[JOSS] Fig 1

[jakelangham]

This issue relates to ongoing reviews at openjournals/joss-reviews#7018.

I have been going through the README and new documentation and got a bit stuck on the first figure. I think the idea is just about clear but it is not presented in a self-contained way and the writing is very terse. (I can see why that might be a good thing in the README but in the documentation you are free to go into detail.) There is a lot of notation in the figure that is not defined so even though I think I've guessed correctly what e.g. l_x and l_y are supposed to mean, I can't be exactly sure.

I've since looked at the paper and here there is a more detailed explanation of the figure, which helped me, though still there is left undefined.

Finally, I think a few brief comment on the properties of the example scheme might be helpful. I suppose the implication of the figure is that the sparse grids combine to give an effective discretisation indicated by the black-and-white diagram in the top right. Can you say anything about how this would compare to discretising over an 'equivalent' full grid (whose resolution was given by the finest scale present in the sparse grids)? Would the sparsely-combined method have the same order of accuracy? Would it be more efficient, or do you need to go to higher dimensional problems to see the benefits? Perhaps this all depends on details outside the scope of the example, but even so, anything you could say could be useful for a reader less familiar with the methods than yourself.

[freifrauvonbleifrei]

@jakelangham this is my first try -- I tried to write it in a way that would not introduce a lot of symbols that are not already present in the image, and to mention only things that are so general I would not have to point to one specific source.

Please let me know what you think! I feel like anything Sparse Grid quickly generates a whole tail of notation, so I am rather unsure if this is the right "depth". If you agree, I would put a similar range of additional info into the paper (or maybe only the complexities?).

[freifrauvonbleifrei]

(see here for the actual rendering -> https://discotec--139.org.readthedocs.build/en/139/combination_technique.html )

[jakelangham]

@freifrauvonbleifrei I find this level of detail very helpful, thanks. I think there are a few remaining typos though:

• The combination coefficients are sometimes c^c and sometimes c^_{\vect{\ell}}. I suppose it should always be the latter.
• \mathcal{I} is not defined and I presume the sum definining f_{SG} should be over \vect{\ell} \in \mathcal{I}.
• In a couple of places the LaTeX is not rendering. There's a \ell missing a backslash and also a \log.

[freifrauvonbleifrei]

Hi @jakelangham, thanks for pointing this out! will update! (also, I am starting to think that my editor renders markdown math differently than github)