We study the discretization of a linear evolution partial differential equation when its Green’s function is known or well approximated. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method that we developed in J. Math. Phys. 51 (2010), n. 10, 103502 (reference [15]). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.
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Iterative subspace algorithms for finite-temperature solution of Dyson equation
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 a number of bond-breaking problems, we found that an easily obtained high-temperature solution with effectively suppressed correlations is a very effective starting point for reaching convergence of the problematic low-temperature solutions through a sequential reduction of temperature during calculations.
more » « less- NSF-PAR ID:
- 10363554
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- The Journal of Chemical Physics
- Volume:
- 156
- Issue:
- 9
- ISSN:
- 0021-9606
- Page Range / eLocation ID:
- Article No. 094101
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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