Editing Inverse problem (section)

==Non-linear inverse problems==

Non-linear inverse problems constitute an inherently more difficult family of inverse problems. Here the forward map <math>F</math> is a non-linear operator. Modeling of physical phenomena often relies on the solution of a partial differential equation (see table above except for gravity law): although these partial differential equations are often linear, the physical parameters that appear in these equations depend in a non-linear way of the state of the system and therefore on the observations we make on it.

=== Some classical non-linear inverse problems ===

==== Inverse scattering problems ====

Whereas linear inverse problems were completely solved from the theoretical point of view at the end of the nineteenth century {{Citation needed|reason=What specific date ? Is there a publication that marks the complete solution of linear inverse problems ?|date=November 2019}}, only one class of nonlinear inverse problems was so before 1970, that of inverse spectral and (one space dimension) [[inverse scattering problem]]s, after the seminal work of the Russian mathematical school ([[Mark Grigoryevich Krein|Krein]], [[Israel Gelfand|Gelfand]], Levitan, [[Vladimir Marchenko|Marchenko]]). A large review of the results has been given by Chadan and Sabatier in their book "Inverse Problems of Quantum Scattering Theory" (two editions in English, one in Russian).

In this kind of problem, data are properties of the spectrum of a linear operator which describe the scattering. The spectrum is made of [[eigenvalue]]s and [[eigenfunction]]s, forming together the "discrete spectrum", and generalizations, called the continuous spectrum. The very remarkable physical point is that scattering experiments give information only on the continuous spectrum, and that knowing its full spectrum is both necessary and sufficient in recovering the scattering operator. Hence we have invisible parameters, much more interesting than the null space which has a similar property in linear inverse problems. In addition, there are physical motions in which the spectrum of such an operator is conserved as a consequence of such motion. This phenomenon is governed by special nonlinear partial differential evolution equations, for example the [[Korteweg–de Vries equation]]. If the spectrum of the operator is reduced to one single eigenvalue, its corresponding motion is that of a single bump that propagates at constant velocity and without deformation, a solitary wave called a "[[soliton]]".

A perfect signal and its generalizations for the Korteweg–de Vries equation or other integrable nonlinear partial differential equations are of great interest, with many possible applications. This area has been studied as a branch of mathematical physics since the 1970s. Nonlinear inverse problems are also currently studied in many fields of applied science (acoustics, mechanics, quantum mechanics, electromagnetic scattering - in particular radar soundings, seismic soundings, and nearly all imaging modalities).

A final example related to the [[Riemann hypothesis]] was given by Wu and Sprung, the idea is that in the [[semiclassical physics|semiclassical]] [[old quantum theory]] the inverse of the potential inside the Hamiltonian is proportional to the [[half-derivative]] of the eigenvalues (energies) counting function&nbsp;''n''(''x'').

====Permeability matching in oil and gas reservoirs====
The goal is to recover the diffusion coefficient in the [[Diffusion equation|parabolic partial differential equation]] that models single phase fluid flows in porous media. This problem has been the object of many studies since a pioneering work carried out in the early seventies.<ref>{{cite journal |last1=Chavent |first1=Guy |last2=Lemonnier |first2=Patrick |last3=Dupuy |first3=Michel |title=History Matching by Use of Optimal Control Theory |journal=Society of Petroleum Engineers Journal |date=1975 |volume=15 |issue=2 |pages=74–86 |doi=10.2118/4627-PA}}</ref> Concerning two-phase flows an important problem is to estimate the relative permeabilities and the capillary pressures.<ref>{{cite journal |last1=Chavent |first1=Guy |last2=Cohen |first2=Gary |last3=Espy |first3=M. |title=Determination of relative permeabilities and capillary pressures by an automatic adjustment method |journal=Society of Petroleum Engineers |date=1980 |issue=January |doi=10.2118/9237-MS}}</ref>

==== Inverse problems in the wave equations ====

The goal is to recover the wave-speeds (P and S waves) and the density distributions from [[seismogram]]s. Such inverse problems are of prime interest in seismology and [[exploration geophysics]].
We can basically consider two mathematical models:
* The [[Wave equation|acoustic wave equation]] (in which S waves are ignored when the space dimensions are 2 or 3)
* The [[Linear elasticity|elastodynamics equation]] in which the P and S wave velocities can be derived from the [[Lamé parameters]] and the density.
These basic [[Hyperbolic partial differential equation|hyperbolic equations]] can be upgraded by incorporating [[attenuation]], [[anisotropy]], ...

