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The mathematical term regularization has two main meanings, both associated with making a function more `regular' or smooth.
Regularization in physics
In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator (for example, the minimal distance <math>\epsilon<math> in space which is useful if the divergences arise from short-distance physical effects). The correct physical result is obtained in the limit in which the regulator goes away (in our example, <math>\epsilon\to 0<math>), but the virtue of the regulator is that for its finite value, the result is finite.
However, the result usually includes terms proportional to expressions like <math>1/ \epsilon<math> which are not well-defined in the limit <math>\epsilon\to 0<math>. Regularization is the first step towards obtaining a completely finite and meaningful result; in quantum field theory it must be usually followed by a related, but independent technique called renormalization. Renormalization is based on the requirement that some physical quantities - expressed by seemingly divergent expressions such as <math>1/ \epsilon<math> - are equal to the observed values. Such a constraint allows one to calculate a finite value for many other quantities that looked divergent.
Regularization of ill-posed problems
Inverse problems are often ill-posed. To solve these problems numerically one must introduce some additional information about the solution, such as an assumption on the smoothness or a bound on the norm.
The same idea arose in many fields of science. A simple form of regularization applied to integral equations, generally termed Tikhonov regularization after Andrey Nikolayevich Tychonoff, is essentially a trade-off between fitting the data and reducing a norm of the solution. In statistics a similar concept was introduced about the same time for finite-dimensional problems, where it is known as ridge regression.
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