Understanding Sobolev Norms: A Beginner's Guide

[emmasaunders12]

Can someone give me in very abstract terms what a sobolev norm is or means?

 

[RUber]

It is a norm like any other. Sobolev norms take into consideration the size of derivatives of the function as well, so smoother functions will have smaller norms in general. I think there are also rules for 1/2 norms and things like that which use maximums, but I don't have my reference text handy. What are you doing with Sobolev norms?

 

[emmasaunders12]

It is a norm like any other. Sobolev norms take into consideration the size of derivatives of the function as well, so smoother functions will have smaller norms in general. I think there are also rules for 1/2 norms and things like that which use maximums, but I don't have my reference text handy. What are you doing with Sobolev norms?

Thanks, image registration, so how can you ensure that a function has a sobolev structure and hence a sobolev norm?

 

[RUber]

I have normally seen the spaces used as a restriction -- saying that solutions will only be sought from inside a sobolev space. Verifying that a function fits within a sobolev space requires taking the norm. If the norm is not infinite, the function should be in the space.

 

[emmasaunders12]

I have normally seen the spaces used as a restriction -- saying that solutions will only be sought from inside a sobolev space. Verifying that a function fits within a sobolev space requires taking the norm. If the norm is not infinite, the function should be in the space.

That's a very simple explanation thanks, how does one calculate the sobolev norm of a function and if a function fits within a sobolev space what does that tell us about the function?

 

[RUber]

What sort of function are you trying to find the norm for? What space do you want it to be in?
The definitions are mostly based on integrals, so if you know the function, you can compute the norm.
The Wikipedia definition is sufficient without getting too deep into the theory. Otherwise, check out Adams' book on Sobolev Spaces to learn more. https://en.wikipedia.org/wiki/Sobolev_space
If you have a function in mind, we can work through finding the norm together.

 

[emmasaunders12]

What sort of function are you trying to find the norm for? What space do you want it to be in?
The definitions are mostly based on integrals, so if you know the function, you can compute the norm.
The Wikipedia definition is sufficient without getting too deep into the theory. Otherwise, check out Adams' book on Sobolev Spaces to learn more. https://en.wikipedia.org/wiki/Sobolev_space
If you have a function in mind, we can work through finding the norm together.

Hi its a vector field for image registration, but apparently the solobev norm allows it to be in the space of diffeomorphisms

 

[emmasaunders12]

Hi its a vector field for image registration, but apparently the solobev norm allows it to be in the space of diffeomorphisms

Would be great if you can assist in helping me understand how to find the sobolev norm of a function, I am not even sure how to start?

 

[RUber]

Let's say you have a function, like ##f(x) = x^2 ## and your domain is defined by ##x \in [0,2]##, and you want to take the (1,2) Sobolev norm, that is, the 2 norm of the function and 1 derivative. In this case, you would write your norm as
##\| f \|_{1,2} =\left( \int_0^2 \left| f(x) \right| ^2 \, dx + \int_0^2 \left| f'(x) \right|^2 \, dx \right) ^{1/2} ##
##\| f \|_{1,2} =\left( \int_0^2 \left| x^2 \right| ^2 \, dx + \int_0^2 \left| 2x \right|^2 \, dx \right) ^{1/2} ##
In this case, these integrals are easy to compute, and your norm would be:
## \| f \|_{1,2} = \left( \frac{2^5}{5}+ \frac{2^5}{3} \right) ^{1/2} \approx 4.13. ##

Clearly, a function like ##x^2## would not be in this space if the domain were the entire real line, since the norm would not be bounded in that case.
If your only goal is to show that a function is in the space, but not actually compute the norms, there are ways to show this with bounding principles and other arguments.

Does that help?

 

[emmasaunders12]

Let's say you have a function, like ##f(x) = x^2 ## and your domain is defined by ##x \in [0,2]##, and you want to take the (1,2) Sobolev norm, that is, the 2 norm of the function and 1 derivative. In this case, you would write your norm as
##\| f \|_{1,2} =\left( \int_0^2 \left| f(x) \right| ^2 \, dx + \int_0^2 \left| f'(x) \right|^2 \, dx \right) ^{1/2} ##
##\| f \|_{1,2} =\left( \int_0^2 \left| x^2 \right| ^2 \, dx + \int_0^2 \left| 2x \right|^2 \, dx \right) ^{1/2} ##
In this case, these integrals are easy to compute, and your norm would be:
## \| f \|_{1,2} = \left( \frac{2^5}{5}+ \frac{2^5}{3} \right) ^{1/2} \approx 4.13. ##

Clearly, a function like ##x^2## would not be in this space if the domain were the entire real line, since the norm would not be bounded in that case.
If your only goal is to show that a function is in the space, but not actually compute the norms, there are ways to show this with bounding principles and other arguments.

Does that help?

Thanks essentially I have been told that the function (v) is constrained to be in the space of diffeomorphisms by ensuring the norm of the velocity field is regularized with a differential operator from fluid mechanics to ensure the transformation lies in the space of differmorphisms. Does this make sense