2,400 research outputs found
Krylov projection methods for linear Hamiltonian systems
We study geometric properties of Krylov projection methods for large and
sparse linear Hamiltonian systems. We consider in particular energy
preservation. We discuss the connection to structure preserving model
reduction. We illustrate the performance of the methods by applying them to
Hamiltonian PDEs.Comment: 16 pages, 17 figure
Large-scale electronic structure theory for simulating nanostructure process
Fundamental theories and practical methods for large-scale electronic
structure calculations are given, in which the computational cost is
proportional to the system size. Accuracy controlling methods for microscopic
freedoms are focused on two practical solver methods, Krylov-subspace method
and generalized-Wannier-state method. A general theory called the
'multi-solver' scheme is also formulated, as a hybrid between different solver
methods. Practical examples are carried out in several insulating and metallic
systems with 10^3-10^5 atoms. All the theories provide general guiding
principles of constructing an optimal calculation for simulating nanostructure
processes, since a nanostructured system consists of several competitive
regions, such as bulk and surface regions, and the simulation is designed to
reproduce the competition with an optimal computational cost.Comment: 19 pages, 6 figures. To appear in J. Phys. Cond. Matt. A preprint PDF
file in better graphics is available at
http://fujimac.t.u-tokyo.ac.jp/lses/index_e.htm
A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems
In this paper, we discuss numerical methods for solving large-scale
continuous-time algebraic Riccati equations. These methods have been the focus
of intensive research in recent years, and significant progress has been made
in both the theoretical understanding and efficient implementation of various
competing algorithms. There are several goals of this manuscript: first, to
gather in one place an overview of different approaches for solving large-scale
Riccati equations, and to point to the recent advances in each of them. Second,
to analyze and compare the main computational ingredients of these algorithms,
to detect their strong points and their potential bottlenecks. And finally, to
compare the effective implementations of all methods on a set of relevant
benchmark examples, giving an indication of their relative performance
Krylov Subspace Method for Molecular Dynamics Simulation based on Large-Scale Electronic Structure Theory
For large scale electronic structure calculation, the Krylov subspace method
is introduced to calculate the one-body density matrix instead of the
eigenstates of given Hamiltonian. This method provides an efficient way to
extract the essential character of the Hamiltonian within a limited number of
basis set. Its validation is confirmed by the convergence property of the
density matrix within the subspace. The following quantities are calculated;
energy, force, density of states, and energy spectrum. Molecular dynamics
simulation of Si(001) surface reconstruction is examined as an example, and the
results reproduce the mechanism of asymmetric surface dimer.Comment: 7 pages, 3 figures; corrected typos; to be published in Journal of
the Phys. Soc. of Japa
On the ADI method for the Sylvester Equation and the optimal- points
The ADI iteration is closely related to the rational Krylov projection
methods for constructing low rank approximations to the solution of Sylvester
equation. In this paper we show that the ADI and rational Krylov approximations
are in fact equivalent when a special choice of shifts are employed in both
methods. We will call these shifts pseudo H2-optimal shifts. These shifts are
also optimal in the sense that for the Lyapunov equation, they yield a residual
which is orthogonal to the rational Krylov projection subspace. Via several
examples, we show that the pseudo H2-optimal shifts consistently yield nearly
optimal low rank approximations to the solutions of the Lyapunov equations
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