Editing Inverse problem (section)

===Some classical linear inverse problems for the recovery of distributed parameters===
The problems mentioned below correspond to different versions of the Fredholm integral: each of these is associated with a specific kernel <math>K</math>.

====Deconvolution====
The goal of [[deconvolution]] is to reconstruct the original image or signal <math>p(x)</math> which appears as noisy and blurred on the data <math>d(x)</math>.<ref>Kaipio, J., & Somersalo, E. (2010). Statistical and computational inverse problems. New York, NY: Springer.</ref>
From a mathematical point of view, the kernel <math>K(x,y)</math> here only depends on the difference between <math>x</math> and <math>y</math>.

====Tomographic methods====
In these methods we attempt at recovering a distributed parameter, the observation consisting in the measurement of the integrals of this parameter carried out along a family of lines. We denote by <math>\Gamma_x</math> the line in this family associated with measurement point <math>x</math>. The observation at <math>x</math> can thus be written as:
<math display="block">d(x) = \int_{\Gamma_x} w(x,y) p(y) \, dy</math>
where <math>s</math> is the arc-length along <math>{\Gamma_x}</math> and <math>w(x,y)</math> a known weighting function. Comparing this equation with the Fredholm integral above, we notice that the kernel <math>K(x,y)</math> is kind of a [[Dirac delta function|delta function]] that peaks on line <math>{\Gamma_x}</math>. With such a kernel, the forward map is not compact.

===== Computed tomography =====
In [[X-ray computed tomography]] the lines on which the parameter is integrated are straight lines: the [[tomographic reconstruction]] of the parameter distribution is based on the inversion of the [[Radon transform]]. Although from a theoretical point of view many linear inverse problems are well understood, problems involving the Radon transform and its generalisations still present many theoretical challenges with questions of sufficiency of data still unresolved. Such problems include incomplete data for the x-ray transform in three dimensions and problems involving the generalisation of the x-ray transform to tensor fields. Solutions explored include [[Algebraic Reconstruction Technique]], [[filtered backprojection]], and as computing power has increased, [[iterative reconstruction]] methods such as [[SAMV (algorithm)|iterative Sparse Asymptotic Minimum Variance]].<ref name=AbeidaZhang>{{cite journal | last1=Abeida | first1=Habti | last2=Zhang | first2=Qilin | last3=Li | first3=Jian | last4=Merabtine | first4=Nadjim | title=Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing | journal=IEEE Transactions on Signal Processing | volume=61 | issue=4 | year=2013 | issn=1053-587X | doi=10.1109/tsp.2012.2231676 | pages=933–944 | url=https://qilin-zhang.github.io/_pages/pdfs/SAMVpaper.pdf | arxiv=1802.03070 | bibcode=2013ITSP...61..933A | s2cid=16276001 }}</ref>

===== Diffraction tomography =====
Diffraction tomography is a classical linear inverse problem in exploration seismology: the amplitude recorded at one time for a given source-receiver pair is the sum of contributions arising from points such that the sum of the distances, measured in traveltimes, from the source and the receiver, respectively, is equal to the corresponding recording time. In 3D the parameter is not integrated along lines but over surfaces. Should the propagation velocity be constant, such points are distributed on an ellipsoid. The inverse problems consists in retrieving the distribution of diffracting points from the seismograms recorded along the survey, the velocity distribution being known.  A direct solution has been originally proposed by [http://amath.colorado.edu/~beylkin/papers/BEYLKI-1983a.pdf Beylkin] and Lambaré et al.:<ref>{{cite journal |last1=Lambaré |first1=Gilles |last2=Virieux |first2=Jean |last3=Madariaga |first3=Raul |last4=Jin |first4=Side |title=Iterative asymptotic inversion in the acoustic approximation |journal=  Geophysics |date=1992 |volume=57 |issue=9 |pages=1138–1154 |doi=10.1190/1.1443328 |bibcode=1992Geop...57.1138L |s2cid=55836067 }}</ref> these works were the starting points of approaches known as amplitude preserved migration (see Beylkin<ref>{{cite journal |last1=Beylkin |first1=Gregory |title=The inversion problem and applications of The generalized Radon transform |journal=Communications on Pure and Applied Mathematics |date=1984 |volume=XXXVII |issue=5 |pages=579–599 |url=http://amath.colorado.edu/faculty/beylkin/papers/BEYLKI-1984.pdf |doi=10.1002/cpa.3160370503}}</ref><ref>{{cite journal |last1=Beylkin |first1=Gregory |title=Imaging of discontinuities in the inverse scaterring problem by inversion of a causal generalized Radon transform |journal=J. Math. Phys. |date=1985 |volume=26 |issue=1 |pages=99–108|doi=10.1063/1.526755 |bibcode=1985JMP....26...99B }}</ref> and Bleistein<ref>{{cite journal |last1=Bleistein |first1=Norman |title=On the imaging of reflectors in the earth |journal=Geophysics |date=1987 |volume=52 |issue=7 |pages=931–942 |doi=10.1190/1.1442363 |bibcode=1987Geop...52..931B |s2cid=5095133 }}</ref>).  Should geometrical optics techniques (i.e. [https://www.encyclopediaofmath.org/index.php/Ray_method rays])  be used for the solving the wave equation, these methods turn out to be closely related to the so-called least-squares migration methods<ref>{{cite journal |last1=Nemeth |first1=Tamas |last2=Wu |first2=Chengjun |last3=Schuster |first3=Gerard |title=Least-squares migration of incomplete reflection data |journal = Geophysics |date=1999 |volume=64 |issue=1 | pages=208–221 | url=https://csim.kaust.edu.sa/web/FWI&LSM_papers/LSM/Geophysics1999Nemeth.pdf |doi=10.1190/1.1444517 |bibcode=1999Geop...64..208N }}</ref> derived from the least-squares approach (see Lailly,<ref name='Proceedings of the international conference on "Inverse Scattering, theory and applications", Tulsa, Oklahoma'>{{cite book |last1=Lailly |first1=Patrick |title=The seismic inverse problem as a sequence of before stack migrations |date=1983 |publisher=SIAM |location=Philadelphia |isbn=0-89871-190-8 |pages=206–220}}</ref> Tarantola<ref>{{Cite journal | doi=10.1190/1.1441754| title=Inversion of seismic reflection data in the acoustic approximation| journal=Geophysics| volume=49| issue=8| pages=1259–1266| year=1984| last1=Tarantola| first1=Albert | bibcode=1984Geop...49.1259T| s2cid=7596552}}</ref>).

