Need a Functional Analysis Pro - Haaalp!!

[zer0snr]

Help!! I need a good mathematician or physicist to put me on the right path......



I have a patch of a surface in 3d space. Assume it is parameterizable, and that its boundaries coincide with the lines of constant parameters.

The question I have is that: how can "create" a basis set on this surface so that I can approximate functions defined on this surface in some optimal fashion (say for example, but not necessarily, for a fixed error by the using the least number of basis functions).



For example, if it were a spherical patch I could use spherical harmonics, and it would be optimal in the least squares sense (ie fitting error).



I realize that as such there are, possibly, an infinite basis sets for such an approximation, and therefore supply a criteria to select one of those many basis sets. At this stage, am not not picky about what that crietria should be (LS or something else).

What, I guess, I'm looking for is how can I create a new basis set which in some fashion also uses the geometry of the surface too. Not clear? Well, I'm not sure what I'm talking about too. But let me try using the following example - above I mentioned the surface is parametrizable, and that the boundaries

were lines of constant parameters, so, I guess I could instead propose to approximate my function directly in the "flat" rectangular parametric space. In which case I have made no use of the geometry of the surface (other than the mapping function).



I'm not sure how to think about "this problem." Perhaps, I haven't posed my question "properly." What I'm looking for is some sort of formalism or framework to help me create new basis sets on a surface in 3d space. Where should I look?





Thanks!!

[zer0snr]

Perhaps, someone could at least tell me, intuitively, whether I'm wasting my time or not? Thanks...

[chiral3]

Sorry man, I can't help.  I am getting hung up on the word "optimal".  I am familiar in the context of finite elements where you interpolate mesh structures with basis functions, and there is some choosing there, but not "optimal".  Splines are another place to look.  You might want to look into radial basis functions.  Since you can parameterize the thing, this might be your best bet.

[zer0snr]

Thanks chiral3. I've been rather sloppy in laying out my thoughts, and also the word "optimal." On a re-read, I've managed to get myself "confused" on the word optimal. Let me get back to you on that.

[bloodninja]

Yes, this is a bit weird. Having said that, let's see if we can sort some of this out by answering a few questions:



1. If you have a 2d surface embedded in 3 dimensions then locally you should be able to write a graph for the points on the surface:



(x,y,z) = (x,y, f(x,y))



This requires a function f(x,y) for the z coordinate.



Now, you can choose whatever orthogonal functions u(x,y) that you want to write



f(x,y) = Sum_n a_n u_n(x,y)



where the coefficients are



a_n =   and <,> is the inner product on the function space.



So, do you have an f(x,y)....I would think that you do since you say you can 'parametrize it'? As you say, there are better and worse orthogonals to use to do this; if the patch is spherical then spherical harmonics seem more appropriate then say fourier series.



2. What do you mean by 'parametrization of the surface'? Do you mean that you have coordinates for the surface; or do you mean that you can write:



f(x,y) = g(x,y, a_i)



for some function g where a_i are model parameters?



3. You say that you want to define a 'basis' *on* the surface and that you want to then define functions 'over' the surface and in terms of this 'basis'. That suggests to be a coordinate basis. In the event that this is what you mean; then you can use the exterior calculus to write



ds^2 = dx^2 + dy^2 + ( f,x dx + f,y dy)^2



and your basis is dx, dy.



Much more can be said about all of this once some of these questions are sorted out.

[Nonius]

I'm not sure I quite understand the question.  If you have a patch of a surface (which in geek terms is called a chart), you have a mapping from a patch of flat two dim space into the surface.  As BN said, it actually can take the form of (x,y)->(x,y,f(x,y)), that is, all sufficiently small patches of surfaces can be realized as pieces of graphs of real valued functions. 



There are many ways of coordinitizing patches of surfaces, as you mention for a patch of the sphere, you can do (x,y,sqrt(1-x^2-y^2)) or you could do (theta1, theta2)-> spherical coordinates. 



Of course, surfaces are not always flat, so the moniker "basis" is slightly misleading, since basis normall means a minimal spanning set of vectors of a flat space.  Thus, I am taking liberty in assuming that you mean the basis in the patch of flat 2- dim space that coordinatizes the patch of the surface. 



I also am assuming that you mean real valued functions defined on the patch of the surface.  Also, that the patch has a boundary, although I don't quite know what you mean by constant on the boundary, maybe you mean functions that are constant when restricted to the boundary?  Anyway, I guess I don't fully understand the question, but, my guess is that such questions probably do not have general answers, simply because of the generality of the situation.

