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I am aware of the definition of a Rate of convergence for iterative methods involving a single point $$ x_{n+1} = \phi(x_n) ~~~;~~\lim_{n \rightarrow \infty} {{ |x_{n+1} -\alpha|} \over{|x_{n} -\alpha|^p}} = C $$

However, what definitions and tools we have when working with calculus of variations, trying to converge a function or a surface to some optimal function or surface?

Thanks in advance.

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When the structures in question are smooth, such as smooth manifolds/smooth maps, there is a well-defined notion of closeness (or, equivalently, convergence) in the $C^k$-topology, with $k\in\mathbb{N}\cup\{0, \infty, \omega\}$. See Whitney topology for maps. This can be used for distance between manifolds as well. In order to compare how close two manifolds are, they must be of the same dimension (and hence locally diffeomorphic). For general sets in metric spaces, there is Hausdorff metric. – William Jan 4 '13 at 15:35
do you have any document\article using these methods to determine the rate of convergence of something? ( post it as an aswer and I will accept it ) – BenMatok Jan 5 '13 at 12:55
Can't think of one off the top of my head, but one could use the fact that the strong Whitney topology is metrizable when the space is compact, and the weak Whitney topology is always metrizable (see here). As soon as you have a metric, you would define rate of convergence as usual. – William Jan 6 '13 at 14:08
Actually, depending on your situation, it may be even easier. For example, if you have submanifolds embedded in some $\mathbb{R}^N$, then you can easily define distance between them as the Hausdorff metric plus the distance between tangent space (for the tangent space part you can use the Grassmanian), or if both submanifolds are of codimension one, then you can simply measure the angle between the tangent spaces (think of surfaces in $\mathbb{R}^3$. Perhaps you should give more info about your specific situation. – William Jan 6 '13 at 14:14
See also the Gauss map (and its generalizations). – William Jan 6 '13 at 14:35

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