661 research outputs found
Google matrix of the citation network of Physical Review
We study the statistical properties of spectrum and eigenstates of the Google
matrix of the citation network of Physical Review for the period 1893 - 2009.
The main fraction of complex eigenvalues with largest modulus is determined
numerically by different methods based on high precision computations with up
to binary digits that allows to resolve hard numerical problems for
small eigenvalues. The nearly nilpotent matrix structure allows to obtain a
semi-analytical computation of eigenvalues. We find that the spectrum is
characterized by the fractal Weyl law with a fractal dimension .
It is found that the majority of eigenvectors are located in a localized phase.
The statistical distribution of articles in the PageRank-CheiRank plane is
established providing a better understanding of information flows on the
network. The concept of ImpactRank is proposed to determine an influence domain
of a given article. We also discuss the properties of random matrix models of
Perron-Frobenius operators.Comment: 25 pages. 17 figures. Published in Phys. Rev.
Krylov subspace techniques for model reduction and the solution of linear matrix equations
This thesis focuses on the model reduction of linear systems and the solution of large
scale linear matrix equations using computationally efficient Krylov subspace techniques.
Most approaches for model reduction involve the computation and factorization of large
matrices. However Krylov subspace techniques have the advantage that they involve only
matrix-vector multiplications in the large dimension, which makes them a better choice
for model reduction of large scale systems. The standard Arnoldi/Lanczos algorithms are
well-used Krylov techniques that compute orthogonal bases to Krylov subspaces and, by
using a projection process on to the Krylov subspace, produce a reduced order model that
interpolates the actual system and its derivatives at infinity. An extension is the rational
Arnoldi/Lanczos algorithm which computes orthogonal bases to the union of Krylov
subspaces and results in a reduced order model that interpolates the actual system and
its derivatives at a predefined set of interpolation points. This thesis concentrates on the
rational Krylov method for model reduction.
In the rational Krylov method an important issue is the selection of interpolation points
for which various techniques are available in the literature with different selection criteria.
One of these techniques selects the interpolation points such that the approximation
satisfies the necessary conditions for H2 optimal approximation. However it is possible
to have more than one approximation for which the necessary optimality conditions are
satisfied. In this thesis, some conditions on the interpolation points are derived, that
enable us to compute all approximations that satisfy the necessary optimality conditions
and hence identify the global minimizer to the H2 optimal model reduction problem.
It is shown that for an H2 optimal approximation that interpolates at m interpolation
points, the interpolation points are the simultaneous solution of m multivariate polynomial
equations in m unknowns. This condition reduces to the computation of zeros of a
linear system, for a first order approximation. In case of second order approximation the
condition is to compute the simultaneous solution of two bivariate polynomial equations.
These two cases are analyzed in detail and it is shown that a global minimizer to the
H2 optimal model reduction problem can be identified. Furthermore, a computationally
efficient iterative algorithm is also proposed for the H2 optimal model reduction problem
that converges to a local minimizer.
In addition to the effect of interpolation points on the accuracy of the rational interpolating
approximation, an ordinary choice of interpolation points may result in a reduced
order model that loses the useful properties such as stability, passivity, minimum-phase and bounded real character as well as structure of the actual system. Recently in the
literature it is shown that the rational interpolating approximations can be parameterized
in terms of a free low dimensional parameter in order to preserve the stability of the
actual system in the reduced order approximation. This idea is extended in this thesis
to preserve other properties and combinations of them. Also the concept of parameterization
is applied to the minimal residual method, two-sided rational Arnoldi method
and H2 optimal approximation in order to improve the accuracy of the interpolating
approximation.
The rational Krylov method has also been used in the literature to compute low rank
approximate solutions of the Sylvester and Lyapunov equations, which are useful for
model reduction. The approach involves the computation of two set of basis vectors in
which each vector is orthogonalized with all previous vectors. This orthogonalization
becomes computationally expensive and requires high storage capacity as the number of
basis vectors increases. In this thesis, a restart scheme is proposed which restarts without
requiring that the new vectors are orthogonal to the previous vectors. Instead, a set of
two new orthogonal basis vectors are computed. This reduces the computational burden
of orthogonalization and the requirement of storage capacity. It is shown that in case
of Lyapunov equations, the approximate solution obtained through the restart scheme
approaches monotonically to the actual solution
Ulam method for the Chirikov standard map
We introduce a generalized Ulam method and apply it to symplectic dynamical
maps with a divided phase space. Our extensive numerical studies based on the
Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator
on a chaotic component converges to a continuous limit. Typically, in this
regime the spectrum of relaxation modes is characterized by a power law decay
for small relaxation rates. Our numerical data show that the exponent of this
decay is approximately equal to the exponent of Poincar\'e recurrences in such
systems. The eigenmodes show links with trajectories sticking around stability
islands.Comment: 13 pages, 13 figures, high resolution figures available at:
http://www.quantware.ups-tlse.fr/QWLIB/ulammethod/ minor corrections in text
and fig. 12 and revised discussio
On the parallel solution of parabolic equations
Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented
On the Generation of Large Passive Macromodels for Complex Interconnect Structures
This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed technique
Time integration and steady-state continuation for 2d lubrication equations
Lubrication equations allow to describe many structurin processes of thin
liquid films. We develop and apply numerical tools suitable for their analysis
employing a dynamical systems approach. In particular, we present a time
integration algorithm based on exponential propagation and an algorithm for
steady-state continuation. In both algorithms a Cayley transform is employed to
overcome numerical problems resulting from scale separation in space and time.
An adaptive time-step allows to study the dynamics close to hetero- or
homoclinic connections. The developed framework is employed on the one hand to
analyse different phases of the dewetting of a liquid film on a horizontal
homogeneous substrate. On the other hand, we consider the depinning of drops
pinned by a wettability defect. Time-stepping and path-following are used in
both cases to analyse steady-state solutions and their bifurcations as well as
dynamic processes on short and long time-scales. Both examples are treated for
two- and three-dimensional physical settings and prove that the developed
algorithms are reliable and efficient for 1d and 2d lubrication equations,
respectively.Comment: 33 pages, 16 figure
Freed by interaction kinetic states in the Harper model
We study the problem of two interacting particles in a one-dimensional
quasiperiodic lattice of the Harper model. We show that a short or long range
interaction between particles leads to emergence of delocalized pairs in the
non-interacting localized phase. The properties of these Freed by Interaction
Kinetic States (FIKS) are analyzed numerically including the advanced Arnoldi
method. We find that the number of sites populated by FIKS pairs grows
algebraically with the system size with the maximal exponent , up to a
largest lattice size reached in our numerical simulations, thus
corresponding to a complete delocalization of pairs. For delocalized FIKS pairs
the spectral properties of such quasiperiodic operators represent a deep
mathematical problem. We argue that FIKS pairs can be detected in the framework
of recent cold atom experiments [M.~Schreiber {\it et al.} Science {\bf 349},
842 (2015)] by a simple setup modification. We also discuss possible
implications of FIKS pairs for electron transport in the regime of
charge-density wave and high superconductivity.Comment: 26 pages, 21 pdf and png figures, additional data and high quality
figures are available at http://www.quantware.ups-tlse.fr/QWLIB/fikspairs/ ,
parts of sections 2 and 3 moved to appendices, manuscript accepted for EPJ
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