1,380 research outputs found
Frozen Gaussian approximation for general linear strictly hyperbolic system: formulation and Eulerian methods
The frozen Gaussian approximation, proposed in [Lu and Yang, [15]], is an
efficient computational tool for high frequency wave propagation. We continue
in this paper the development of frozen Gaussian approximation. The frozen
Gaussian approximation is extended to general linear strictly hyperbolic
systems. Eulerian methods based on frozen Gaussian approximation are developed
to overcome the divergence problem of Lagrangian methods. The proposed Eulerian
methods can also be used for the Herman-Kluk propagator in quantum mechanics.
Numerical examples verify the performance of the proposed methods
Fast algorithm for periodic density fitting for Bloch waves
We propose an efficient algorithm for density fitting of Bloch waves for
Hamiltonian operators with periodic potential. The algorithm is based on column
selection and random Fourier projection of the orbital functions. The
computational cost of the algorithm scales as , where is number of
spatial grid points, is the number of sampling -points in first
Brillouin zone, and is the number of bands under consideration. We validate
the algorithm by numerical examples in both two and three dimensions
Detecting localized eigenstates of linear operators
We describe a way of detecting the location of localized eigenvectors of a
linear system for eigenvalues with
comparatively large. We define the family of functions where is a parameter and
is the th standard basis vector. We
prove that eigenvectors associated to eigenvalues with large absolute value
localize around local maxima of : the metastable states in the
power iteration method (slowing down its convergence) can be used to predict
localization. We present a fast randomized algorithm and discuss different
examples: a random band matrix, discretizations of the local operator and the nonlocal operator
Bold Diagrammatic Monte Carlo in the Lens of Stochastic Iterative Methods
This work aims at understanding of bold diagrammatic Monte Carlo (BDMC)
methods for stochastic summation of Feynman diagrams from the angle of
stochastic iterative methods. The convergence enhancement trick of the BDMC is
investigated from the analysis of condition number and convergence of the
stochastic iterative methods. Numerical experiments are carried out for model
systems to compare the BDMC with related stochastic iterative approaches
Improved sampling and validation of frozen Gaussian approximation with surface hopping algorithm for nonadiabatic dynamics
In the spirit of the fewest switches surface hopping, the frozen Gaussian
approximation with surface hopping (FGA-SH) method samples a path integral
representation of the non-adiabatic dynamics in the semiclassical regime. An
improved sampling scheme is developed in this work for FGA-SH based on birth
and death branching processes. The algorithm is validated for the standard test
examples of non-adiabatic dynamics.Comment: 14 pages, 9 figure
Orbital minimization method with regularization
We consider a modification of the OMM energy functional which contains an
penalty term in order to find a sparse representation of the low-lying
eigenspace of self-adjoint operators. We analyze the local minima of the
modified functional as well as the convergence of the modified functional to
the original functional. Algorithms combining soft thresholding with gradient
descent are proposed for minimizing this new functional. Numerical tests
validate our approach. As an added bonus, we also prove the unanticipated and
remarkable property that every local minimum the OMM functional without the
term is also a global minimum.Comment: 22 pages, 6 figure
Phase Space Sketching for Crystal Image Analysis based on Synchrosqueezed Transforms
Recent developments of imaging techniques enable researchers to visualize
materials at the atomic resolution to better understand the microscopic
structures of materials. This paper aims at automatic and quantitative
characterization of potentially complicated microscopic crystal images,
providing feedback to tweak theories and improve synthesis in materials
science. As such, an efficient phase-space sketching method is proposed to
encode microscopic crystal images in a translation, rotation, illumination, and
scale invariant representation, which is also stable with respect to small
deformations. Based on the phase-space sketching, we generalize our previous
analysis framework for crystal images with simple structures to those with
complicated geometry
Continuum limit and preconditioned Langevin sampling of the path integral molecular dynamics
We investigate the continuum limit that the number of beads goes to infinity
in the ring polymer representation of thermal averages. Studying the continuum
limit of the trajectory sampling equation sheds light on possible
preconditioning techniques for sampling ring polymer configurations with large
number of beads. We propose two preconditioned Langevin sampling dynamics,
which are shown to have improved stability and sampling accuracy. We present a
careful mode analysis of the preconditioned dynamics and show their connections
to the normal mode, the staging coordinate and the Matsubara mode
representation for ring polymers. In the case where the potential is quadratic,
we show that the continuum limit of the preconditioned mass modified Langevin
dynamics converges to its equilibrium exponentially fast, which suggests that
the finite-dimensional counterpart has a dimension-independent convergence
rate. In addition, the preconditioning techniques can be naturally applied to
the multi-level quantum systems in the nonadiabatic regime, which are
compatible with various numerical approaches
Cauchy-Born rule and spin density wave for the spin-polarized Thomas-Fermi-Dirac-von Weizsacker model
The electronic structure (electron charges and spins) of a perfect crystal
under external magnetic field is analyzed using the spin-polarized
Thomas-Fermi-Dirac-von Weizsacker model. An extension of the classical
Cauchy-Born rule for crystal lattices is established for the electronic
structure under sharp stability conditions on charge density wave and spin
density wave. A Landau-Lifschitz type micromagnetic energy functional is
derived.Comment: 24 pages; dated June 17, 201
Analysis of multiscale integrators for multiple attractors and irreversible Langevin samplers
We study multiscale integrator numerical schemes for a class of stiff
stochastic differential equations (SDEs). We consider multiscale SDEs with
potentially multiple attractors that behave as diffusions on graphs as the
stiffness parameter goes to its limit. Classical numerical discretization
schemes, such as the Euler-Maruyama scheme, become unstable as the stiffness
parameter converges to its limit and appropriate multiscale integrators can
correct for this. We rigorously establish the convergence of the numerical
method to the related diffusion on graph, identifying the appropriate choice of
discretization parameters. Theoretical results are supplemented by numerical
studies on the problem of the recently developing area of introducing
irreversibility in Langevin samplers in order to accelerate convergence to
equilibrium
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