pinecone/core-2020-05-10-deduplication huggingface dataset:
An Analog Characterization of the Grzegorczyk Hierarchy

AbstractWe study a restricted version of Shannon's general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n−3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions
Kolmogorov Width of Classes of Smooth Functions on the Sphere Sd−1

AbstractLet Sd−1≔{(x1,…,xd)∈Rd:x21+···+x2d=1} be the unit sphere of the d-dimensional Euclidean space Rd. For r>0, we denote by Brp (1⩽p⩽∞) the class of functions f on Sd−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on Sd−1,λ=d−22 and Pλk(t) is the ultraspherical polynomial.For 1⩽p,q⩽∞, the Kolmogorov N-width of Brp in Lq(Sd−1) is given bythe left-most infimum being taken over all N-dimensional subspaces XN of Lq(Sd−1).The main result in this paper is that for r⩾2(d−1)2,where AN≍BN means that there exists a positive constant C, independent of N, such that C−1AN⩽BN⩽CAN.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions
Decoupling Approximations Applied to an Infinite Array of Fluid Loaded Baffled Membranes

The article of record as published may be found at http://dx.doi.org/10.1006/jcph.1994.1064Application of a time dependent. nonlocal radiation boundary condition. used in conjunction
with the finite difference technique is applied to the acoustic problem of scattering
from an infinite array of baffled. fluid loaded membranes. The new boundary operator is
compared with the second order Engquist and Majda boundary operator, and with several
fluid/structure decoupling approximation techniques in the determination of scattering
amplitudes
Time-Symmetric ADI and Causal Reconnection: Stable Numerical Techniques
  for Hyperbolic Systems on Moving Grids

Moving grids are of interest in the numerical solution of hydrodynamical
problems and in numerical relativity. We show that conventional integration
methods for the simple wave equation in one and more than one dimension exhibit
a number of instabilities on moving grids. We introduce two techniques, which
we call causal reconnection and time-symmetric ADI, which together allow
integration of the wave equation with absolute local stability in any number of
dimensions on grids that may move very much faster than the wave speed and that
can even accelerate. These methods allow very long time-steps, are fully
second-order accurate, and offer the computational efficiency of
operator-splitting.Comment: 45 pages, 19 figures. Published in 1994 but not previously available
  in the electronic archive
Simulation of Cavity Flow by the Lattice Boltzmann Method

A detailed analysis is presented to demonstrate the capabilities of the
lattice Boltzmann method. Thorough comparisons with other numerical solutions
for the two-dimensional, driven cavity flow show that the lattice Boltzmann
method gives accurate results over a wide range of Reynolds numbers. Studies of
errors and convergence rates are carried out. Compressibility effects are
quantified for different maximum velocities, and parameter ranges are found for
stable simulations. The paper's objective is to stimulate further work using
this relatively new approach for applied engineering problems in transport
phenomena utilizing parallel computers.Comment: Submitted to J. Comput. Physics, late
Momentum Lattice Simulation on a Small Lattice Using Stochastic
  Quantization

We have studied the scalar $\phi^4$-model in the symmetric phase and the
non--compact $U(1)$ gauge theory on a momentum lattice using the Langevin
equation for generating configurations. In the $\phi^4$-model we have analyzed
the renormalized mass and in the $U(1)$-model we have analyzed the Wilson loop
operator. We used a second order algorithm for solving the Langevin equation,
and we looked for the convergence rate of the method. We studied the stochastic
time needed to generate equilibrium configurations and compared first and
second order schemes for both models.Comment: LaTeX, 6 pages plus 3 figures included, Laval preprint
  LAVAL-PHY-31-9
Useful Bases for Problems in Nuclear and Particle Physics

A set of exactly computable orthonormal basis functions that are useful in
computations involving constituent quarks is presented. These basis functions
are distinguished by the property that they fall off algebraically in momentum
space and can be exactly Fourier-Bessel transformed. The configuration space
functions are associated Laguerre polynomials multiplied by an exponential
weight, and their Fourier-Bessel transforms can be expressed in terms of Jacobi
polynomials in $\Lambda^2/(k^2 + \Lambda^2)$. A simple model of a meson
containing a confined quark-antiquark pair shows that this basis is much better
at describing the high-momentum properties of the wave function than the
harmonic-oscillator basis.Comment: 12 pages LaTeX/revtex, plus 2 postscript figure
Numerically implemented perturbation method for the nonlinear magnetic
  moment of an anisotropic superconductor

We present a method to compute the magnetic moment of a bulk, finite-size,
three-dimensional, anisotropic superconductor. Our numerically implemented
perturbative procedure is based on a solution of the nonlinear Maxwell- London
equations, where we include the nonlinear relation between current and gauge
invariant velocity. The method exploits the small ratio of penetration depth to
sample size. We show how to treat the open boundary conditions over an infinite
domain and the continuity requirement at the interface. We demonstrate how our
method substantially reduces the computational work required and discuss its
implementation to an oblate spheroid. The numerical solution is obtained from a
finite difference method. We briefly discuss the relevance of this work to
similar problems in other fields.Comment: 43 pages RevTex ms and four postscript figures. To appear in Journal
  of Computational Physic
Common Molecular Dynamics Algorithms Revisited: Accuracy and Optimal
  Time Steps of Stoermer-Leapfrog Integrators

