Pole Expansion for Solving a Type of Parametrized Linear Systems in Electronic Structure Calculations

@article{Damle2014PoleEF,
  title={Pole Expansion for Solving a Type of Parametrized Linear Systems in Electronic Structure Calculations},
  author={Anil Damle and Lin Lin and Lexing Ying},
  journal={SIAM J. Sci. Comput.},
  year={2014},
  volume={36}
}
We present a new method for solving parametrized linear systems. Under certain assumptions on the parametrization, solutions to the linear systems for all parameters can be accurately approximated by linear combinations of solutions to linear systems for a small set of fixed parameters. Combined with either direct solvers or preconditioned iterative solvers for each linear system with a fixed parameter, the method is particularly suitable for situations when solutions to a large number of… Expand
Preconditioning Orbital Minimization Method for Planewave Discretization
TLDR
Numerical results validate the performance of the new preconditioner for the orbital minimization method, in particular, the iteration number is reduced to $\mathcal{O}(1)$ and often only a few iterations are enough for convergence. Expand

References

SHOWING 1-10 OF 53 REFERENCES
A fast direct solver for boundary integral equations in two dimensions
We describe an algorithm for the direct solution of systems of linear algebraic equations associated with the discretization of boundary integral equations with non-oscillatory kernels in twoExpand
A partial Padé-via-Lanczos method for reduced-order modeling
Abstract The classical Lanczos process can be used to efficiently generate Pade approximants of the transfer function of a given single-input single-output time-invariant linear dynamical system.Expand
The Solution of Parametrized Symmetric Linear Systems
  • K. Meerbergen
  • Mathematics, Computer Science
  • SIAM J. Matrix Anal. Appl.
  • 2003
TLDR
The Lanczos and MINRES methods for the solution of symmetric linear systems with a linear parameter arising from structural engineering are compared and an error estimation for the problem in finite precision arithmetic is proposed. Expand
Self-Consistent Equations Including Exchange and Correlation Effects
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or highExpand
Restarted GMRES for Shifted Linear Systems
TLDR
This work develops a variant of the restarted GMRES method exhibiting the same advantage and investigates its convergence for positive real matrices in some detail and applies it to speed up "multiple masses" calculations arising in lattice gauge computations in quantum chromodynamics, one of the most time-consuming supercomputer applications. Expand
A rational Lanczos algorithm for model reduction
TLDR
A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padé approximant) for the large-scale system. Expand
KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations
TLDR
KSSOLV, a MATLAB toolbox for solving a class of nonlinear eigenvalue problems known as the Kohn-Sham equations, is described, designed to enable researchers in computational and applied mathematics to investigate the convergence properties of the existing algorithms. Expand
An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
The present investigation designs a systematic method for finding the latent roots and the principal axes of a matrix, without reducing the order of the matrix. It is characterized by a wide field ofExpand
Efficient linear circuit analysis by Pade´ approximation via the Lanczos process
TLDR
PVL, an algorithm for computing the Pad6 approximation of Laplace-domain transfer functions of large linear networks via a Lanczos process, has significantly superior numerical stability and renders unnecessary many of the heuristics that AWE and its derivatives had to employ. Expand
The SIESTA method for ab initio order-N materials simulation
We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set,Expand
...
1
2
3
4
5
...