One-particle Green’s functions obtained from the self-consistent solution of the Dyson equation can be employed in the evaluation of spectroscopic and thermodynamic properties for both molecules and solids. However, typical acceleration techniques used in the traditional quantum chemistry self-consistent algorithms cannot be easily deployed for the Green’s function methods because of a non-convex grand potential functional and a non-idempotent density matrix. Moreover, the optimization problem can become more challenging due to the inclusion of correlation effects, changing chemical potential, and fluctuations of the number of particles. In this paper, we study acceleration techniques to target the self-consistent solution of the Dyson equation directly. We use the direct inversion in the iterative subspace (DIIS), the least-squared commutator in the iterative subspace (LCIIS), and the Krylov space accelerated inexact Newton method (KAIN). We observe that the definition of the residual has a significant impact on the convergence of the iterative procedure. Based on the Dyson equation, we generalize the concept of the commutator residual used in DIIS and LCIIS and compare it with the difference residual used in DIIS and KAIN. The commutator residuals outperform the difference residuals for all considered molecular and solid systems within both GW and GF2. For amore »
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Inexact rational Krylov subspace method for eigenvalue problems
An inexact rational Krylov subspace method is studied to solve large-scale nonsymmetric eigenvalue problems. Each iteration (outer step) of the rational Krylov subspace method requires solution to a shifted linear system to enlarge the subspace, performed by an iterative linear solver for large-scale problems. Errors are introduced at each outer step if these linear systems are solved approx- imately by iterative methods (inner step), and they accumulate in the rational Krylov subspace. In this article, we derive an upper bound on the errors intro- duced at each outer step to maintain the same convergence as exact rational Krylov subspace method for approximating an invariant subspace. Since this bound is inversely proportional to the current eigenresidual norm of the target invariant subspace, the tolerance of iterative linear solves at each outer step can be relaxed with the outer iteration progress. A restarted variant of the inexact rational Krylov subspace method is also proposed. Numerical experiments show the effectiveness of relaxing the inner tolerance to save computational cost.
- Ye, Qiang
- Publication Date:
- NSF-PAR ID:
- Journal Name:
- Numerical Linear Algebra with Applications
- Sponsoring Org:
- National Science Foundation
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