Inverse problem
The 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 via manipulation of observed data.
The inverse problem can be formulated as follows:
- Data→ Model parameters Eq. 1
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 number.
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.
Linear inverse problems
A linear inverse problem can be described by:
- d=Gm Eq. 2
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.
Examples
One central example of a linear inverse problem is provided by a Fredholm first kind integral equation.
- <math> d(x) = \int_a^b g(x,y)\,m(y)\,dy <math>
For sufficiently smooth <math>g<math> the operator defined above is compact on reasonable Banach spaces such as <math>L^2(a,b)<math>. Even if the mapping is bijective its inverse will be not be continuous. Thus small errors in the data <math>d<math> are greatly amplified in the solution <math>m<math>. In this sense the inverse problem of inferring <math>m<math> from measured <math>d<math> 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 [[Condition number|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.
Non-linear inverse problems
The other, considerably more complex, set of inverse problems is the class collectively referred to as non-linear problems.
Non-linear inverse problems have a more complex relationship between data and model, represented by the equation:
- d=G(m) Eq. 3
Here g is a non-linear operator and cannot be algebraically separated from the model parameters that form m.
