how do you deal with multidimensional problems?

[floswald]

Hi there,
just wanted to ask how you interpolate a 3D problem, for example, in the Quadratic BSpline case. Do you take the tensor product of the univariate Spline basis functions in each dimension? I couldn't find it in the code. I'd like to talk about this package in a class I'm teaching, so I need to know a bit more.

thanks!

[tomasaschan]

Yes, that is the basic idea. The introductory sections of this paper (referenced in the document under doc/latex in this repo) are a good read.

It's also instructive to look at the output of the following code for various dimensions:

using Interpolations, Base.Cartesian
macroexpand(Interpolations.index_gen(typeof(BSpline(Quadratic(Flat()))), 1))

Try that, then change the dimension to 2 and 3 etc to see how the actual interpolation is implemented. You'll get an expression consisting of variables on the form cx_d, where x indicates a step along the current dimension d (e.g. m for -1, p for +1), as well as indexing expressions into itp.coefs, which is the data to be interpolated (with the pre-filter described in the paper applied already).

[floswald]

Excellent, I see. So you place the data on the vertices of the hypercube to
compute the coefficients, and then in get_index[x] you compute some form
of linear combination of basis functions, depending on the segment that x
comes to lie in?
It is dawning on me that I did hardcode all of this in ApproXD.jl. :-/

On 30 November 2015 at 11:58, Tomas Lycken notifications@github.com wrote:

Yes, that is the basic idea. The introductory sections of this paper
http://bigwww.epfl.ch/publications/thevenaz0002.pdf (referenced in the
document under doc/latex in this repo) are a good read.

It's also instructive to look at the output of the following code for
various dimensions:

using Interpolations, Base.Cartesian
macroexpand(Interpolations.index_gen(typeof(BSpline(Quadratic(Flat()))), 1))

Try that, then change the dimension to 2 and 3 etc to see how the actual
interpolation is implemented. You'll get an expression consisting of
variables on the form cx_d, where x indicates a step along the current
dimension d (e.g. m for -1, p for +1), as well as indexing expressions
into itp.coefs, which is the data to be interpolated (with the pre-filter
described in the paper applied already).

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[tomasaschan]

Yes, exactly.

And it's not too late to make ApproXD.jl use Interpolations.jl instead ;)