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Localized density matrix minimization and linear-scaling algorithms

Publication ,  Journal Article
Lai, R; Lu, J
Published in: Journal of Computational Physics
June 15, 2016

We propose a convex variational approach to compute localized density matrices for both zero temperature and finite temperature cases, by adding an entry-wise ℓ1 regularization to the free energy of the quantum system. Based on the fact that the density matrix decays exponentially away from the diagonal for insulating systems or systems at finite temperature, the proposed ℓ1 regularized variational method provides an effective way to approximate the original quantum system. We provide theoretical analysis of the approximation behavior and also design convergence guaranteed numerical algorithms based on Bregman iteration. More importantly, the ℓ1 regularized system naturally leads to localized density matrices with banded structure, which enables us to develop approximating algorithms to find the localized density matrices with computation cost linearly dependent on the problem size.

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Published In

Journal of Computational Physics

DOI

EISSN

1090-2716

ISSN

0021-9991

Publication Date

June 15, 2016

Volume

315

Start / End Page

194 / 210

Related Subject Headings

  • Applied Mathematics
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences
 

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Lai, R., & Lu, J. (2016). Localized density matrix minimization and linear-scaling algorithms. Journal of Computational Physics, 315, 194–210. https://doi.org/10.1016/j.jcp.2016.02.076
Lai, R., and J. Lu. “Localized density matrix minimization and linear-scaling algorithms.” Journal of Computational Physics 315 (June 15, 2016): 194–210. https://doi.org/10.1016/j.jcp.2016.02.076.
Lai R, Lu J. Localized density matrix minimization and linear-scaling algorithms. Journal of Computational Physics. 2016 Jun 15;315:194–210.
Lai, R., and J. Lu. “Localized density matrix minimization and linear-scaling algorithms.” Journal of Computational Physics, vol. 315, June 2016, pp. 194–210. Scopus, doi:10.1016/j.jcp.2016.02.076.
Lai R, Lu J. Localized density matrix minimization and linear-scaling algorithms. Journal of Computational Physics. 2016 Jun 15;315:194–210.
Journal cover image

Published In

Journal of Computational Physics

DOI

EISSN

1090-2716

ISSN

0021-9991

Publication Date

June 15, 2016

Volume

315

Start / End Page

194 / 210

Related Subject Headings

  • Applied Mathematics
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences