Editing Inverse problem (section)

==History==
Starting with the effects to discover the causes has concerned physicists for centuries. A historical example is the calculations of [[John Couch Adams|Adams]] and [[Urbain Le Verrier|Le Verrier]] which led to the discovery of [[Neptune]] from the perturbed trajectory of [[Uranus]]. However, a formal study of inverse problems was not initiated until the 20th century.

One of the earliest examples of a solution to an inverse problem was discovered by [[Hermann Weyl]] and published in 1911, describing the asymptotic behavior of eigenvalues of the [[Laplace–Beltrami operator]].<ref>{{cite journal |last=Weyl |first=Hermann |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=63048 |title=Über die asymptotische Verteilung der Eigenwerte |journal=Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen |pages=110–117 |year=1911 |access-date=2018-05-14 |archive-url=https://web.archive.org/web/20130801090504/http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=63048 |archive-date=2013-08-01 |url-status=dead }}</ref> Today known as ''[[Weyl's law]]'', it is perhaps most easily understood as an answer to the question of whether it is possible to [[Hearing the shape of a drum|hear the shape of a drum]]. Weyl conjectured that the eigenfrequencies of a drum would be related to the area and perimeter of the drum by a particular equation, a result improved upon by later mathematicians.

The field of inverse problems was later touched on by [[Soviet Union|Soviet]]-[[Armenians|Armenian]] physicist, [[Viktor Ambartsumian]].<ref>[http://ambartsumian.ru/en/papers/epilogue-ambartsumian’-s-paper/ » Epilogue — Ambartsumian’ s paper Viktor Ambartsumian<!-- Bot generated title -->]</ref><ref>{{cite journal|title=A life in astrophysics. Selected papers of Viktor A. Ambartsumian| first=Rouben V.| last=Ambartsumian| journal=Astrophysics| volume=41| issue=4| pages=328–330| doi=10.1007/BF02894658| year = 1998| s2cid=118952753}}</ref>

While still a student, Ambartsumian thoroughly studied the theory of atomic structure, the formation of energy levels, and the [[Schrödinger equation]] and its properties, and when he mastered the theory of [[eigenvalues and eigenvectors|eigenvalues]] of [[differential equation]]s, he pointed out the apparent analogy between discrete energy levels and the eigenvalues of differential equations. He then asked: given a family of eigenvalues, is it possible to find the form of the equations whose eigenvalues they are? Essentially Ambartsumian was examining the inverse [[Sturm–Liouville problem]], which dealt with determining the equations of a vibrating string. This paper was published in 1929 in the German physics journal ''[[Zeitschrift für Physik]]'' and remained in obscurity for a rather long time. Describing this situation after many decades, Ambartsumian said, "If an astronomer publishes an article with a mathematical content in a physics journal, then the most likely thing that will happen to it is oblivion."

Nonetheless, toward the end of the Second World War, this article, written by the 20-year-old Ambartsumian, was found by Swedish mathematicians and formed the starting point for a whole area of research on inverse problems, becoming the foundation of an entire discipline.

Then important efforts have been devoted to a "direct solution" of the inverse scattering problem especially by [[Marchenko equation|Gelfand and Levitan]] in the Soviet Union.<ref name="sciencedirect.com">{{cite journal |last1=Burridge |first1=Robert |title=The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems |journal=Wave Motion |date=1980 |volume=2 |issue=4 |pages=305–323 |doi=10.1016/0165-2125(80)90011-6 |bibcode=1980WaMot...2..305B }}</ref> They proposed an analytic constructive method for determining the solution.  When computers became available, some authors have investigated the possibility of applying their approach to similar problems such as the inverse problem in the 1D wave equation. But it rapidly turned out that the inversion is an unstable process: noise and errors can be tremendously amplified making a direct solution hardly practicable.
Then, around the seventies, the least-squares and probabilistic approaches came in and turned out to be very helpful for the determination of parameters involved in various physical systems. This approach met a lot of success. Nowadays inverse problems are also investigated in fields outside physics, such as chemistry, economics, and computer science. Eventually, as numerical models become prevalent in many parts of society, we may expect an inverse problem associated with each of these numerical models.