Regularization

The construction of approximate solutions of ill-posed problems that are stable with respect to small perturbations of the initial data (see also Regularization method).

Comments

The concept of "regularization" in mathematics is a quite general one, which extends far beyond regularization methods as are used to deal with ill-posed problems. It encompasses at least the following two intermingling ideas.

1) The systematic replacement of a mathematical object $ A $ by a more regular one $ A ^ { \mathop{\rm reg} } $, usually in such a way that $ ( A ^ { \mathop{\rm reg} } ) ^ { \mathop{\rm reg} } = A ^ { \mathop{\rm reg} } $.

2) The definition of a value of a function or other concept for objects where that value or concept is a priori undefined (or infinite, undetermined, $ \dots $). This is often done by placing the object in a suitable family (a deformation) in such a way that the function value or concept is defined for all objects in the family near the original one, and then taking a suitable limit. Another technique consists in the removal of "systematic infinities" . The details of various regularization methods that are used depend very much on the particular context. Instead of the word "regularization" , one also finds such methods and techniques labelled by words and phrases like "normalization" , "renormalization" , "desingularization" , "resolution of singularities" , $ \dots $.

Examples of regularizations in the sense of 1) or 2) above (or both) are: regularized sequences (cf. Regularization of sequences), regularized operators and regularized solutions (cf. Ill-posed problems; Regularization method; Integral equations, numerical methods; Fredholm equation, numerical methods), penalty function and other regularization techniques in optimization theory (cf. Mathematical programming; Penalty functions, method of), various renormalization schemes (cf. Renormalization), the normalization and desingularization of schemes and varieties (cf. Normal scheme; Resolution of singularities), the regularization of distributions (cf. Generalized function), the regularized trace of a Sturm–Liouville operator (cf. Sturm–Liouville problem), and the regularized characteristic determinant of a Hilbert–Schmidt operator.

Still another example is the zeta-function regularization used to define certain (quotients of) infinite determinants in functional integration and quantum field theory. This goes as follows. Let $ A $ be a suitable operator, e.g. a Laplace or Laplace–Beltrami operator. Define its generalized zeta-function

$$ \zeta_{A} (s) = \sum_n \lambda_{n}^{-s} , $$

where $ \lambda _ {n} $ runs over the spectrum of $ A $( counting multiplicities). At least formally, $ \zeta ^ \prime ( s) \mid_{s=0} = - \sum _ {n} \mathop{\rm log} ( \lambda _ {n} ) $, which provides the opportunity to try to define the zeta-function regularized determinant by

$$ \mathop{\rm det} ( A) = \mathop{\rm exp} ( - \zeta ^ \prime ( s)) \mid_{s=0} . $$

For more details (and other related schemes) cf. [a1], [a2].

Two somewhat different uses of the word "regularizing" in mathematics are as follows.

If $ K $ is a bounded linear operator between normed spaces, then a bounded linear operator $ R $ is called a "regularizer of a bounded linear operatorregularizer of K" if there are compact operators $ A, B $ such that $ RK = I- A $, $ KR = I- B $. This concept is of importance in the context of singular integral operators, cf. e.g. [a3]. I.e. $ R $ is an inverse of $ K $ modulo compact operators.

A similar idea, but with deviating terminology, occurs in the theory of pseudo-differential operators. In that context a (pseudo-differential, integral) operator is called regularizing if it takes (extends to an operator that takes) distributions to smooth functions. Given a pseudo-differential operator $ P $, an operator $ R $ is called a right (left) parametrix of $ P $ if $ PR = I+ K $( $ RP = I+ K ^ \prime $), where $ K $( respectively, $ K ^ \prime $) is regularizing; cf. [a4] for a variety of precise statements and results concerning parametrices.

References

[a1]	S.W. Hawking, "Zeta function regularization of path integrals" Comm. Math. Phys. , 55 (1977) pp. 133–148 MR0524257 Zbl 0407.58024
[a2]	R.E. Gamboa Saravi, M.A. Muschietti, J.E. Solomin, "On the quotient of the regularized determinant of two elliptic operators" Comm. Math. Phys. , 110 (1987) pp. 641–654 MR895221 Zbl 0648.35086
[a3]	R. Kress, "Linear integral equations" , Springer (1989) pp. Chapt. 5 MR1007594 Zbl 0671.45001
[a4]	F. Trèves, "Pseudodifferential and Fourier integral operators" , 1–2 , Plenum (1980) MR0597145 MR0597144 Zbl 0453.47027