Inverse problem

An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because it starts with the effects and then calculates the causes. It is the inverse of a forward problem, which starts with the causes and then calculates the effects. An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because it starts with the effects and then calculates the causes. It is the inverse of a forward problem, which starts with the causes and then calculates the effects. Inverse problems are some of the most important mathematical problems in science and mathematics because they tell us about parameters that we cannot directly observe. They have wide application in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, and many other fields. Starting with the effects to discover the causes has concerned physicists for centuries. A historical example is the calculations of Adams and 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. Today known as Weyl's law, it is perhaps most easily understood as an answer to the question of whether it is possible to 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, which has later been improved upon by many mathematicians. The field of inverse problems was later touched on by Soviet-Armenian physicist, Viktor Ambartsumian. 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 of differential equations, 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.'