Editing Inverse problem (section)

==== Computation of the gradient of the objective function ====

Inverse problems, especially in infinite dimension, may be large size, thus requiring important computing time. When the forward map is nonlinear, the computational difficulties increase and minimizing the objective function can be difficult. Contrary to the linear situation, an explicit use of the Hessian matrix for solving the normal equations does not make sense here: the Hessian matrix varies with models. Much more effective is the evaluation of the gradient of the objective function for some models. Important computational effort can be saved when we can avoid the very heavy computation of the [[Jacobian matrix and determinant|Jacobian]] (often called "[[Fréchet derivative]]s"): the adjoint state method, proposed by Chavent and Lions,<ref>{{cite book |last1=Chavent |first1=Guy |title=Identification de coefficients répartis dans les équations aux dérivées partielles |date=1971 |publisher=Thèse d'Etat |location=Université Paris 6}}</ref> is aimed to avoid this very heavy computation. It is now very widely used.<ref>{{cite journal |last1=Plessix |first1=René |title=A review of the adjoint-state method for computing the gradient of a functional with geophysical applications |journal=Geophysical Journal International |date=2006 |volume=167 |issue=2 |pages=495–503 |doi=10.1111/j.1365-246X.2006.02978.x |bibcode=2006GeoJI.167..495P |doi-access=free }}</ref>
[[File:Inversion algorithm.jpg|thumb|352x352px|An inversion algorithm (published Under a Creative Commons license, CC BY-NC-ND by Elsevier) <ref name=":0" />]]