1,599 research outputs found
Unsymmetric Lanczos model reduction and linear state function observer for flexible structures
This report summarizes part of the research work accomplished during the second year of a two-year grant. The research, entitled 'Application of Lanczos Vectors to Control Design of Flexible Structures' concerns various ways to use Lanczos vectors and Krylov vectors to obtain reduced-order mathematical models for use in the dynamic response analyses and in control design studies. This report presents a one-sided, unsymmetric block Lanczos algorithm for model reduction of structural dynamics systems with unsymmetric damping matrix, and a control design procedure based on the theory of linear state function observers to design low-order controllers for flexible structures
The symmetric heavy-light ansatz
The symmetric heavy-light ansatz is a method for finding the ground state of
any dilute unpolarized system of attractive two-component fermions.
Operationally it can be viewed as a generalization of the Kohn-Sham equations
in density functional theory applied to N-body density correlations. While the
original Hamiltonian has an exact Z_2 symmetry, the heavy-light ansatz breaks
this symmetry by skewing the mass ratio of the two components. In the limit
where one component is infinitely heavy, the many-body problem can be solved in
terms of single-particle orbitals. The original Z_2 symmetry is recovered by
enforcing Z_2 symmetry as a constraint on N-body density correlations for the
two components. For the 1D, 2D, and 3D attractive Hubbard models the method is
in very good agreement with exact Lanczos calculations for few-body systems at
arbitrary coupling. For the 3D attractive Hubbard model there is very good
agreement with lattice Monte Carlo results for many-body systems in the limit
of infinite scattering length.Comment: 38 pages, 13 figures, revised manuscript includes results for 1D, 2D,
and 3
Solving large sparse eigenvalue problems on supercomputers
An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require the use of the matrix in the form of matrix by vector multiplications. The implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method are compared. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, methods of performing this operation efficiently are discussed. The advantages and the disadvantages of each method are compared and implementation aspects are discussed. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. Possible parallel implementations are also discussed
Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems
In these lecture notes, we present a pedagogical review of a number of
related {\it numerically exact} approaches to quantum many-body problems. In
particular, we focus on methods based on the exact diagonalization of the
Hamiltonian matrix and on methods extending exact diagonalization using
renormalization group ideas, i.e., Wilson's Numerical Renormalization Group
(NRG) and White's Density Matrix Renormalization Group (DMRG). These methods
are standard tools for the investigation of a variety of interacting quantum
systems, especially low-dimensional quantum lattice models. We also survey
extensions to the methods to calculate properties such as dynamical quantities
and behavior at finite temperature, and discuss generalizations of the DMRG
method to a wider variety of systems, such as classical models and quantum
chemical problems. Finally, we briefly review some recent developments for
obtaining a more general formulation of the DMRG in the context of matrix
product states as well as recent progress in calculating the time evolution of
quantum systems using the DMRG and the relationship of the foundations of the
method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical
review appearing in the proceedings of the "IX. Training Course in the
Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri
sul Mare (Salerno, Italy, October 2004
Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media
Maxwell's equations for propagation of electromagnetic waves in dispersive
and absorptive (passive) media are represented in the form of the Schr\"odinger
equation , where is a linear
differential operator (Hamiltonian) acting on a multi-dimensional vector
composed of the electromagnetic fields and auxiliary matter fields describing
the medium response. In this representation, the initial value problem is
solved by applying the fundamental solution to the initial field
configuration. The Faber polynomial approximation of the fundamental solution
is used to develop a numerical algorithm for propagation of broad band wave
packets in passive media. The action of the Hamiltonian on the wave function
is approximated by the Fourier grid pseudospectral method. The algorithm
is global in time, meaning that the entire propagation can be carried out in
just a few time steps. A typical time step is much larger than that in finite
differencing schemes, . The accuracy and stability
of the algorithm is analyzed. The Faber propagation method is compared with the
Lanczos-Arnoldi propagation method with an example of scattering of broad band
laser pulses on a periodic grating made of a dielectric whose dispersive
properties are described by the Rocard-Powels-Debye model. The Faber algorithm
is shown to be more efficient. The Courant limit for time stepping, , is exceeded at least in 3000 times in the Faber propagation
scheme.Comment: Latex, 17 pages, 4 figures (separate png files); to appear in J.
Comput. Phy
Krylov-space approach to the equilibrium and the nonequilibrium single-particle Green's function
The zero-temperature single-particle Green's function of correlated fermion
models with moderately large Hilbert-space dimensions can be calculated by
means of Krylov-space techniques. The conventional Lanczos approach consists of
finding the ground state in a first step, followed by an approximation for the
resolvent of the Hamiltonian in a second step. We analyze the character of this
approximation and discuss a numerically exact variant of the Lanczos method
which is formulated in the time domain. This method is extended to get the
nonequilibrium single-particle Green's function defined on the
Keldysh-Matsubara contour in the complex time plane. The proposed method will
be important as an exact-diagonalization solver in the context of
self-consistent or variational cluster-embedding schemes. For the recently
developed nonequilibrium cluster-perturbation theory, we discuss the efficient
implementation and demonstrate the feasibility of the Krylov-based solver. The
dissipation of a strong local magnetic excitation into a non-interacting bath
is considered as an example for applications.Comment: 20 pages, 5 figures, v2 with minor corrections, JPCM in pres
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