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Spline interpolation is a rich and extensively studied field, dating back to at least the 1940s. Here are a few comments about related work. The interpolation a

by jethkl 3y ago

Spline interpolation is a rich and extensively studied field, dating back to at least the 1940s. Here are a few comments about related work. The interpolation algorithm itself can be adjusted in numerous ways. One effective algorithm for reducing overshooting artifacts is Akima spline interpolation, but there are many others. Cubic splines have noteworthy theoretical properties; you can search for "energy minimizer cubic spline" to find out more. As already mentioned in @creata's comment, instead of using Gaussian elimination, you can take advantage of the fact that the matrix is banded and solve it using a banded solve algorithm (search for DGBSV in LAPACK). There is also research that explores the relationship between shift-invariant spaces and functions that can be expressed as a linear combination of splines. Finally, multivariate interpolation is an extension of the 1-d theory to n-d. This is used in image and surface interpolation, for example.

algas 3y ago
I was hoping someone would mention the variational properties of splines, so I'm glad to see you bring up their energy minimization. In Strang's Intro to Applied Mathematics, he briefly mentions (at the end of the section on splines) that a spline interpolation can be found by minimum principles, since it's essentially passing a beam in bending through the control points. I was curious if anyone has used that practically, eg writing a finite element spline solver?
- jethkl 3y ago
  I am unaware of any effort in that direction, but I did find a 1987 paper by Höllig [1] that seems to do a kind of converse (approximates FEM solutions using splines. I haven't read the paper, but that's my quick take). Recovering splines using a FEM might be an interesting learning exercise, since the analytic solution is known and the model seems simple. The motivating physical model is described by flat splines [2], and Carl de Boor has photographs of physical splines in action [3]. [1] https://www.semanticscholar.org/paper/Finite-element-methods-with-B-splines-Höllig/aa9f8256ed0f334882f597d7771837c6ecd605e0 https://www.semanticscholar.org/paper/Finite-element-methods... [2] https://en.wikipedia.org/wiki/Flat_spline https://en.wikipedia.org/wiki/Flat_spline [3] https://pages.cs.wisc.edu/~deboor/draftspline.html https://pages.cs.wisc.edu/~deboor/draftspline.html

by jethkl 3y ago

algas 3y ago

I was hoping someone would mention the variational properties of splines, so I'm glad to see you bring up their energy minimization. In Strang's Intro to Applied Mathematics, he briefly mentions (at the end of the section on splines) that a spline interpolation can be found by minimum principles, since it's essentially passing a beam in bending through the control points. I was curious if anyone has used that practically, eg writing a finite element spline solver?

jethkl 3y ago
I am unaware of any effort in that direction, but I did find a 1987 paper by Höllig [1] that seems to do a kind of converse (approximates FEM solutions using splines. I haven't read the paper, but that's my quick take). Recovering splines using a FEM might be an interesting learning exercise, since the analytic solution is known and the model seems simple. The motivating physical model is described by flat splines [2], and Carl de Boor has photographs of physical splines in action [3]. [1] https://www.semanticscholar.org/paper/Finite-element-methods-with-B-splines-Höllig/aa9f8256ed0f334882f597d7771837c6ecd605e0 https://www.semanticscholar.org/paper/Finite-element-methods... [2] https://en.wikipedia.org/wiki/Flat_spline https://en.wikipedia.org/wiki/Flat_spline [3] https://pages.cs.wisc.edu/~deboor/draftspline.html https://pages.cs.wisc.edu/~deboor/draftspline.html