Timeline for Thin-Plate-Spline understanding and solution
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 31, 2023 at 10:31 | history | edited | user8469759 | CC BY-SA 4.0 |
edited body
|
| May 31, 2023 at 10:24 | comment | added | user8469759 | I gave it more thought and I wrote an update. Would be nice if someone checked I did not write rubbish. | |
| May 31, 2023 at 10:24 | history | edited | user8469759 | CC BY-SA 4.0 |
added 2588 characters in body
|
| May 26, 2023 at 6:33 | comment | added | user8469759 | I think this is also a good reference: google.co.kr/books/edition/Spline_Models_for_Observational_Data/… | |
| May 22, 2023 at 15:23 | history | edited | gmvh |
Replaced "approximation-theory" by more pertinent "splines" tag
|
|
| May 22, 2023 at 11:14 | answer | added | rych | timeline score: 3 | |
| May 22, 2023 at 10:36 | comment | added | rych | You're asking for the reference. I've tried to find a nice write-up on this too. Meanwhile, your solution looks okay to me. You've successfully linked the energy functional and biharmonic equation for the bending of a thin plate pinned at points. Indeed, the $\omega$ is the delta function in this case and so a solution will be a sum of (Green's) radial functions which you found in math.stackexchange.com/questions/4702167/… by introducing variables and so on (optionally using Fourier transform) and it is indeed the TPS plus a low-degree polynomial. | |
| May 22, 2023 at 7:36 | comment | added | user8469759 | I'll wait for a more conclusive answer. | |
| May 22, 2023 at 6:50 | comment | added | user378654 | I don't know, and maybe someone can answer more conclusively (it's clear you can minimize over some homogeneous space like $\dot{H}^2$, but you have to check whether you get unique solutions that way). I'd suggest thinking through the 1D Laplacian analogy: minimize $\int_{\mathbb{R}} (u')^2$ over $u$ with some finite number of points prescribed. You should get a piecewise linear function which has slope $0$ at the rightmost and leftmost (infinite) intervals as the only minimizer, so the energy form "forced" this nontrivial boundary condition at infinity. | |
| May 22, 2023 at 4:37 | comment | added | user8469759 | What if I assume maybe $f \in W(\mathbb{R}^2)^{3,2} \cap C^3(\mathbb{R}^2)$? (so first 3 order derivatives in $L^2$ but also continuous?) Is it too restrictive? | |
| May 22, 2023 at 4:26 | comment | added | user378654 | Actually minimizing would perhaps require more care, though, or at least some consideration of how your functions behave at infinity. | |
| May 22, 2023 at 4:22 | comment | added | user378654 | Part of this is presumably related to the property of the biharmonic equation in the plane where isolated points have positive capacity, i.e. you can prescribe a boundary condition at points, unlike the Laplace equation (this property can be read off the fundamental solution: it's continuous at the origin). So you can minimize over the class of functions with prescribed values at the given set of points, and you'll get a minimizer in this class (and it will satisfy the EL equation on the complement of the set of points, analogously to what you derive with Lagrange multipliers formally). | |
| May 22, 2023 at 1:48 | history | asked | user8469759 | CC BY-SA 4.0 |