Extrapolation not working for some BCs

[tomasaschan]

There are still some boundary conditions for which InterpCubic (from #19) isn't working correctly. See e.g. this comment on #29.

[tomasaschan]

Acutally, it appears that it's not only for cubic interpolation we have problems with some BCs:

using Grid

f(x) = sin(2pi * (x-1) / 10)
yirefl = InterpGrid(f(1:11),BCreflect,InterpLinear)
yiperi = InterpGrid(f(1:11),BCperiodic,InterpLinear)

df = vcat(
DataFrame(x=-9:.1:40,y=[yirefl[x]+2 for x in -9:.1:40],t="BCreflect"),
DataFrame(x=-9:.1:40,y=[yiperi[x]-1 for x in -9:.1:40],t="BCperiodic"),
)
plot(df,x=:x,y=:y,color=:t,Geom.path,Guide.colorkey("Boundary condition"))

There's a weird shift in BCreflect, stemming from reflecting around the half-point outside the data set rather than around the edge, and similarly in BCperiodic. Also, neither seem to work at all for indices x<1, i.e. to the left of the given data set.

(If the final datapoint is left out from the BCperiodic case, i.e. f(1:10) instead of f(1:11), it looks good for large indices. Still bad for x<1 though.)

[timholy]

This definition of reflection is consistent with a reflection of a vector x begin defined as [x, x[end:-1:1]]. It's also the definition used by matlab in their image transformations.

The thing below 1 is just weird, gotta look into that.

[tomasaschan]

@timholy Yeah, I guess it makes a lot of sense for imaging applications, but it really doesn't for physics applications. For example, interpolating the electric field along a ray path, when the signal hits a reflecting surface could be done using BCreflect if the reflection point was the last node, rather than at the half-point outside. Maybe Interpolations.jl should have two reflecting boundary conditions, which do this differently (one nice for images and one nice for physics)?

[timholy]

I'm not sure it's physics vs images. Who says the reflecting surface is positioned exactly at a node? Perhaps I might have set my grid points to be a half-node away from the edge.

But flexibility would be good. I'm not sure we want to support any choice in the interval [0,1], but certainly on-node and half-node seem reasonable.

[tomasaschan]

Yeah, I didn't like the naming either, but I had to distinguish between them somehow :P Supporting both seems like the target then. Hopefully, on-node reflection will be good enough for cases when the current behavior between the two near-edge nodes is unsatisfactory for some reason.

[timholy]

Sounds good to me.

[cmundi]

Joining very late. In most applications of field theories (EM, QM, solid state, etc.) it is essential to be able to place the boundary at node or mid-node. There are even some edge (pun!) cases where quarter-intervals are numerically convenient. Having node and mid-node will cover most cases and help users write more transparently correct code.

[tomasaschan]

@cmundi: Both node and mid-node are supported in Interpolations.jl in terms of the GridBehavior type argument - if the boundaries should be on the outermost datapoint, use OnGrid, while if you want the boundary a half-interval outside the otuermost datapoint, use OnCell.

[cmundi]

Perfect! Thanks for your reply.
On Mar 17, 2015 9:48 AM, "Tomas Lycken" notifications@github.com wrote:

@cmundi https://github.com/cmundi: Both node and mid-node are supported
in Interpolations.jl in terms of the GridBehavior type argument - if the
boundaries should be on the outermost datapoint, use OnGrid, while if you
want the boundary a half-interval outside the otuermost datapoint, use
OnCell.

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