tea.mathoverflow.net - Discussion Feed (What about numerics?)

Nilima comments on "What about numerics?" (14589)

2011-05-23T18:29:38-07:00

Michael, I'd suggest you consider sending a few people your manuscript. I would be happy to read it, and circulate it to colleagues to ask for their views.

There's a cautionary tale about numerics, efficiency and accuracy - if you look up the history of the conjugate gradient method, it emerged 'too early'. Round-off errors rendered this subtle algorithm impractical, and it was almost consigned to the dustbins. Arrive double-precision arithmetic, and voila! Every numerical analyst learns about Krylov subspace iterations now.

In other words, without reading your paper I cannot tell if the speed issue outweighs the accuracy one. Here's the test: fix the precision for a well-posed saddle point problem on a manifold. how does your method compare to existing ones, when trying to achieve this precision?

an_mo_user comments on "What about numerics?" (14545)

2011-05-18T12:34:34-07:00

Michael Kissner, you are welcome. I assumed (wrongly it seems) there was somebody who supervises your research-work; part of some master-thesis, say. However, as Will Jagy said, in your case the 'go-to' professor seems like the perfect person to talk to. Whether he/she is an expert in the field or not is not so relevant, the crucial thing is familiarity with your individual situation, as well as the general (country) and local (the institution and those close by) academic environment; this is key to give usefuld advice. Good luck, and hoping to see some of your mathematical questions on main!

Will Jagy comments on "What about numerics?" (14533)

2011-05-17T15:24:47-07:00

Michael, in the American system, students getting a Master's or a Ph.D. will typically have a single professor who is regarded as their adviser and takes the most direct responsibility. There have been enough history discussions on MO to show that this is not necessarily the case elsewhere, and probably was not always the case in the U.S. You do really need to ask your go-to professor. By and large, people on MO do not read or offer judgement on any sort of manuscript posted online. If anyone seems especially interested, you might consider sending a pdf to some individual by email. Still, it is hard to imagine an interested stranger offering the kind of hands-on support you require at this stage.

Michael Kissner comments on "What about numerics?" (14530)

2011-05-17T13:55:18-07:00

Thanks an_mo_user. Sadly, as a normal master student I don't have an Advisor. I will however ask my "go-to" Professor on what he thinks (or is that what is meant by an advisor?)

an_mo_user comments on "What about numerics?" (14529)

2011-05-17T13:26:35-07:00

I will try to give a very general answer to your 2nd (now only) question.

First, you should talk to somebody more senior who knows the field and the precise situation. In view of your age (as given in your user profile) and as you say it was supposed to be your first paper, I assume you have some kind of advisor or senior collegue with whom you are working with. If she/he thinks it is a good idea to publish it then try it, if not then not. The idea that somebody else will improve your work seems unlikely to me; except for a 'local' person, say the person advising you, somebody else in the 'group', somebody else this person knows. To find some good collaborator you do not already know (at least indirectly) for an ongoing project on-the-fly seems very unlikely. So, again, you should talk to somebody more senior who knows the field and the precise situation.

Finally, while you retracted it, let me add: you really should not worry about generalities some other mathematicians say on another field. For essentially any pair of maths subfields (A,B) you will find somebody that will explain you why A is way more important than B (and yes often based on highly incomplete information). If an expert in your field tells you that your precise research direction does not seem so fruitful, you should pay attention. But which subfield is 'the best,' is a question of the type which [insert random sport]-team is the best; discuss it if you enjoy it, if not, ignore the discussion.

Michael Kissner comments on "What about numerics?" (14528)

2011-05-17T12:48:32-07:00

"since it seems to be an opinion survey"

Actually, this is kind of what I was hoping for. I'm quite biased, as our University is all about applied Physics/Engineering. Which means that all the math we do, at some point turns into some form of Numerics.
There is no pure Mathematics departement here. It has only been recently that I have learned of this "new" world of pure mathematics.

The reason I wanted to know what the MO-Community thinks about Numerics is rather simple. I've recently had to "defend" Numerics against Friends/Students from another Munich University (See below).
I just wanted to know if this was an isolated case, or that the general opinion of Numerics was rather low?

But you're right, this is probably not the right type of question for MO. I've edited this part out of my original post.

"[...] a specific class of problems where you can bound the error in terms of mesh size, and for that class, your method has better error asymptotics"

Yes, I can reduce the overall error using my alternate method of FEM.

Roughly speaking, current Methods approximate Meshes using linear or quadratic elements (or NURBS, as you pointed out, in Nilima's answer).
I try to minimize this error, which sadly made my method alot more complex

"I'm not a computer scientist, but as far as I know, the only known advantages of quantum computers are derived from database searching and Fourier transforms in black-box abelian groups"

The Quantum computer argument was made by a friend of mine (my 2nd post). I'm no computer scientist either, so I have no idea what QC are able and not able to do.

