39,590 research outputs found
Multi-Layer Potfit: An Accurate Potential Representation for Efficient High-Dimensional Quantum Dynamics
The multi-layer multi-configuration time-dependent Hartree method (ML-MCTDH)
is a highly efficient scheme for studying the dynamics of high-dimensional
quantum systems. Its use is greatly facilitated if the Hamiltonian of the
system possesses a particular structure through which the multi-dimensional
matrix elements can be computed efficiently. In the field of quantum molecular
dynamics, the effective interaction between the atoms is often described by
potential energy surfaces (PES), and it is necessary to fit such PES into the
desired structure. For high-dimensional systems, the current approaches for
this fitting process either lead to fits that are too large to be practical, or
their accuracy is difficult to predict and control.
This article introduces multi-layer Potfit (MLPF), a novel fitting scheme
that results in a PES representation in the hierarchical tensor (HT) format.
The scheme is based on the hierarchical singular value decomposition, which can
yield a near-optimal fit and give strict bounds for the obtained accuracy.
Here, a recursive scheme for using the HT-format PES within ML-MCTDH is
derived, and theoretical estimates as well as a computational example show that
the use of MLPF can reduce the numerical effort for ML-MCTDH by orders of
magnitude, compared to the traditionally used Potfit representation of the PES.
Moreover, it is shown that MLPF is especially beneficial for high-accuracy PES
representations, and it turns out that MLPF leads to computational savings
already for comparatively small systems with just four modes.Comment: Copyright (2014) American Institute of Physics. This article may be
downloaded for personal use only. Any other use requires prior permission of
the author and the American Institute of Physic
On the Spectrum of Superspheres
Sigma models on coset superspaces, such as odd dimensional superspheres, play
an important role in physics and in particular the AdS/CFT correspondence. In
this work we apply recent general results on the spectrum of coset space models
and on supergroup WZNW models to study the conformal sigma model with target
space S^{3|2}. We construct its vertex operators and provide explicit formulas
for their anomalous dimensions, at least to leading order in the sigma model
coupling. The results are used to revisit a non-perturbative duality between
the supersphere and the OSP(4|2) Gross-Neveu model that was conjectured by
Candu and Saleur. With the help of powerful all-loop results for 1/2 BPS
operators in the Gross-Neveu model we are able to recover the entire zero mode
spectrum of the sigma model at a certain finite value of the Gross-Neveu
coupling. In addition, we argue that the sigma model constraints and equations
of motion are implemented correctly in the dual Gross-Neveu description. On the
other hand, high(er) gradient operators of the sigma model are not all
accounted for. It is possible that this discrepancy is related to an
instability from high gradient operators that has previously been observed in
the context of Anderson localization
Total Variation Regularized Tensor RPCA for Background Subtraction from Compressive Measurements
Background subtraction has been a fundamental and widely studied task in
video analysis, with a wide range of applications in video surveillance,
teleconferencing and 3D modeling. Recently, motivated by compressive imaging,
background subtraction from compressive measurements (BSCM) is becoming an
active research task in video surveillance. In this paper, we propose a novel
tensor-based robust PCA (TenRPCA) approach for BSCM by decomposing video frames
into backgrounds with spatial-temporal correlations and foregrounds with
spatio-temporal continuity in a tensor framework. In this approach, we use 3D
total variation (TV) to enhance the spatio-temporal continuity of foregrounds,
and Tucker decomposition to model the spatio-temporal correlations of video
background. Based on this idea, we design a basic tensor RPCA model over the
video frames, dubbed as the holistic TenRPCA model (H-TenRPCA). To characterize
the correlations among the groups of similar 3D patches of video background, we
further design a patch-group-based tensor RPCA model (PG-TenRPCA) by joint
tensor Tucker decompositions of 3D patch groups for modeling the video
background. Efficient algorithms using alternating direction method of
multipliers (ADMM) are developed to solve the proposed models. Extensive
experiments on simulated and real-world videos demonstrate the superiority of
the proposed approaches over the existing state-of-the-art approaches.Comment: To appear in IEEE TI
Multiconfiguration time-dependent Hartree impurity solver for nonequilibrium dynamical mean-field theory
Nonequilibrium dynamical mean-field theory (DMFT) solves correlated lattice
models by obtaining their local correlation functions from an effective model
consisting of a single impurity in a self-consistently determined bath. The
recently developed mapping of this impurity problem from the Keldysh time
contour onto a time-dependent single-impurity Anderson model (SIAM) [C. Gramsch
et al., Phys. Rev. B 88, 235106 (2013)] allows one to use wave function-based
methods in the context of nonequilibrium DMFT. Within this mapping, long times
in the DMFT simulation become accessible by an increasing number of bath
orbitals, which requires efficient representations of the time-dependent SIAM
wave function. These can be achieved by the multiconfiguration time-dependent
Hartree (MCTDH) method and its multi-layer extensions. We find that MCTDH
outperforms exact diagonalization for large baths in which the latter approach
is still within reach and allows for the calculation of SIAMs beyond the system
size accessible by exact diagonalization. Moreover, we illustrate the
computation of the self-consistent two-time impurity Green's function within
the MCTDH second quantization representation.Comment: 12 pages, 8 figure
Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables
Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety
of engineering and scientific fields. Dynamic mode decomposition (DMD), which
is a numerical algorithm for the spectral analysis of Koopman operators, has
been attracting attention as a way of obtaining global modal descriptions of
NLDSs without requiring explicit prior knowledge. However, since existing DMD
algorithms are in principle formulated based on the concatenation of scalar
observables, it is not directly applicable to data with dependent structures
among observables, which take, for example, the form of a sequence of graphs.
In this paper, we formulate Koopman spectral analysis for NLDSs with structures
among observables and propose an estimation algorithm for this problem. This
method can extract and visualize the underlying low-dimensional global dynamics
of NLDSs with structures among observables from data, which can be useful in
understanding the underlying dynamics of such NLDSs. To this end, we first
formulate the problem of estimating spectra of the Koopman operator defined in
vector-valued reproducing kernel Hilbert spaces, and then develop an estimation
procedure for this problem by reformulating tensor-based DMD. As a special case
of our method, we propose the method named as Graph DMD, which is a numerical
algorithm for Koopman spectral analysis of graph dynamical systems, using a
sequence of adjacency matrices. We investigate the empirical performance of our
method by using synthetic and real-world data.Comment: 34 pages with 4 figures, Published in Neural Networks, 201
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