1,966 research outputs found
Time-Dependent Density Matrix Renormalization Group Algorithms for Nearly Exact Absorption and Fluorescence Spectra of Molecular Aggregates at Both Zero and Finite Temperature
We implement and apply time-dependent density matrix renormalization group
(TD-DMRG) algorithms at zero and finite temperature to compute the linear
absorption and fluorescence spectra of molecular aggregates. Our implementation
is within a matrix product state/operator framework with an explicit treatment
of the excitonic and vibrational degrees of freedom, and uses the locality of
the Hamiltonian in the zero-exciton space to improve the efficiency and
accuracy of the calculations. We demonstrate the power of the method by
calculations on several molecular aggregate models, comparing our results
against those from multi-layer multiconfiguration time- dependent Hartree and
n-particle approximations. We find that TD-DMRG provides an accurate and
efficient route to calculate the spectrum of molecular aggregates.Comment: 10 figure
Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study
Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly
being used for a variety of model analyses in areas such as partial
differential equations, nonautonomous differentiable dynamical systems, and
random dynamical systems. These vectors identify spatially varying directions
of specific asymptotic growth rates and obey equivariance principles. In recent
years new computational methods for approximating Oseledets vectors have been
developed, motivated by increasing model complexity and greater demands for
accuracy. In this numerical study we introduce two new approaches based on
singular value decomposition and exponential dichotomies and comparatively
review and improve two recent popular approaches of Ginelli et al. (2007) and
Wolfe and Samelson (2007). We compare the performance of the four approaches
via three case studies with very different dynamics in terms of symmetry,
spectral separation, and dimension. We also investigate which methods perform
well with limited data
Randomized Dynamic Mode Decomposition
This paper presents a randomized algorithm for computing the near-optimal
low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging
techniques to compute low-rank matrix approximations at a fraction of the cost
of deterministic algorithms, easing the computational challenges arising in the
area of `big data'. The idea is to derive a small matrix from the
high-dimensional data, which is then used to efficiently compute the dynamic
modes and eigenvalues. The algorithm is presented in a modular probabilistic
framework, and the approximation quality can be controlled via oversampling and
power iterations. The effectiveness of the resulting randomized DMD algorithm
is demonstrated on several benchmark examples of increasing complexity,
providing an accurate and efficient approach to extract spatiotemporal coherent
structures from big data in a framework that scales with the intrinsic rank of
the data, rather than the ambient measurement dimension. For this work we
assume that the dynamics of the problem under consideration is evolving on a
low-dimensional subspace that is well characterized by a fast decaying singular
value spectrum
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