2,516 research outputs found
Random field Ising model: dimensional reduction or spin-glass phase?
The stability of the random field Ising model (RFIM) against spin glass (SG)
fluctuations, as investigated by M\'ezard and Young, is naturally expressed via
Legendre transforms, stability being then associated with the non-negativeness
of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is
found that the signal for the occurrence of the SG transition will manifest
itself in free-energy {\sl fluctuations\/} only, and not in the free energy
itself. Eigenvalues of the inverse SG susceptibility matrix is then approached
by the Rayleigh Ritz method which provides an upper bound. Coming from the
paramagnetic phase {\sl on the Curie line,\/} one is able to use a virial-like
relationship generated by scaling the {\sl single\/} unit length in
higher dimension a new length sets in, the inverse momentum cut off).
Instability towards a SG phase being probed on pairs of {\sl distinct\/}
replicas, it follows that, despite the repulsive coupling of the RFIM the
effective pair coupling is {\sl attractive\/} (at least for small values of the
parameter the coupling and the
effective random field fluctuation). As a result, \lq\lq bound states\rq\rq\
associated with replica pairs (negative eigenvalues) provide the instability
signature. {\sl Away from the Curie line\/}, the attraction is damped out till
the SG transition line is reached and paramagnetism restored. In the
SG transition always precedes the ferromagnetic one, thus the domain in
dimension where standard dimensional reduction would apply (on the Curie line)
shrinks to zero.Comment: te
Parallel eigensolvers in plane-wave Density Functional Theory
We consider the problem of parallelizing electronic structure computations in
plane-wave Density Functional Theory. Because of the limited scalability of
Fourier transforms, parallelism has to be found at the eigensolver level. We
show how a recently proposed algorithm based on Chebyshev polynomials can scale
into the tens of thousands of processors, outperforming block conjugate
gradient algorithms for large computations
A robust and efficient implementation of LOBPCG
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely
used to compute eigenvalues of large sparse symmetric matrices. The algorithm
can suffer from numerical instability if it is not implemented with care. This
is especially problematic when the number of eigenpairs to be computed is
relatively large. In this paper we propose an improved basis selection strategy
based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence
criterion which is backward stable to enhance the robustness. We also suggest
several algorithmic optimizations that improve performance of practical LOBPCG
implementations. Numerical examples confirm that our approach consistently and
significantly outperforms previous competing approaches in both stability and
speed
Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies
Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are useed to argue convergence and establish rates of convergence. An example and numerical results are discussed
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
Linear Stochastic Models of Nonlinear Dynamical Systems
We investigate in this work the validity of linear stochastic models for
nonlinear dynamical systems. We exploit as our basic tool a previously proposed
Rayleigh-Ritz approximation for the effective action of nonlinear dynamical
systems started from random initial conditions. The present paper discusses
only the case where the PDF-Ansatz employed in the variational calculation is
``Markovian'', i.e. is determined completely by the present values of the
moment-averages. In this case we show that the Rayleigh-Ritz effective action
of the complete set of moment-functions that are employed in the closure has a
quadratic part which is always formally an Onsager-Machlup action. Thus,
subject to satisfaction of the requisite realizability conditions on the noise
covariance, a linear Langevin model will exist which reproduces exactly the
joint 2-time correlations of the moment-functions. We compare our method with
the closely related formalism of principal oscillation patterns (POP), which,
in the approach of C. Penland, is a method to derive such a linear Langevin
model empirically from time-series data for the moment-functions. The
predictive capability of the POP analysis, compared with the Rayleigh-Ritz
result, is limited to the regime of small fluctuations around the most probable
future pattern. Finally, we shall discuss a thermodynamics of statistical
moments which should hold for all dynamical systems with stable invariant
probability measures and which follows within the Rayleigh-Ritz formalism.Comment: 36 pages, 5 figures, seceq.sty for sequential numbering of equations
by sectio
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