Fréchet derivative

The Fréchet derivative ( after Maurice René Fréchet ) generalizes the notion of derivation from the usual differential calculus in normed spaces on. For maps between finite dimensional spaces resulting from this Differenzierbarkeitsbegriff the usual notion of total differentiability.

• 2.1 Linear Operators
• 2.2 Real-valued functions
• 2.3 Integral Operator

Definition

Let and two normed spaces and an open subset. An operator means Fréchet differentiable at the point when there is a limited linear operator such that

Applies. The operator is called the Fréchet derivative of at the point. If there is the Fréchet derivative for all, then that means the picture on with the Fréchet derivative of. With the space of continuous linear maps is called from to.

Equivalent definition

An equivalent definition is:

Are available for each one so that

For all with. This is also briefly review with the help of the Landau symbols:

Examples

Linear operators

For finite-dimensional normed spaces, all linear operators are Fréchet - differentiable with constant dissipation. At any point is the derivative of the linear operator itself.

In the infinite-dimensional case are Fréchet differentiable under the linear operators exactly the bounded ( = continuous ). Unrestricted linear operators are not Fréchet - differentiable.

Real-valued functions

If a real-valued function defined on an open set, and has continuous partial derivatives, then is also Fréchet differentiable. The derivation of the place will be lodged by the usual gradient of given:

This example shows the connection to the usual differential calculus in. The Fréchet derivative is, in fact a generalization of the differential calculus on normed spaces.

Integral operator

Be, steady and continuous and continuously differentiable in the second argument. The non-linear integral operator defined by

Is Fréchet differentiable. Its derivation is

Due to the mean value theorem of differential calculus applies namely

Applies with and because of the uniform continuity of on

For. So for true

What the representation of the derivation proves.

Calculation rules

It can be the usual rules for computing the total derivative in the show for the Fréchet derivative. The following equations are valid as long as they are meaningful in the sense of the above definition, ie in particular the illustrations are occurring differentiable at the corresponding locations:

• .
• Chain rule. The product here is in the sense of multiplication ( sequential execution ) to understand linear maps.
• Is a continuous, linear operator, then A is differentiable everywhere and it applies. Together with the chain rule results from the conclusion that we must pull steady, linear operators from the derivative: and.
• Product rule is a continuous, linear mapping n times, then

Relationship between Fréchet and Gâteaux - derivative

Be on the site Fréchet differentiable, then there exists the Gâteaux differential for any direction and we have:

The converse is not true in general.

In addition, there exists the Gâteaux - derivative of at the point, which is referred to below, and we have:

Again, the converse is not true in general. Under the following conditions the converse also holds:

If it is in an environment of Gâteaux - differentiable, that is, the Gâteaux differential at each point of interest is continuous and linear, and the picture

Continuous at the point with respect to the operator norm, so is Fréchet differentiable at the point.

This condition is not necessary. Some already exist in the one-dimensional totally differentiable functions that are not continuously differentiable.

Example of use

The Frechet derivative may for example be used for solving the so-called inverse boundary value problems in the context of a Newton's method. As an example of this application, we consider an inverse boundary value problem for the Laplace equation:

There was an unknown territory. We consider the exterior Dirichlet problem in which the boundary values ​​are given by a source to the point. Then satisfies the bounded and twice continuously differentiable function in the Laplace equation:

And the Dirichlet boundary condition:

With we denote the fundamental solution of the Laplace equation, which describes a point source at the point.

When inverse boundary value problem, we expect a second (known) from area containing. On the edge of we measure the values ​​of the solution of the direct Dirichlet problem. So we know the track. Our goal now is to reconstruct the unknown boundary from the knowledge of this track.

This problem can be formally described by an operator that maps the unknown boundary onto the famous track. So we need to solve the following nonlinear equation:

This equation can be linearized, for example, using the Newton method. We restrict ourselves to regions whose boundary can be represented as follows:

So now we are looking for the unknown radius function. The linearized equation ( Newton's method ) is then as follows:

Here denotes the Fréchet derivative of the operator ( The existence of the Fréchet derivative for it can be shown and can be determined from a direct boundary value problem ). This equation is then dissolved, and we have found with a new approximation to the unknown desired edge. Can then be iterated with this approximation method.