[zer0snr]

bloodninja,

While the example, (x,y,z) = (x,y,f(x,y)), you give is possible, I was thinking of a

more general case, (x,y,z) = (f(u,v), g(u,v), h(u,v)). And by parameterization I am

referring to the "surface coordinates" (u,v) (an example for (u,v) would be arc lengths

in appropriate directions). I think the usage "parameterization" is possibly a CAD term, but am

not sure where I picked it up (possibly the differential geometry text by D. Struik).



As you note, I could use, in my notation (u,v) above, some basis set in the rectangular

(u,v)-space (rectangular 'cause I assumed that the 4 borders correspond to lines of constant

u & v). So for example, if the surface were a part of the sphere, then (u,v) would correspond

to the latitude & longitudes, but if I used the "standard" Fourier basis set my convergence

would suffer, and sph. harmonics would do better (ie fewer terms in a truncated expansion)

at approximating some function on the surface. In this particular case, the reason why

one choice (sph harm.'s) does better than the other (vanilla Fourier) is because the former

matches exactly the singular parametrization at the poles of the sphere. And, hence exhibits

better convergence.



chiral3,

I guess the above example is probably the best way I can explain what I exactly meant by optimal.

Taking the example of standard Fourier vs. sph. harm. over a sphere, for a fixed error (say

error averaged over the domain of approximation) one can approximate a

function (defined on the sphere) with fewer terms (basis functions) using sph. harm.'s than

using standard Fourier. And thus optimal (of course, I'm assuming that sph. harm's are the

best amongst possibly infinite other basis sets, which, I don't know for sure. But, sph.

harm's are better than standard Fourier for approx over a sphere).



With respect to splines and other local approximants such as used in FE's, I'm trying to avoid

local approximation (at this stage). In the past, I've used B-splines with great success but want

to avoid them momentarily.....currently I want to explore global approximants....



Radial basis sets? I need to look into that, are they local or global? Will do a google

on them....



nonius,

just saw you post and will think about what you say....



Thanks all for thinking about it...

[kr]

no, zer0 you're trying to do too much.  With the charts, you just pull back to the flat thing and use any old basis you like.  If the jacobian doesn't vary too much, you're fine, and if it does, then just split up your charts a little more... if there's a serious kink then you will get crap convergence in the flat pullback-space.



Beyond the charts, there is the overall topology of the space that's an issue without a decent generic solution.  Basically you can piece together all the little charts, but how they come together in a global sense is a complicated problem with a lot of variety.  Spheres are a lot different than flat open space, so much so that certain properties of high-dim spheres are actually not known.  So, analysis people just don't go there. 



What they are more likely to do is start with a nice smooth 'partition of unity'.  Let's say X is your overall space, let's assume it looks like a bottle of jug wine.  Let 1 = 1(x) be the function that is always 1, no matter what x in X.  What you want to do is write



1(x) = sum_i b_i(x)



where b_i(x) is zero almost everywhere, except in some localized region where it looks like a pimple.  For instance, it may be 0 on every chart but the i'th one, where it is mostly one (except that it tapers down smoothly to zero on the edges).



Now, if you can do a basis projection on a specific chart, i.e. f = sum_j a_ij f_ij(x) on the i'th chart, you just piece all these guys together with your partition of unity, i.e.



g(x) = sum_i,j a_ij b_i(x) f_ij(x)



Hopefully the # of i's is finite, then if you take g_j = sum_i b_i.f_ij then you have some basis functions.  It ain't pretty, but it works.  In math we don't really care if it's finite, as long as we get the kind of convergence we require, but for practical applications, we try to do better than just say "there exists a set of charts" and engineer things to work really well.  I.e. for a sphere, we cut it in half, and think about the northern hemisphere and the southern hemisphere, overlapping just a very little bit, and do plain old flat euclidean analysis on each part separately.

[chiral3]

I am so fucking piussed off.  I just spent 10 minutes in the  equation editor only to have it crap out and then I lost the response I started.... eeeerrgrgrgrgvrhbg.



Short version:  Check out RBFs.



I think what bloodninja is referring to is different.  If you want to enbed a S^3 in R^2 you would use a fundemental form



M_ij = g_mn x^m_,i x^n_,j



where you would parameterize



x1 = rcos(theta)sin(theta), x2=........ so that x1^s+x2^2+x3^2=r^2 for Dead r, theta, phi)



That is what I think that he is referring to, which is more the diff geom point of view

[dgn2]

yeah, I cannot figure out how to make the equation editor work...not too clever.