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 can be found in system identification, optics, radar, acoustics, communication theory, signal processing, medical imaging, computer vision, geophysics, oceanography, meteorology, astronomy, remote sensing, natural language processing, machine learning, nondestructive testing, slope stability analysis and many other fields.
History 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, a result improved upon by later 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." 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 Gelfand and Levitan in the Soviet Union. 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.
Conceptual understanding Since Newton, scientists have extensively attempted to model the world. In particular, when a mathematical model is available (for instance, Newton's gravitational law or Coulomb's equation for electrostatics), we can foresee, given some parameters that describe a physical system (such as a distribution of mass or a distribution of electric charges), the behavior of the system. This approach is known as mathematical modeling and the above-mentioned physical parameters are called the model parameters or simply the model. To be precise, we introduce the notion of state of the physical system: it is the solution of the mathematical model's equation. In optimal control theory, these equations are referred to as the state equations. In many situations we are not truly interested in knowing the physical state but just its effects on some objects (for instance, the effects the gravitational field has on a specific planet). Hence we have to introduce another operator, called the observation operator, which converts the state of the physical system (here the predicted gravitational field) into what we want to observe (here the movements of the considered planet). We can now introduce the so-called forward problem, which consists of two steps:
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