<ref name=Chavent>{{cite book |last1=Chavent |first1=Guy |title=Nonlinear Least Squares for Inverse problems |date=2010 |publisher=Springer |isbn=978-90-481-2785-6}}</ref> proposed to conceptually split the minimization of the data misfit function into two consecutive steps (<math>P_\text{adm}</math> is the subset of admissible models):
* projection step: given <math>d_\text{obs}</math> find a projection on <math>F(P_\text{adm})</math> (nearest point on <math>F(P_\text{adm})</math> according to the distance involved in the definition of the objective function)
* given this projection find one pre-image that is a model whose image by operator <math>F</math> is this projection.
Difficulties can - and usually will - arise in both steps:
# operator <math>F</math> is not likely to be one-to-one, therefore there can be more than one pre-image,
# even when <math>F</math> is one-to-one, its inverse may not be continuous over <math>F(P)</math>,
# the projection on <math>F(P_\text{adm})</math> may not exist, should this set be not closed,
# the projection on <math>F(P_\text{adm})</math> can be non-unique and not continuous as this can be non-convex due to the non-linearity of <math>F</math>.
We refer to Chavent<ref name=Chavent/> for a mathematical analysis of these points.