29,741 research outputs found
Determination of the rank of an integration lattice
The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations.
In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ⊥. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B
Abelian and non-abelian symmetries in infinite projected entangled pair states
We explore in detail the implementation of arbitrary abelian and non-abelian
symmetries in the setting of infinite projected entangled pair states on the
two-dimensional square lattice. We observe a large computational speed-up;
easily allowing bond dimensions in the square lattice Heisenberg model
at computational effort comparable to calculations at without
symmetries. We also find that implementing an unbroken symmetry does not
negatively affect the representative power of the state and leads to identical
or improved ground-state energies. Finally, we point out how to use symmetry
implementations to detect spontaneous symmetry breaking.Comment: 18 pages, submitted to SciPost Physic
Crystal lattice properties fully determine short-range interaction parameters for alkali and halide ions
Accurate models of alkali and halide ions in aqueous solution are necessary
for computer simulations of a broad variety of systems. Previous efforts to
develop ion force fields have generally focused on reproducing experimental
measurements of aqueous solution properties such as hydration free energies and
ion-water distribution functions. This dependency limits transferability of the
resulting parameters because of the variety and known limitations of water
models. We present a solvent-independent approach to calibrating ion parameters
based exclusively on crystal lattice properties. Our procedure relies on
minimization of lattice sums to calculate lattice energies and interionic
distances instead of equilibrium ensemble simulations of dense fluids. The gain
in computational efficiency enables simultaneous optimization of all parameters
for Li+, Na+, K+, Rb+, Cs+, F-, Cl-, Br-, and I- subject to constraints that
enforce consistency with periodic table trends. We demonstrate the method by
presenting lattice-derived parameters for the primitive model and the
Lennard-Jones model with Lorentz-Berthelot mixing rules. The resulting
parameters successfully reproduce the lattice properties used to derive them
and are free from the influence of any water model. To assess the
transferability of the Lennard-Jones parameters to aqueous systems, we used
them to estimate hydration free energies and found that the results were in
quantitative agreement with experimentally measured values. These
lattice-derived parameters are applicable in simulations where coupling of ion
parameters to a particular solvent model is undesirable. The simplicity and low
computational demands of the calibration procedure make it suitable for
parametrization of crystallizable ions in a variety of force fields.Comment: 9 pages, 5 table
Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds
In this paper, we study the efficient numerical integration of functions with
sharp gradients and cusps. An adaptive integration algorithm is presented that
systematically improves the accuracy of the integration of a set of functions.
The algorithm is based on a divide and conquer strategy and is independent of
the location of the sharp gradient or cusp. The error analysis reveals that for
a function (derivative-discontinuity at a point), a rate of convergence
of is obtained in . Two applications of the adaptive integration
scheme are studied. First, we use the adaptive quadratures for the integration
of the regularized Heaviside function---a strongly localized function that is
used for modeling sharp gradients. Then, the adaptive quadratures are employed
in the enriched finite element solution of the all-electron Coulomb problem in
crystalline diamond. The source term and enrichment functions of this problem
have sharp gradients and cusps at the nuclei. We show that the optimal rate of
convergence is obtained with only a marginal increase in the number of
integration points with respect to the pure finite element solution with the
same number of elements. The adaptive integration scheme is simple, robust, and
directly applicable to any generalized finite element method employing
enrichments with sharp local variations or cusps in -dimensional
parallelepiped elements.Comment: 22 page
Local Decoders for the 2D and 4D Toric Code
We analyze the performance of decoders for the 2D and 4D toric code which are
local by construction. The 2D decoder is a cellular automaton decoder
formulated by Harrington which explicitly has a finite speed of communication
and computation. For a model of independent and errors and faulty
syndrome measurements with identical probability we report a threshold of
for this Harrington decoder. We implement a decoder for the 4D toric
code which is based on a decoder by Hastings arXiv:1312.2546 . Incorporating a
method for handling faulty syndromes we estimate a threshold of for
the same noise model as in the 2D case. We compare the performance of this
decoder with a decoder based on a 4D version of Toom's cellular automaton rule
as well as the decoding method suggested by Dennis et al.
arXiv:quant-ph/0110143 .Comment: 22 pages, 21 figures; fixed typos, updated Figures 6,7,8,
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