3,527 research outputs found
Introduction to stochastic error correction methods
We propose a method for eliminating the truncation error associated with any
subspace diagonalization calculation. The new method, called stochastic error
correction, uses Monte Carlo sampling to compute the contribution of the
remaining basis vectors not included in the initial diagonalization. The method
is part of a new approach to computational quantum physics which combines both
diagonalization and Monte Carlo techniques.Comment: 11 pages, 1 figur
Lanczos algorithm with Matrix Product States for dynamical correlation functions
The density-matrix renormalization group (DMRG) algorithm can be adapted to
the calculation of dynamical correlation functions in various ways which all
represent compromises between computational efficiency and physical accuracy.
In this paper we reconsider the oldest approach based on a suitable
Lanczos-generated approximate basis and implement it using matrix product
states (MPS) for the representation of the basis states. The direct use of
matrix product states combined with an ex-post reorthogonalization method
allows to avoid several shortcomings of the original approach, namely the
multi-targeting and the approximate representation of the Hamiltonian inherent
in earlier Lanczos-method implementations in the DMRG framework, and to deal
with the ghost problem of Lanczos methods, leading to a much better convergence
of the spectral weights and poles. We present results for the dynamic spin
structure factor of the spin-1/2 antiferromagnetic Heisenberg chain. A
comparison to Bethe ansatz results in the thermodynamic limit reveals that the
MPS-based Lanczos approach is much more accurate than earlier approaches at
minor additional numerical cost.Comment: final version 11 pages, 11 figure
Density Matrix Renormalization Group and Reaction-Diffusion Processes
The density matrix renormalization group (DMRG) is applied to some
one-dimensional reaction-diffusion models in the vicinity of and at their
critical point. The stochastic time evolution for these models is given in
terms of a non-symmetric ``quantum Hamiltonian'', which is diagonalized using
the DMRG method for open chains of moderate lengths (up to about 60 sites). The
numerical diagonalization methods for non-symmetric matrices are reviewed.
Different choices for an appropriate density matrix in the non-symmetric DMRG
are discussed. Accurate estimates of the steady-state critical points and
exponents can then be found from finite-size scaling through standard
finite-lattice extrapolation methods. This is exemplified by studying the
leading relaxation time and the density profiles of diffusion-annihilation and
of a branching-fusing model in the directed percolation universality class.Comment: 16 pages, latex, 5 PostScript figures include
Text Summarization
With the overwhelming amount of textual information available in electronic formats on the web, there is a need for an efficient text summarizer capable of condensing large bodies of text into shorter versions while keeping the relevant information intact. Such a technology would allow users to get their information in a shortened form, saving valuable time. Since 1997, Microsoft Word has included a summarizer for documents, and currently there are companies that summarize breaking news and send SMS for mobile phones. I wish to create a text summarizer to provide condensed versions of original documents. My focus is on blogs, because people are increasingly using this mode of communication to express their opinions on a variety of topics. Consequently, it will be very useful for a reader to be able to employ a concise summary, tailored to his or her own interests to quickly browse through volumes of opinions relevant to any number of topics. Although many summarization methods exist, my approach involves employing the Lanczos algorithm to compute eigenvalues and eigenvectors of a large sparse matrix and SVD (Singular Value Decomposition) as a means of identifying latent topics hidden in contexts; and the next phase of the process involves taking a high-dimensional set of data and reducing it to a lower-dimensional set. This procedure makes it possible to identify the best approximation of the original text. Since SQL makes it possible to allow analyzing data sets and take advantage of the parallel processing available today, in most database management systems, SQL is employed in my project. The utilization of SQL without external math libraries, however, adds to challenge in the computation of the SVD and the Lanczos algorithm
A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential
We present a method for the numerical calculation of derivatives of functions
of general complex matrices. The method can be used in combination with any
algorithm that evaluates or approximates the desired matrix function, in
particular with implicit Krylov-Ritz-type approximations. An important use case
for the method is the evaluation of the overlap Dirac operator in lattice
Quantum Chromodynamics (QCD) at finite chemical potential, which requires the
application of the sign function of a non-Hermitian matrix to some source
vector. While the sign function of non-Hermitian matrices in practice cannot be
efficiently approximated with source-independent polynomials or rational
functions, sufficiently good approximating polynomials can still be constructed
for each particular source vector. Our method allows for an efficient
calculation of the derivatives of such implicit approximations with respect to
the gauge field or other external parameters, which is necessary for the
calculation of conserved lattice currents or the fermionic force in Hybrid
Monte-Carlo or Langevin simulations. We also give an explicit deflation
prescription for the case when one knows several eigenvalues and eigenvectors
of the matrix being the argument of the differentiated function. We test the
method for the two-sided Lanczos approximation of the finite-density overlap
Dirac operator on realistic gauge field configurations on lattices with
sizes as large as and .Comment: 26 pages elsarticle style, 5 figures minor text changes, journal
versio
Improving the Efficiency of FP-LAPW Calculations
The full-potential linearized augmented-plane wave (FP-LAPW) method is well
known to enable most accurate calculations of the electronic structure and
magnetic properties of crystals and surfaces. The implementation of atomic
forces has greatly increased it's applicability, but it is still generally
believed that FP-LAPW calculations require substantial higher computational
effort compared to the pseudopotential plane wave (PPW) based methods.
In the present paper we analyse the FP-LAPW method from a computational point
of view. Starting from an existing implementation (WIEN95 code), we identified
the time consuming parts and show how some of them can be formulated more
efficiently. In this context also the hardware architecture plays a crucial
role. The remaining computational effort is mainly determined by the setup and
diagonalization of the Hamiltonian matrix. For the latter, two different
iterative schemes are compared. The speed-up gained by these optimizations is
compared to the runtime of the ``original'' version of the code, and the PPW
approach. We expect that the strategies described here, can also be used to
speed up other computer codes, where similar tasks must be performed.Comment: 20 pages, 3 figures. Appears in Comp. Phys. Com. Other related
publications can be found at http://www.rz-berlin.mpg.de/th/paper.htm
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