His argument went as follows:
QC are able to solve Matrix equations and find eigenvalues alot faster => We can invert Matrices instead of using Iterative Methods => No more use for Numerics.

This seems to be the common misconception that all of numerics is about solving Ax = b.
And he forgot one crucial point, if we can solve a big equation in a short time, we will just want to make it bigger.

Scott Carnahan comments on "What about numerics?" (14527)

2011-05-17T10:24:43-07:00

As I write this, there are 190 questions tagged as numerical analysis, and many of them have received insightful answers. Questions about numerics are welcome on MathOverflow, as long as they are well-defined mathematical questions and at an appropriate level. The first question you are asking does not seem to fit these criteria, since it seems to be an opinion survey. I can't speak for others, but I doubt there will be a "general MO-view" of numerics, since the users form a rather diverse group.

I'm not a computer scientist, but as far as I know, the only known advantages of quantum computers are derived from database searching and Fourier transforms in black-box abelian groups. In particular, I don't know of any evidence that quantum computers will make the typical numerical problem any easier to solve.

Regarding the second question, what do you mean when you say that your numerical method is "more accurate" than existing methods? My wild guess is that you have in mind a specific class of problems where you can bound the error in terms of mesh size, and for that class, your method has better error asymptotics. Is that an accurate interpretation? If so, your results are probably publishable. I see that you asked a question about this last month. Have you followed up on the reference in Nilima's answer?

Michael Kissner comments on "What about numerics?" (14526)

2011-05-17T10:23:01-07:00

"Is the thing with the quantum computers a trick-question?"

I've had a similar discussion (regarding my first question) with some friends, and this was the argument given by one of them. While not meant as a trick question (but re-reading my question, sure seems like one)
it was rather meant to point out that I'm also interested in "false beliefs" or other arguments you've heard in relation to numerics.

After reading your point a), it actually now makes good sense. I just realised that the only reason why numerics "stuck out" to me was, that my other interests are represented rather well here (Topology & Riemann Geometry). But you're right, there are other under-represented fields here on MO (Statistics, OR...).

an_mo_user comments on "What about numerics?" (14524)

2011-05-17T08:00:45-07:00

Regarding 1. (I cannot answer 2., on the one hand, for lack of expertise in this field, and on the other hand it will be difficult to answer without detailed information; perhaps you can ask somebody senior you know for advice?)

I do not know the general MO-view and I am not around since the start of this site, but it is my understanding that, yes, numerics is rare on MO put merely due to the fact that relatively few regular users of MO have this as their center of interest and thus few question/answers come up and thus few people with these interest become regular users and thus...a vicious circle.

But, numerics is by no means the only field which is not present on MO relative to its general relevance. I believe there are two phenomena:

a. as described above, if many people in a given subfield are one MO, then even more arrive. (I do not know the initial distribution of the user base well; but I beleive that simply due to the respective interests of the people who started the site there is some bias that propagated. Not because somebody wants it to be like this, but just because it happended like this and essentially cannot be changed. In an alternate world where some people in numerics of PDEs rather than algebraic geometers or topologists had started the site it might look very different.)

b. I believe some fields are better suited for the format. In fields with a lot of theory there are more questions somebody might have that somebody else can (easily) answer. However, in fact, based on my vague understanding, I always imagined numerics could be a good fit for this type of questions as it is (in certain aspects) a bit vague and thus experience (that can be communicated) is important.

Regarding numerics in general my personal view. It is not my field, but I am aware of some things I find very interesting (mathematically) and I am convinced there are many more (which I simply have never heard of). Certainly not a 'border group.' Is the thing with the quantum computers a trick-question?

Michael Kissner comments on "What about numerics?" (14523)

2011-05-17T07:26:29-07:00

(This question is aimed at every mathematician)
I didn't want to post this as a MO-Question, as I don't know if it would be appropriate?

I've been working on a new numerical method this past half year (Geodesic FEM on Riemannian Manifolds).
I have however hit a brick wall. My method seems to work, it's actually even more accurate than current Methods. BUT, it's slower (only by a constant), hard to use and in alot of cases near impossible to implement.

The question is: Is it worth publishing? I've put alot of heart and soul into this and this was supposed to be my first paper. It really makes me sad to think that it's not even worth publishing.
My thinking was, that perhaps someone else could "improve" on it and make it worthwhile? Or is that just wishfull thinking? (HONEST! Answer please)

Edit: Removed the other question, as was pointed out to me, it was rather opinion based and I have to agree. It's probably best left to the coffee table.