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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), remote sensing, nondestructive testing and astronomy.
Inverse problems are typically ill posed, as opposed to the well-posed problems more typical when modelling physical situations where the model parameters or material properties are known. Of the three conditions for a well-posed problem suggested by Jacques Hadamard it is the condition of stability of solution that is most often violated. In the sense of functional analysis, the inverse problem is represented by an unbounded mapping between Banach spaces. While Inverse Problems are often formulated in infinite dimensional spaces, limitations of a finite number measurements, and the practical consideration of recovering only a finite number of unknown parameters, lead to the problems being recast in discrete form. In this case the inverse problem will typically be ill-conditioned. See condition numberIn numerical analysis, the condition number associated with a numerical problem is a measure of that quantity's amenability to digital computation, that is, how well-posed the problem is. A problem with a low condition number is said to be well-conditione.
Inverse modelling is a term applied to describe the group of methods used to gain information about a physical system based on observations of that system. In other words, it is an attempt to solve the inverse problem.
A linear inverse problem can be described by:
where G is an operator, which represents the explicit relationship between data and model parameters and is a representation of the `physical system' in Equation 1 above.
One central example of a linear inverse problem is provided by a FredholmIn mathematics, the Fredholm integral equation introduced by Ivar Fredholm gives rises to a Fredholm operator. From the point of view of functional analysis it therefore has a well-understood abstract eigenvalue theory. In this case that is supported by a first kind integral equationIn mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way. Integral equations ar.
For sufficiently smooth the operator defined above is compactOperator theory In functional analysis, a compact operator is a linear operator L from a Banach space X to another Banach space Y such that the image under L of any bounded subset of X is a relatively compact subset of Y''. Such an operator is necessarily on reasonable Banach spaces such as In mathematics, the Lp and spaces are spaces of p-power integrable functions and corresponding sequence spaces''. They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. See also root mean square. Even if the mapping is bijectiveIn mathematics, a bijection bijective function or one-to-one correspondence is a function that is both injective ("one-to-one") and surjective ("onto"), and therefore bijections are also called one-to-one and onto . Intuitively, a bijective function creat its inverseIn mathematical analysis, an inverse function is in simple terms a function which "does the reverse" of a given function. More formally, if f is a function with domain X, then f -1 is its inverse function if and only if for every we have: :. For example, will be not be continuous. Thus small errors in the data are greatly amplified in the solution . In this sense the inverse problem of inferring from measured is ill-posed.
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 ill-conditioned.
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 X-ray computerized tomography.