662 research outputs found
T3NS: three-legged tree tensor network states
We present a new variational tree tensor network state (TTNS) ansatz, the
three-legged tree tensor network state (T3NS). Physical tensors are
interspersed with branching tensors. Physical tensors have one physical index
and at most two virtual indices, as in the matrix product state (MPS) ansatz of
the density matrix renormalization group (DMRG). Branching tensors have no
physical index, but up to three virtual indices. In this way, advantages of
DMRG, in particular a low computational cost and a simple implementation of
symmetries, are combined with advantages of TTNS, namely incorporating more
entanglement. Our code is capable of simulating quantum chemical Hamiltonians,
and we present several proof-of-principle calculations on LiF, N and the
bis(-oxo) and peroxo isomers of
.Comment: 14 pages, 8 figure
New Approaches for ab initio Calculations of Molecules with Strong Electron Correlation
Reliable quantum chemical methods for the description of molecules with
dense-lying frontier orbitals are needed in the context of many chemical
compounds and reactions. Here, we review developments that led to our
newcomputational toolbo x which implements the quantum chemical density matrix
renormalization group in a second-generation algorithm. We present an overview
of the different components of this toolbox.Comment: 19 pages, 1 tabl
Fluctuating Currents in Stochastic Thermodynamics II. Energy Conversion and Nonequilibrium Response in Kinesin Models
Unlike macroscopic engines, the molecular machinery of living cells is
strongly affected by fluctuations. Stochastic Thermodynamics uses Markovian
jump processes to model the random transitions between the chemical and
configurational states of these biological macromolecules. A recently developed
theoretical framework [Wachtel, Vollmer, Altaner: "Fluctuating Currents in
Stochastic Thermodynamics I. Gauge Invariance of Asymptotic Statistics"]
provides a simple algorithm for the determination of macroscopic currents and
correlation integrals of arbitrary fluctuating currents. Here, we use it to
discuss energy conversion and nonequilibrium response in different models for
the molecular motor kinesin. Methodologically, our results demonstrate the
effectiveness of the algorithm in dealing with parameter-dependent stochastic
models. For the concrete biophysical problem our results reveal two interesting
features in experimentally accessible parameter regions: The validity of a
non-equilibrium Green--Kubo relation at mechanical stalling as well as negative
differential mobility for superstalling forces.Comment: PACS numbers: 05.70.Ln, 05.40.-a, 87.10.Mn, 87.16.Nn. An accompanying
publication "Fluctuating Currents in Stochastic Thermodynamics I. Gauge
Invariance of Asymptotic Statistics" is available at
http://arxiv.org/abs/1407.206
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
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