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Currently I am trying to apply to a graduate program overseas. They require me to include a research proposal in my application. My topic of interest is computational materials science.

So, one application of ab initio methods for modelling matter is density functional theory. In it, a central component is a formulation of the many-body Schrodinger equation known as the Kohm-Sham equation. The equation is

$$ \left( \frac{-\hbar^2\nabla^2}{2m} + \mathcal{V}_N(\textbf{r}) + \mathcal{V}_H(\textbf{r}) + \mathcal{V}_{xc}[\rho(\textbf{r})] \right) \phi_i(\textbf{r}) = \textit{E}_i \phi_i (\textbf{r}) $$

To solve this equation, one can apply the self consistency principle and iteratively solve the equation until convergence:

  1. Pick initial $\rho(\textbf{r})$ and decide the exchange-correlation functional
  2. Calculate potentials
  3. Solve the eigenvalue problem
  4. Get new $\rho(\textbf{r})$ and check for convergence
  5. Go to step 2 if necessary

The eigenproblem step can get computationally expensive for large systems. The proposed research would speed up this step for certain systems by applying parallelization and relevant numerical techniques like spectral approximation, polynomial filtering, and spectral slicing. We also propose looking into modifying known iterative algorithms for solving the eigenvalue problem.

EDIT: I feel like I'm throwing a dart while blindfolded with writing this research proposal for a graduate school application and would like some input.

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    $\begingroup$ Welcome to our site! $\endgroup$ – Camps Jan 9 at 11:30
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    $\begingroup$ I faced the same situation several times. What I always did was: first, look for the supervisor research lines and second, look at the literature for the state of the art and then made a project. The idea you wrote above is not only related to Material Science, but is broad. $\endgroup$ – Camps Jan 9 at 11:35
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    $\begingroup$ I agree with the comment of @camps and want to add that it might be a good idea to get into informal contact with such a PI and tell him on an abstract level that you are interested in performing research work on speeding up such calculations. You already name specific approaches but I think you don't realize that the applicability of these approaches depends on the specific DFT implementation and many DFT codes use several of these approaches since years. Therefore you should first listen to the ideas the PI has when asked about such a project. $\endgroup$ – Gregor Michalicek Jan 9 at 16:49
  • $\begingroup$ Thanks @Camps for the very quick reply. I looked up several researchers. $\endgroup$ – Jonathan Jan 10 at 0:29
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    $\begingroup$ Since this has been pushed to the front page I take the opportunity to make another remark: Having a look at what kind of software libraries are available for the diagonalization one kind of application is rather scarcely covered. This is solving the dense generalized eigenvalue problem for memory distributed calculations on GPUs, i.e., if the problem is distributed over multiple nodes of an HPC system. I think people have started working on this but at the moment this situation is kind of a showstopper. I could imagine that there are still many interesting research questions in that area. $\endgroup$ – Gregor Michalicek Jun 3 at 14:09
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I agree with Camps that your best bet is to look at a potential advisor and see what they study. Your given example might be a bit of a stretch for some computational materials science advisors, due to its very fundamental nature of solving the problem rather than applying the problem. This could be exactly what one advisor works on though so you should look around the department.

You should look to see who does applied versus methods work for example, you would want to work with a methods orientated person for that project. Machine learning is also a vague methods based project topic as well.

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