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I'm not sure if this is the right place to ask, but:

I'm a final year undergraduate maths student, and I'm soon to begin a project based on some paper or some topic of interest to me. I haven't been able to come up with a project title thus far.

My request is: could someone direct me to some papers/articles that I could study in detail and maybe base my project on? I've taken two separate numerical methods modules and enjoyed both of them, so I'd like to do it on something out of this area. Specifically, I'd be interested in:

higher order RK-methods for solving ODEs
numerical methods for solving PDEs in $2, 3, 4$ dimensions
methods for estimating eigenvalues of large matrices
Newton's method in higher dimensions

Thanks in advance.

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  • $\begingroup$ Non-answer: Numerical methods for ODEs are a pretty solved area of numerical mathematics. As for PDEs, much of the current research is focused on proving convergence, obtaining bounds on the rates of convergence, etc. If you have some basic knowledge of numerical methods and numerical linear algebra, I would recommend learning about a more rapidly growing area instead. An example that comes to mind is randomized numerical linear algebra. $\endgroup$ – parsiad Nov 26 '17 at 21:46
  • $\begingroup$ When I took numerical lin alg, the professor who taught it specialized in it. That is, it was his area of research/specialty. I will forward this to him to see what he says. Is there anything else specific you would like me to ask him? $\endgroup$ – David Reed Nov 26 '17 at 22:17
  • $\begingroup$ @parsiad thanks for your reply, would you be able to direct me to a useful article/paper relating to that? I understand that it's a well covered area but I was hoping for maybe a specific method/result that I could study and understand well to use as the basis for my project. I have updated my original question $\endgroup$ – davkav9 Nov 26 '17 at 22:45
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    $\begingroup$ @vanaghka: M. Mahoney's lecture notes (arxiv.org/abs/1608.04481) are a self-contained intro to RandNLA. In my opinion, implementing any of the algorithms in those notes (e.g., least squares, low rank approximation, etc.) would be a good undergraduate course project. $\endgroup$ – parsiad Nov 27 '17 at 0:08

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