3,632 research outputs found
A Lanczos Method for Approximating Composite Functions
We seek to approximate a composite function h(x) = g(f(x)) with a global
polynomial. The standard approach chooses points x in the domain of f and
computes h(x) at each point, which requires an evaluation of f and an
evaluation of g. We present a Lanczos-based procedure that implicitly
approximates g with a polynomial of f. By constructing a quadrature rule for
the density function of f, we can approximate h(x) using many fewer evaluations
of g. The savings is particularly dramatic when g is much more expensive than f
or the dimension of x is large. We demonstrate this procedure with two
numerical examples: (i) an exponential function composed with a rational
function and (ii) a Navier-Stokes model of fluid flow with a scalar input
parameter that depends on multiple physical quantities
Approximating spectral densities of large matrices
In physics, it is sometimes desirable to compute the so-called \emph{Density
Of States} (DOS), also known as the \emph{spectral density}, of a real
symmetric matrix . The spectral density can be viewed as a probability
density distribution that measures the likelihood of finding eigenvalues near
some point on the real line. The most straightforward way to obtain this
density is to compute all eigenvalues of . But this approach is generally
costly and wasteful, especially for matrices of large dimension. There exists
alternative methods that allow us to estimate the spectral density function at
much lower cost. The major computational cost of these methods is in
multiplying with a number of vectors, which makes them appealing for
large-scale problems where products of the matrix with arbitrary vectors
are relatively inexpensive. This paper defines the problem of estimating the
spectral density carefully, and discusses how to measure the accuracy of an
approximate spectral density. It then surveys a few known methods for
estimating the spectral density, and proposes some new variations of existing
methods. All methods are discussed from a numerical linear algebra point of
view
Block bond-order potential as a convergent moments-based method
The theory of a novel bond-order potential, which is based on the block
Lanczos algorithm, is presented within an orthogonal tight-binding
representation. The block scheme handles automatically the very different
character of sigma and pi bonds by introducing block elements, which produces
rapid convergence of the energies and forces within insulators, semiconductors,
metals, and molecules. The method gives the first convergent results for
vacancies in semiconductors using a moments-based method with a low number of
moments. Our use of the Lanczos basis simplifies the calculations of the band
energy and forces, which allows the application of the method to the molecular
dynamics simulations of large systems. As an illustration of this convergent
O(N) method we apply the block bond-order potential to the large scale
simulation of the deformation of a carbon nanotube.Comment: revtex, 43 pages, 11 figures, submitted to Phys. Rev.
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
A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process
We discuss a Krylov-Schur like restarting technique applied within the symplectic Lanczos algorithm for the Hamiltonian eigenvalue problem. This allows to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic Lanczos algorithm. The Krylov-Schur-like restarting is based on the SR algorithm. Some ingredients of the latter need to be adapted to the structure of the symplectic Lanczos recursion. We demonstrate the efficiency of the new method for several Hamiltonian eigenproblems
Application of vector-valued rational approximations to the matrix eigenvalue problem and connections with Krylov subspace methods
Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage
Many Masses on One Stroke: Economic Computation of Quark Propagators
The computational effort in the calculation of Wilson fermion quark
propagators in Lattice Quantum Chromodynamics can be considerably reduced by
exploiting the Wilson fermion matrix structure in inversion algorithms based on
the non-symmetric Lanczos process. We consider two such methods: QMR (quasi
minimal residual) and BCG (biconjugate gradients). Based on the decomposition
of the Wilson mass matrix, using QMR, one can carry
out inversions on a {\em whole} trajectory of masses simultaneously, merely at
the computational expense of a single propagator computation. In other words,
one has to compute the propagator corresponding to the lightest mass only,
while all the heavier masses are given for free, at the price of extra storage.
Moreover, the symmetry can be used to cut
the computational effort in QMR and BCG by a factor of two. We show that both
methods then become---in the critical regime of small quark
masses---competitive to BiCGStab and significantly better than the standard MR
method, with optimal relaxation factor, and CG as applied to the normal
equations.Comment: 17 pages, uuencoded compressed postscrip
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