Editing Inverse problem (section)

==== A non-convex data misfit function ====
The forward map being nonlinear, the data misfit function is likely to be non-convex, making local minimization techniques inefficient. Several approaches have been investigated to overcome this difficulty: 
* use of global optimization techniques such as sampling of the posterior density function and [[Metropolis–Hastings algorithm|Metropolis algorithm]] in the inverse problem probabilistic framework,<ref>{{cite journal |last1=Koren |first1=Zvi |last2=Mosegaard |first2=Klaus |last3=Landa |first3=Evgeny |last4=Thore |first4=Pierre |last5=Tarantola |first5=Albert |title=Monte Carlo Estimation and Resolution Analysis of Seismic Background Velocities |journal=Journal of Geophysical Research |date=1991 |volume=96 |issue=B12|pages=20289–20299 |doi=10.1029/91JB02278 |bibcode=1991JGR....9620289K }}</ref> genetic algorithms (alone or in combination with Metropolis algorithm: see<ref>{{cite journal| last1=Tahmasebi|first1=Pejman| last2=Javadpour|first2=Farzam| last3=Sahimi|first3=Muhammad| title=Stochastic shale permeability matching: Three-dimensional characterization and modeling| journal=International Journal of Coal Geology| date=August 2016| volume=165| pages=231–242| doi=10.1016/j.coal.2016.08.024|bibcode=2016IJCG..165..231T | url=https://www.researchgate.net/publication/307626119}}</ref> for an application to the determination of permeabilities that match the existing permeability data), neural networks, regularization techniques including multi scale analysis;
* reformulation of the least-squares objective function so as to make it smoother (see<ref name="ReferenceB"/><ref name="ReferenceC"/> for the inverse problem in the wave equations.)