Timeline for Approximating a $C^1$ function in $Lip$ norm with piecewise linear
Current License: CC BY-SA 3.0
10 events
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| Sep 12, 2016 at 18:42 | comment | added | Willie Wong | If you have to use linear approximations, then naturally you want to decompose your space using the appropriate dimensional simplexes. Doing a quick search I found this article, which proves $L^\infty$ bounds on the differences of the first derivatives (between the approximated and approximating functions) assuming $C^{1,1}$ bounds on the original function. I didn't read carefully enough to see if what they proved can be adapted to what you want. But maybe that's a starting point to look in the literature. | |
| Sep 12, 2016 at 18:08 | comment | added | Piero D'Ancona | Yes this sounds easier to obtain. But for the application I have in mind piecewise linear would be much better | |
| Sep 12, 2016 at 15:54 | comment | added | Willie Wong | I think the "applied" solution to this is that instead of using piecewise linear approximations, they use piecewise $n$-linear approximations for most direct interpolation needs. (Or piecewise $n$-cubic if they want something that is $C^1$.) For 2D this would be bilinear approximations (4 data points and 4 unknowns) on the square. // I need to think a bit about whether Lip convergence holds; it should follow from the fact that $C^1$ implies strong differentiability. | |
| Sep 12, 2016 at 9:47 | comment | added | Duchamp Gérard H. E. | @PieroD'Ancona O.K. | |
| Sep 12, 2016 at 8:46 | comment | added | Piero D'Ancona | But let me repeat: I'm asking here because this must be a very well known problem, so probably some expert can answer without duplication of effort. | |
| Sep 12, 2016 at 8:45 | comment | added | Piero D'Ancona | Yes but how to integrate? not obvious how to get a continuous piecewise linear function | |
| Sep 12, 2016 at 8:40 | comment | added | Duchamp Gérard H. E. | Does'nt it do your job to approximate the derivative the way you describe and then integrate ? | |
| Sep 12, 2016 at 7:50 | comment | added | Piero D'Ancona | But what about the convergence in Lip? | |
| Sep 12, 2016 at 7:49 | comment | added | Qiaochu Yuan | In 2D a natural analogue of your procedure is to subdivide $[a, b]^2$ into triangles and join the corresponding points in the graph by the unique triangle passing through them. | |
| Sep 12, 2016 at 7:38 | history | asked | Piero D'Ancona | CC BY-SA 3.0 |