What is different between point-collocation and spectral projection method?

[nguyenhoaiThanhbk2811]

Hi you. I am doing a comparison between two method point collocation and spectral project method. What is different between the two methods? and which the method has the best result? when do we use them? I have read the tutorial of yours. your conclusion is the spectral Project have the result more than point collocation. but when I try with my research then I have the opposite conclusion with you. have you the paper about this issue?

I have been used the tool NISP on the Scilab software to implement the spectral Projection method to compare with the point collocation method in your tool.

"https://chaospy.readthedocs.io/en/master/tutorial.html"

I am also reading the paper may be the paper using your tool but it is not compared between the two methods. "https://www.biorxiv.org/content/biorxiv/early/2018/03/05/274779.full.pdf" thank you. HoaiThanh

[jonathf]

First of all, if you used NISP, you haven't used spectral projection you have ben using psuedo-spectral projection (as I call it) or Non-intrusive spectral projection (as they call it). Actual spectral projection, often called intrusive Galerkin method.

The difference between the two methods it can be summarized as follows:

• Collocation method estimate the Fourier coefficients through least squares regression of evaluations of a model.
• Psuedo-spectral method estimates the same coefficients through nummerical integration by noting that each coefficient have the form: <u,P>/<P,P> and applying quadrature rules.

As a rule of thumb, pseudo-spectral method has the potential for higher accuracy, but is also more fragile. You are more likely to get stable good results with collocation. This is especially true for highly non-linear problems.

I am assuming that you are doing this for research, and the fact that you are asking this question, you are looking to get an overview. If that is correct, I can highly recommend the book "Numerical Methods for Stochastic Computations -- A Spectral Method Aproach" by Dongbin Xiu. It explains the concepts a lot better and in details.

[jonathf]

No, I am saying that spectral projection is an ambiguous term and you shouldn't use the term without "pseudo-" or "non-intrusive" prefix is in order to not mistake it from the Galerkin methods.

Sounds about right.

I would like to note that I think we have gotten to the point that it is worth saying using an issue tracker of a software you are using might not be the correct place for such a conversation. I am here to help you if you need help implementing a specific method using Chaospy. I am not here to help you understand the underlying theory. That is why I recommended the book.