Nonlocal operator explained

In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined solely from the values of the input function in any neighbourhood of any point. An example of a nonlocal operator is the Fourier transform.

Formal definition

Let

X be a topological space, Y a set, F(X) a function space containing functions with domain X , and G(Y) a function space containing functions with domain Y . Two functions u and v in F(X) are called equivalent at x\in X if there exists a neighbourhood N of x such that u(x') = v(x') for all x'\in N . An operator A: F(X) \to G(Y) is said to be local if for every y\in Y there exists an x\in X such that Au(y) = Av(y) for all functions u and v in F(X) which are equivalent at x . A nonlocal operator is an operator which is not local.

For a local operator it is possible (in principle) to compute the value

Au(y) using only knowledge of the values of u in an arbitrarily small neighbourhood of a point x . For a nonlocal operator this is not possible.

Examples

Differential operators are examples of local operators. A large class of (linear) nonlocal operators is given by the integral transforms, such as the Fourier transform and the Laplace transform. For an integral transform of the form

(Au)(y) = \int \limits _{X} u(x) K(x, y) dx, where K is some kernel function, it is necessary to know the values of u almost everywhere on the support of K( ⋅ , y) in order to compute the value of Au at y .

An example of a singular integral operator is the fractional Laplacian

( - \Delta) ^{sf(x)} = c _{d,s} \int\limits _{R ^{d}} (f(x) - f(y))/(|x - y| ^{d+2s}) dy. The prefactor c _{d,s} : = (4 ^{s\Gamma(d/2+s)})/(\pi ^{d/2} |\Gamma( - s)|) involves the Gamma function and serves as a normalizing factor. The fractional Laplacian plays a role in, for example, the study of nonlocal minimal surfaces.^[1]

Applications

Some examples of applications of nonlocal operators are:

• Time series analysis using Fourier transformations
• Analysis of dynamical systems using Laplace transformations
• Image denoising using non-local means^[2]
• Modelling Gaussian blur or motion blur in images using convolution with a blurring kernel or point spread function

See also

• Fractional calculus
• Linear map
• Nonlocal Lagrangian
• Action at a distance

External links

• Nonlocal equations wiki

Notes and References

1. Caffarelli . L. . Roquejoffre . J.-M. . Savin . O. . 2010 . Nonlocal minimal surfaces . Communications on Pure and Applied Mathematics . 63 . 9 . 1111–1144 . en . 10.1002/cpa.20331. 0905.1183 . 10480423 .
2. Book: A Non-Local Algorithm for Image Denoising . Buades . A. . Coll . B. . Morel . J.-M. . 2005 . 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) . IEEE . 9780769523729 . 2 . San Diego, CA, USA . 60–65 . 10.1109/CVPR.2005.38. 11206708 .