3 ms·

Thanks for the answer and the SIAM reference, I did manage to pick up something interesting from there, I think. Smoothness, dimensionality and also adaptivity

by gerty 11y ago

Thanks for the answer and the SIAM reference, I did manage to pick up something interesting from there, I think. Smoothness, dimensionality and also adaptivity are vast topics indeed. Good luck with the future work!

mccourt 11y ago
I tried to do some digging to find easy to access (both through the web, but also not terribly complicated) references on error bounds for kernels interpolation. I don't think there is one ... although the internet will surely correct me if I'm wrong. The original source that I often return to (even before I look in my book) is Wendland's 2005 book "Scattered Data Approximation". But that book is really tough to read, even for me. Lots of tough math. Fasshauer's 2007 book "Meshfree Approximation Methods in Matlab" is much easier to read, but references Wendland's book for most of the heavy lifting. There is a newer branch of study on convergence dealing with what are called "sampling inequalities". In many ways, the math behind these is even worse as it usually requires a bunch of polynomial theory, but the results are more easily accessible. In particular, Christian Rieger and Barbara Wolmuth have outstanding content on this which is readily available on the web. In particular, I was able to find Christian's PhD thesis (https://www.deutsche-digitale-bibliothek.de/binary/JOWXLCGSV4NBOC5ZMLW553MRS6X65BFB/full/1.pdf https://www.deutsche-digitale-bibliothek.de/binary/JOWXLCGSV...) which provides a good journey from the start to actual results. Sorry I can't provide a cleaner statement about this. Maybe the most basic point I can make about convergence here would be to reference the early part of Christian's thesis where he alludes to the theorem that says the quality of an interpolant sort of looks like: error of interpolant = O(h^{k-d}) That is a gross simplification, but gives the gist of the result. h is the "fill distance" (or grid width for structured data) k is the smoothness of the kernel and d is the dimension of the data. This indicates why smooth kernels (large k) are needed to get the convergence for high d.

by gerty 11y ago

mccourt 11y ago

I tried to do some digging to find easy to access (both through the web, but also not terribly complicated) references on error bounds for kernels interpolation. I don't think there is one ... although the internet will surely correct me if I'm wrong. The original source that I often return to (even before I look in my book) is Wendland's 2005 book "Scattered Data Approximation". But that book is really tough to read, even for me. Lots of tough math. Fasshauer's 2007 book "Meshfree Approximation Methods in Matlab" is much easier to read, but references Wendland's book for most of the heavy lifting. There is a newer branch of study on convergence dealing with what are called "sampling inequalities". In many ways, the math behind these is even worse as it usually requires a bunch of polynomial theory, but the results are more easily accessible. In particular, Christian Rieger and Barbara Wolmuth have outstanding content on this which is readily available on the web. In particular, I was able to find Christian's PhD thesis (https://www.deutsche-digitale-bibliothek.de/binary/JOWXLCGSV4NBOC5ZMLW553MRS6X65BFB/full/1.pdf https://www.deutsche-digitale-bibliothek.de/binary/JOWXLCGSV...) which provides a good journey from the start to actual results. Sorry I can't provide a cleaner statement about this. Maybe the most basic point I can make about convergence here would be to reference the early part of Christian's thesis where he alludes to the theorem that says the quality of an interpolant sort of looks like: error of interpolant = O(h^{k-d}) That is a gross simplification, but gives the gist of the result. h is the "fill distance" (or grid width for structured data) k is the smoothness of the kernel and d is the dimension of the data. This indicates why smooth kernels (large k) are needed to get the convergence for high d.