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| Hauptseite > Publikationsdatenbank > Sequences of generalized eigenproblems in DFT |
| Hauptseite > Publikationsdatenbank > Sequences of generalized eigenproblems in DFT |
| Conference Presentation | PreJuSER-15613 |
;
2011
Abstract: Research in several branches of chemistry and material science relies on large numerical simulations. Many of these simulations are based on Density Functional Theory (DFT) models that lead to sequences of generalized eigenproblems $\{P^{(i)}\}$. Every simulation normally requires the solution of hundreds of sequences, each comprising dozens of large and dense eigenproblems; in addition, the problems at iteration $i+1$ of $\{P^{(i)}\}$ are constructed manipulating the solution of the problems at iteration $i$. The size of each problem ranges from 10 to 40 thousand and the interest lays in the eigenpairs corresponding to the lower 10-30\% part of the spectrum. Due to the dense nature of the eigenproblems and the large portion of the spectrum requested, iterative solvers are not competitive; as a consequence, current simulation codes uniquely use direct methods. In this talk we present a study that highlights how eigenproblems in successive iterations are strongly correlated to one another. In order to understand this result, we need to stress the importance of the basis wave functions, which constitute the building blocks of any DFT scheme. Indeed, the matrix entries of each problem in $\{P^{(i)}\}$ are calculated through superposition integrals of a set of basis wave functions. Moreover, the state wave functions---describing the quantum states of the material---are linear combinations of basis wave functions with coefficients given by the eigenvectors of the problem. Since a new set of basis wave functions is determined at each iteration of the simulation, the eigenvectors between adjacent iterations are only loosely linked with one another. In light of these considerations it is surprising to find such a deep correlation between the eigenvectors of successive problems. We set up a mechanism to track the evolution over iterations $i=1,\dots,n$ of the angle between eigenvectors $x^{(i)}$ and $x^{(i+1)}$ corresponding to the $j^{th}$ eigenvalue. In all cases the angles decrease noticeably after the first few iterations and become almost negligible, even though the overall simulation is not close to convergence. Even the state of the art direct eigensolvers cannot exploit this behavior in the solutions. In contrast, we propose a 2-step approach in which the use of direct methods is limited to the first few iterations, while iterative methods are employed for the rest of the sequence. The choice of the iterative solver is dictated by the large number of eigenpairs required in the simulation. For this reason we envision the Subspace Iterations Method---despite its slow convergence rate---to be the method of choice. Nested at the core of the method lays an inner loop $V \leftarrow A V$; due to the observed correlation between eigenvectors, the convergence is reached in a limited number of steps. In summary, we propose evidence in favor of a mixed solver in which direct and iterative methods are combined together.
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