The solution of the inverse problem in the 1D wave equation has been the object of many studies. It is one of the very few non-linear inverse problems for which we can prove the uniqueness of the solution.<ref name="sciencedirect.com"/> The analysis of the stability of the solution was another challenge.<ref name="ReferenceA">{{cite journal |last1=Bamberger |first1=Alain |last2=Chavent |first2=Guy |last3=Lailly |first3=Patrick |title=About the stability of the inverse problem in the 1D wave equation, application to the interpretation of the seismic profiles |journal=Journal of Applied Mathematics and Optimization |date=1979 |volume=5 |pages=1–47 |doi=10.1007/bf01442542 |s2cid=122428594 }}</ref> Practical applications, using the least-squares approach, were developed.<ref name="ReferenceA"/><ref>{{cite journal |last1=Macé |first1=Danièle |last2=Lailly |first2=Patrick |title=Solution of the VSP one dimensional inverse problem |journal=Geophysical Prospecting |date=1986 |volume=34 |issue=7 |pages=1002–1021 |osti=6901651 |doi=10.1111/j.1365-2478.1986.tb00510.x |bibcode=1986GeopP..34.1002M }}</ref>
Extension to 2D or 3D problems and to the elastodynamics equations was attempted since the 80's but turned out to be very difficult ! This problem often referred to as Full Waveform Inversion (FWI), is not yet completely solved: among the main difficulties are the existence of non-Gaussian noise into the seismograms, cycle-skipping issues (also known as phase ambiguity), and the chaotic behavior of the data misfit function.<ref>{{cite journal |last1=Virieux |first1=Jean |last2=Operto |first2=Stéphane |title=An overview of full-waveform inversion in exploration geophysics |journal= Geophysics|date=2009 |volume=74 |issue=6 |pages=WCC1–WCC26 |url=https://www.researchgate.net/publication/228078264 |doi=10.1190/1.3238367}}</ref> Some authors have investigated the possibility of reformulating the inverse problem so as to make the objective function less chaotic than the data misfit function.<ref name="ReferenceB">{{cite journal |last1=Clément |first1=François |last2=Chavent |first2=Guy |last3=Gomez |first3=Suzana |title=Migration-based traveltime waveform inversion of 2-D simple structures: A synthetic example |journal= Geophysics |date=2001 |volume=66 |issue=3 |pages=845–860|doi=10.1190/1.1444974 |bibcode=2001Geop...66..845C }}</ref><ref name="ReferenceC">{{cite journal |last1=Symes |first1=William |last2=Carrazone |first2=Jim |title=Velocity inversion by Differential semblance optimization |journal= Geophysics |date=1991 |volume=56 |issue=5 |pages=654–663 |doi=10.1190/1.1443082 |bibcode=1991Geop...56..654S }}</ref>

==== Travel-time tomography ====

Realizing how difficult is the inverse problem in the wave equation, seismologists investigated a simplified approach making use of geometrical optics. In particular they aimed at inverting for the propagation velocity distribution, knowing the arrival times of wave-fronts observed on seismograms. These wave-fronts can be associated with direct arrivals or with reflections associated with reflectors whose geometry is to be determined, jointly with the velocity distribution.

The arrival time distribution <math>{\tau}(x)</math> (<math>x</math> is a point in physical space) of a wave-front issued from a point source, satisfies the [[Eikonal equation]]:
<math display="block">\|\nabla \tau (x)\| =  s(x),</math>
where <math>s(x)</math> denotes the [[Slowness (seismology)|slowness]] (reciprocal of the velocity) distribution. The presence of <math>\| \cdot \| </math> makes this equation nonlinear. It is classically solved by shooting [[Ray tracing (physics)|rays]] (trajectories about which the arrival time is stationary) from the point source.

This problem is tomography like: the measured arrival times are the integral along the ray-path of the slowness. But this tomography like problem is nonlinear, mainly because the unknown ray-path geometry depends upon the velocity (or slowness) distribution. In spite of its nonlinear character, travel-time tomography turned out to be very effective for determining the propagation velocity in the Earth or in the subsurface, the latter aspect being a key element for seismic imaging, in particular using methods mentioned in Section "Diffraction tomography".

=== 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.

=== Computational aspects ===

==== 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.)

==== 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" />]]