To the best of my knowledge the book "Verification and Validation in Computational Science and Engineering" (1998) by Patrick Roache is one the most comprehensive engineering books detailing systematic studies to verify and validate numerical work.
I would like to find something equivalent related to spectral methods. What are the techniques to verify that a spectral method code is solving the differential equations correctly and how to validate that the model implemented spectral method code consistently replicate reality?
I'm not aware of any text/publication on V&V that seeks to address this in a systematic way when using spectral methods. There are numerous such studies (most on verification) in the literature, but no systematic study. On the other hand, standard techniques for V&V as outlined in the text by Roache are method agnostic and can be applied equally well to spectral methods as so more traditional methods such as FEM.
Jan, thanks for your response. On validation' I'm in agreement. The ultimate test is how well a model replicates reality.
On verification is where I hesitate. In theory, I agree with you. But Prof. Roache cautions the reader on extending his GCI approach to SM. I'm not an expert in SM but I was exposed to it on an elective course and the research work of one of my PhD committee member.s Polynomial convergence is somewhat different from grid convergence.
Again, Thank You!
All texts on spectral methods discuss the issue of convergence, that's a major part of the work in the area. That's what verification and validation mean-one tests the method on a problem whose solution is known and studies how the error evolves. The verification and validation of the numerical method itself takes place *before* any statement about how well it applies to an unknown problem is made, for it's its calibration on a known problem that provides the meaning that the method works at all-same as in any experimental procedure: first one models the apparatus, then one calibrates it on known data and then one can start checking that hitherto unknown data are described as expected by the model.
Thanks. Do you have very good reference on SM to recommend?
Not ALL texts address the issue of convergence in a methodical way. Europe has been ahead of us on the issue of quantifying uncertainties, either in the mathematics (verification) or in the physics (validation).
As indicated in my question, the book by Roache on V&V is so far the most comprehensive work I have read and studied. I'm reviewing the literature and came across the following reference:
Verification and validation benchmarks. .W.L. Oberkampf, T.G. Trucano / Nuclear Engineering and Design 238 (2008) 716–743
It is a well written compilation on the history of V&V and the different methods. Their discussion was helpful in confirming Roache was a pioneer and that his work has been fundamental in the progress of V&V. It confirms what Prof. Hesthaven shared in his initial response about the universality of Roche's approach.
Again Thank You!
Well a nice reference, to get started, is Boyd, cf. http://perso.uclouvain.be/alphonse.magnus/num1a/boyd01.pdf
After that it's a matter of taste involving the theory, i.e. mathematical aspects, the applications and the coding strategy. Apparently a lot of effort has gone into developing interfaces, like in Matlab. For efficiency, in the long term, though, coding in C is much more efficient, of course.
Basically most of the applications involve ``pseudospectral'' methods-it's very rare that one is expanding in eigenfunctions of the spatial operator; most of the time the solution is expanded in a basis of functions, that satisfy the boundary conditions and on which fast transforms act as their name implies. So the fundamental tools are fast transforms and solvers of the resulting system of ordinary differential equations-that's where most of the effort, in practice lies, since it is a good idea not to squander on the solver the advantages gained from using the transforms.
Thank You!
The paraphrasing line below the book dedication is inspiration to keep working harder on what we do....
The concept of 'verification' proposed by Roache is far from universally accepted.
In the widely cited paper
N. Oreskes, K. Shrader-Frechette, K. Belitz, Verification, Validation, and Confirmation of Numerical Models in the Earth Sciences, Science, Vol. 263, No. 5147 (Feb. 4, 1994), pp. 641-646
the authors propose arguments against the use of this term and rather talk about benchmarking.
Does it really matter, whether it's *called* ``verification'' or ``benchmarking'', or something else? The important point is to check that the code does work on known examples and its errors remain under control in the cases of interest.
Concerning the comment by Stam Nicolis, sometimes the choice of words has some implicit implications for a given scientific community, which is not always so easy to make explicit for people in different application areas. Saying what it means that a code 'works' is getting to be a rather complicated busines (think only of how can tries to assess that climate models 'work'), so I believe that some debate on how code quality assessment is carried out and some shared terminology among different communities can help reducing the potential (and actual) confusion.
Well, the code isn't that important, since it depends on compiler and so on-it's the ``machine-independent'' properties that matter. Given some (pseudo-)spectral method it's a machine independent statement to say that the error behaves in a certain way as a function of the number of terms in the expansion in a given basis. The statement that a well-defined measure of the error attains 10^(-16) is something that can be checked independently of any one implementation, for instance.
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