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I read the notes on online about Regularized matrix computation. It said

The truncated SVD solution has “ringing,” e.g., Gibbs’s phenomenon in truncated Fourier series

I haven't seen any work related that has SVD and prodcuing the Gibbs's phenomenon, since it does not assume any kind of continuity, so I don't get why there is Gibbs's phenomena?

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    $\begingroup$ SVD deals with linear projections. We all know linear approximations to step functions perform badly across the entire domain. Gibb's Phenomenon has to do with higher order (polynomial) approximations. So it seems unrelated to me. On the other hand, if the matrix is just a polynomial basis, then I doubt truncation will necessarily get rid of ringing. $\endgroup$ – AdamO May 23 '18 at 21:21
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Well, I just find an example that uses DMD (dynamic mode decomposition) with truncated SVD, that reveals the problem of Gibbs phenomenon.

It is a simple linear advection problem but the initial condition is discontinous. So any finite truncation of frequency that leads to a finite set of travelling waves will have Gibbs phenomenon.

I have to say it is surprise to me. But since someone has proved that DMD is nothing but FFT in periodic domain, then it is natural.

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  • $\begingroup$ Do you have a reference or a link to an example? $\endgroup$ – kjetil b halvorsen Oct 30 '18 at 12:16
  • $\begingroup$ @kjetilbhalvorsen You can write a very simple DMD testing case yourself for square wave. I don't think that will take you more than 10 mins. $\endgroup$ – ArtificiallyIntelligence Oct 30 '18 at 14:47

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