===== {{anchor|Doppler tomography}}Doppler tomography (astrophysics) =====

If we consider a rotating stellar object, the spectral lines we can observe on a spectral profile will be shifted due to Doppler effect. Doppler tomography aims at converting the information contained in spectral monitoring of the object into a 2D image of the emission (as a function of the radial velocity and of the phase in the periodic rotation movement) of the stellar atmosphere. As explained by [[Tom Marsh (astronomer)|Tom Marsh]]<ref>{{cite journal |last1=Marsh |first1=Tom |title=Doppler tomography |journal=Astrophysics and Space Science |volume=296 |date=2005 |issue=1–4 |pages=403–415 |doi=10.1007/s10509-005-4859-3 |arxiv=astro-ph/0011020 |bibcode=2005Ap&SS.296..403M |s2cid=15334110 }}</ref> this linear inverse problem is tomography like: we have to recover a distributed parameter which has been integrated along lines to produce its effects in the recordings.

==== Inverse heat conduction ====

Early publications on inverse heat conduction arose from determining surface heat flux during atmospheric re-entry from buried temperature sensors.<ref>{{Cite journal | author1 = Shumakov, N. V. | title = A method for the experimental study of the process of heating a solid body | journal = Soviet Physics –Technical Physics (Translated by American Institute of Physics) | volume = 2 | pages = 771 | year =  1957 }}</ref><ref>{{Cite journal | author1 = Stolz, G. Jr. | title =  Numerical solutions to an inverse problem of heat conduction for simple shapes | journal = Journal of Heat Transfer | volume =  82 | pages = 20–26 | year = 1960 | doi = 10.1115/1.3679871 }}</ref>
Other applications where surface heat flux is needed but surface sensors are not practical include: inside reciprocating engines, inside rocket engines; and, testing of nuclear reactor components.<ref>{{Cite book | author1 = Beck, J. V. | author2 =  Blackwell, B.  |author3 =  St. Clair, C. R. Jr. | title = Inverse Heat Conduction. Ill-Posed Problems | location =  New York | publisher = J. Wiley & Sons | year = 1985 | isbn =  0471083194 }}</ref> A variety of numerical techniques have been developed to address the ill-posedness and sensitivity to measurement error caused by damping and lagging in the temperature signal.<ref>{{cite journal | author1 = Beck, J. V. | author2 =  Blackwell, B. |author3 =  Haji-Sheikh, B. | title =  Comparison of some inverse heat conduction methods using experimental data | journal =  International Journal of Heat and Mass Transfer | volume = 39 | issue = 17 | year =  1996  | pages = 3649–3657 | doi =  10.1016/0017-9310(96)00034-8
| bibcode = 1996IJHMT..39.3649B }}</ref><ref>{{cite book | author1 = Ozisik, M. N. |author2 =  Orlande, H. R. B.
| title =  Inverse Heat Transfer, Fundamentals and Applications | publisher = CRC Press | year = 2021 | isbn = 9780367820671 | edition = 2nd }}</ref><ref>{{cite book | title = Inverse Engineering Handbook, edited by K. A. Woodbury | publisher = CRC Press | year = 2002 | isbn = 9780849308611 }}</ref>