An inverse problem is the task that often occurs in many branches of science and mathematics where the values of some model parameter(s) must be obtained from the observed data.
The inverse problem can be formulated as follows:
The transformation from data to model parameters is a result of the interaction of a physical system, e.g., the Earth, the atmosphere, gravity etc. Inverse problems arise for example in geophysics, medical imaging (such as computed axial tomography and EEG/ERP), remote sensing, ocean acoustic tomography, nondestructive testing, and astronomy.
Inverse problems are typically ill posed, as opposed to the wellposed problems more typical when modeling physical situations where the model parameters or material properties are known. Of the three conditions for a wellposed problem suggested by Jacques Hadamard (existence, uniqueness, stability of the solution or solutions) the condition of stability is most often violated. In the sense of functional analysis, the inverse problem is represented by a mapping between metric spaces. While inverse problems are often formulated in infinite dimensional spaces, limitations to a finite number of measurements, and the practical consideration of recovering only a finite number of unknown parameters, may lead to the problems being recast in discrete form. In this case the inverse problem will typically be illconditioned. In these cases, regularization may be used to introduce mild assumptions on the solution and prevent overfitting. Many instances of regularized inverse problems can be interpreted as special cases of Bayesian inference.
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An inverse problem is to find m such that (at least approximately)
where G is an operator describing the explicit relationship between data d and model parameters m, and is a representation of the physical system. In various contexts, the operator G is called forward operator, observation operator, or observation function.
A linear inverse problem can be described by:
where the forward operator G is a linear operator describing the explicit relationship between data and model parameters, and is a representation of the physical system. In the case of a discrete linear inverse problem describing a linear system, d and m are vectors, and the problem can be written as
where G is a matrix, often called the observation matrix.
One central example of a linear inverse problem is provided by a Fredholm first kind integral equation.
For sufficiently smooth g the operator defined above is compact on reasonable Banach spaces such as L^{p} spaces. Even if the mapping is injective its inverse will not be continuous. (However, by the bounded inverse theorem, if the mapping is bijective, then the inverse will be bounded (ie. continuous).) Thus small errors in the data d are greatly amplified in the solution m. In this sense the inverse problem of inferring m from measured d is illposed.
To obtain a numerical solution, the integral must be approximated using quadrature, and the data sampled at discrete points. The resulting system of linear equations will be illconditioned.
Another example is the inversion of the Radon transform. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This is precisely the problem solved in image reconstruction for Xray computerized tomography. 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 xray transform in three dimensions and problems involving the generalisation of the xray transform to tensor fields.
An inherently more difficult family of inverse problems are collectively referred to as nonlinear inverse problems.
Nonlinear inverse problems have a more complex relationship between data and model, represented by the equation:
Here G is a nonlinear operator and cannot be separated to represent a linear mapping of the model parameters that form m into the data. In such research, the first priority is to understand the structure of the problem and to give a theoretical answer to the three Hadamard questions (so that the problem is solved from the theoretical point of view). It is only later in a study that regularization and interpretation of the solution's (or solutions', depending upon conditions of uniqueness) dependence upon parameters and data/measurements (probabilistic ones or others) can be done. Hence the corresponding following sections do not really apply to these problems. Whereas linear inverse problems were completely solved from the theoretical point of view at the end of the nineteenth century, only one class of nonlinear inverse problems was so before 1970, that of inverse spectral and (one space dimension) inverse scattering problems, after the seminal work of the Russian mathematical school (Krein, Gelfand, Levitan, 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 problems, data are properties of the spectrum of a linear operator which describe the scattering. The spectrum is made of eigenvalues and eigenfunctions, 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 "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).
There are four main academic journals covering inverse problems in general.
In addition there are many journals on medical imaging, geophysics, nondestructive testing etc that are dominated by inverse problems in those areas.
