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Studying behaviour of the following differential equation
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I derived the following differential equation from a problem in signal analysis.
$$
0 = f - u - frac12lambdanabla^Tcdotleft(e^-lVert nabla u rVert/lambda fracnabla ulVert nabla urVertright)
$$
here $u = u(x,y)$, $f$ is infinitely times differentiable, the domain is a rectangle $(x,y) in [0,X]times[0,Y]$.
The question is... if $nabla u$ has very large magnitude compared to $lambda > 0$ then the solution is roughly $u = f$, however I do struggle to understand what happens when $nabla u$ is very small. Is there something I can say even roughly? It seems to me it would resemble a diffusion equation, however there's the gradient normalized (so it's a unitary vector field) so I can't really say much about it.
Any help would be appreciated.
differential-equations asymptotics calculus-of-variations image-processing
asked Aug 1 at 22:11
user8469759
1,4271513
add a comment |Â
up vote
1
down vote
favorite
I derived the following differential equation from a problem in signal analysis.
$$
0 = f - u - frac12lambdanabla^Tcdotleft(e^-lVert nabla u rVert/lambda fracnabla ulVert nabla urVertright)
$$
here $u = u(x,y)$, $f$ is infinitely times differentiable, the domain is a rectangle $(x,y) in [0,X]times[0,Y]$.
The question is... if $nabla u$ has very large magnitude compared to $lambda > 0$ then the solution is roughly $u = f$, however I do struggle to understand what happens when $nabla u$ is very small. Is there something I can say even roughly? It seems to me it would resemble a diffusion equation, however there's the gradient normalized (so it's a unitary vector field) so I can't really say much about it.
Any help would be appreciated.
differential-equations asymptotics calculus-of-variations image-processing
asked Aug 1 at 22:11
user8469759
1,4271513
Where did you get this equation? It looks interesting, and I'm curious. Thanks. Cheers!
– Robert Lewis
Aug 1 at 22:43
Are you trying to find $u$?
– Robert Lewis
Aug 1 at 22:44
@Robert Lewis, dsp.stackexchange.com/q/50943/17762. I want to find u using gradient descent. But I need to guess at least a bit how the solution ia going to behave asymptotically.
– user8469759
Aug 2 at 5:37
Thanks let me check it out and get back to you.
– Robert Lewis
Aug 2 at 5:38
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I derived the following differential equation from a problem in signal analysis.
$$
0 = f - u - frac12lambdanabla^Tcdotleft(e^-lVert nabla u rVert/lambda fracnabla ulVert nabla urVertright)
$$
here $u = u(x,y)$, $f$ is infinitely times differentiable, the domain is a rectangle $(x,y) in [0,X]times[0,Y]$.
The question is... if $nabla u$ has very large magnitude compared to $lambda > 0$ then the solution is roughly $u = f$, however I do struggle to understand what happens when $nabla u$ is very small. Is there something I can say even roughly? It seems to me it would resemble a diffusion equation, however there's the gradient normalized (so it's a unitary vector field) so I can't really say much about it.
Any help would be appreciated.
differential-equations asymptotics calculus-of-variations image-processing
asked Aug 1 at 22:11
user8469759
1,4271513
I derived the following differential equation from a problem in signal analysis.
$$
0 = f - u - frac12lambdanabla^Tcdotleft(e^-lVert nabla u rVert/lambda fracnabla ulVert nabla urVertright)
$$
here $u = u(x,y)$, $f$ is infinitely times differentiable, the domain is a rectangle $(x,y) in [0,X]times[0,Y]$.
The question is... if $nabla u$ has very large magnitude compared to $lambda > 0$ then the solution is roughly $u = f$, however I do struggle to understand what happens when $nabla u$ is very small. Is there something I can say even roughly? It seems to me it would resemble a diffusion equation, however there's the gradient normalized (so it's a unitary vector field) so I can't really say much about it.
Any help would be appreciated.
differential-equations asymptotics calculus-of-variations image-processing
asked Aug 1 at 22:11
user8469759
1,4271513
asked Aug 1 at 22:11
user8469759
1,4271513
asked Aug 1 at 22:11
user8469759
1,4271513
asked Aug 1 at 22:11
user8469759
1,4271513
1,4271513
Where did you get this equation? It looks interesting, and I'm curious. Thanks. Cheers!
– Robert Lewis
Aug 1 at 22:43
Are you trying to find $u$?
– Robert Lewis
Aug 1 at 22:44
@Robert Lewis, dsp.stackexchange.com/q/50943/17762. I want to find u using gradient descent. But I need to guess at least a bit how the solution ia going to behave asymptotically.
– user8469759
Aug 2 at 5:37
Thanks let me check it out and get back to you.
– Robert Lewis
Aug 2 at 5:38
add a comment |Â
Where did you get this equation? It looks interesting, and I'm curious. Thanks. Cheers!
– Robert Lewis
Aug 1 at 22:43
Are you trying to find $u$?
– Robert Lewis
Aug 1 at 22:44
@Robert Lewis, dsp.stackexchange.com/q/50943/17762. I want to find u using gradient descent. But I need to guess at least a bit how the solution ia going to behave asymptotically.
– user8469759
Aug 2 at 5:37
Thanks let me check it out and get back to you.
– Robert Lewis
Aug 2 at 5:38
Where did you get this equation? It looks interesting, and I'm curious. Thanks. Cheers!
– Robert Lewis
Aug 1 at 22:43
Where did you get this equation? It looks interesting, and I'm curious. Thanks. Cheers!
– Robert Lewis
Aug 1 at 22:43
Are you trying to find $u$?
– Robert Lewis
Aug 1 at 22:44
Are you trying to find $u$?
– Robert Lewis
Aug 1 at 22:44
@Robert Lewis, dsp.stackexchange.com/q/50943/17762. I want to find u using gradient descent. But I need to guess at least a bit how the solution ia going to behave asymptotically.
– user8469759
Aug 2 at 5:37
@Robert Lewis, dsp.stackexchange.com/q/50943/17762. I want to find u using gradient descent. But I need to guess at least a bit how the solution ia going to behave asymptotically.
– user8469759
Aug 2 at 5:37
Thanks let me check it out and get back to you.
– Robert Lewis
Aug 2 at 5:38
Thanks let me check it out and get back to you.
– Robert Lewis
Aug 2 at 5:38
add a comment |Â
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Where did you get this equation? It looks interesting, and I'm curious. Thanks. Cheers!
– Robert Lewis
Aug 1 at 22:43
Are you trying to find $u$?
– Robert Lewis
Aug 1 at 22:44
@Robert Lewis, dsp.stackexchange.com/q/50943/17762. I want to find u using gradient descent. But I need to guess at least a bit how the solution ia going to behave asymptotically.
– user8469759
Aug 2 at 5:37
Thanks let me check it out and get back to you.
– Robert Lewis
Aug 2 at 5:38