The Stoermer-Verlet-leapfrog group of integrators commonly used in molecular
dynamics simulations has long become a textbook subject and seems to have been
studied exhaustively. There are, however, a few striking effects in performance
of algorithms which are well-known but have not received adequate attention in
the literature. A closer view of these unclear observations results in
unexpected conclusions. It is shown here that contrary to the conventional
point of view, the leapfrog scheme is distinguished in this group both in terms
of the order of truncation errors and the conservation of the total energy. In
this case the characteristic square growth of fluctuations of the total energy
with the step size, commonly measured in numerical tests, results from
additional interpolation errors with no relation to the accuracy of the
computed trajectory. An alternative procedure is described for checking energy
conservation of leapfrog-like algorithms which is free from interpolation
errors. Preliminary tests on a representative model system suggest that
standard step size values used at present are lower than necessary for accurate
sampling.Comment: 11 pages, 4 figures, RevTeX/LaTeX. J. Comput. Phys., in pres
Lagrange-Galerkin methods on spherical geodesic grids

The article of record as published may be located at http://dx.doi.org/10.1006/jcph.1997.5771Lagrange-Galerkin finite element methods that are high-order accurate, exactly integrable, and highly efficient are presented. This paper derives generalized natural Cartesian coordinates in three dimensions for linear triangles on the surface of the sphere. By using these natural coordinates as the finite element basis functions we can integrate the corresponding integrals exactly thereby achieving a high level of accuracy and efficiency for modeling physical problems on the sphere. The discretization of the sphere is achieved by the use of a spherical geodesic triangular grid. A tree data structure that is inherent to this grid is introduced; this tree data structure exploits the property of the spherical geodesic grid, allowing for rapid searching of departure points which is essential to the Lagrange-Galerkin method. The generalized natural coordinates are also used for determining in which element the departure points lie. A comparison of the Lagrange-Galerkin method with an Euler-Galerkin method demonstrates the impressive lever of high order accuracy achieved by the Lagrange-Galerkin method at computational costs comparable or better than the Euler-Galerkin method. In addition, examples using advancing front unstructured grids illustrate the flexibility of the Lagrange-Galerkin method on different grid types, By introducing generalized natural coordinates and the tree data structure for the spherical geodesic grid, the Lagrange-Galerkin method can be used for solving practical problems on the sphere more accurately than current methods, yet requiring less computer time. (C) 1997 Academic Press
The 3-dimensional Fourier grid Hamiltonian method

A method to compute the bound state eigenvalues and eigenfunctions of a
Schr\"{o}dinger equation or a spinless Salpeter equation with central
interaction is presented. This method is the generalization to the
three-dimensional case of the Fourier grid Hamiltonian method for
one-dimensional Schr\"{o}dinger equation. It requires only the evaluation of
the potential at equally spaced grid points and yields the radial part of the
eigenfunctions at the same grid points. It can be easily extended to the case
of coupled channel equations and to the case of non-local interactions.Comment: 13 pages, 1 figure. RevTeX file. To appear in J. Comput. Phy
Finite-temperature evaluation of the Fermi density operator

A rational expansion of the Fermi density operator is proposed. This approach
allows to calculate efficiently physical properties of fermionic systems at
finite temperatures without solving an eigenvalue problem. Using N evaluations
of the Green's function, the Fermi density operator can be approximated,
subject to a given precision, in the energy interval from -A to infinity with A
proportional to N. The presented method may become especially useful for
electronic structure calculations involving the calculation of charge
densities.Comment: 6 pages, 4 Postscript figures, submitted to J. Comp. Phy
Multiscale Computation with Interpolating Wavelets

Multiresolution analyses based upon interpolets, interpolating scaling
functions introduced by Deslauriers and Dubuc, are particularly well-suited to
physical applications because they allow exact recovery of the multiresolution
representation of a function from its sample values on a finite set of points
in space. We present a detailed study of the application of wavelet concepts to
physical problems expressed in such bases. The manuscript describes algorithms
for the associated transforms which, for properly constructed grids of variable
resolution, compute correctly without having to introduce extra grid points. We
demonstrate that for the application of local homogeneous operators in such
bases, the non-standard multiply of Beylkin, Coifman and Rokhlin also proceeds
exactly for inhomogeneous grids of appropriate form. To obtain less stringent
conditions on the grids, we generalize the non-standard multiply so that
communication may proceed between non-adjacent levels. The manuscript concludes
with timing comparisons against naive algorithms and an illustration of the
scale-independence of the convergence rate of the conjugate gradient solution
of Poisson's equation using a simple preconditioning, suggesting that this
approach leads to an O(n) solution of this equation.Comment: 33 pages, figures available at
  http://laisla.mit.edu/muchomas/Papers/nonstand-figs.ps . Updated: (1) figures
  file (figs.ps) now appear with the posting on the server; (2) references got
  lost in the last submissio
A comparison between Fast Multipole Algorithm and Tree-Code to evaluate gravitational forces in 3-D

We present tests of comparison between our versions of the Fast Multipole Algorithm (FMA) and ``classic'' tree-code to evaluate gravitational forces in particle systems. We have optimized the Greengard's original version of FMA allowing for a more efficient criterion of well-separation between boxes, to improve the adaptivity of the method (which is very important in highly inhomogeneous situations) and to permit the smoothing of gravitational interactions. The results of our tests indicate that the tree-code is almost three times faster than FMA for both a homogeneous and a clumped distribution, at least in the interval of N (N< 10^5) here investigated and at the same level of accuracy (error ~ 10^{-3)). This order of accuracy is generally considered as the best compromise between CPU-time consumption and precision for astrophysical simulation. Moreover, the claimed linear dependence on N of the CPU-time of FMA is not confirmed and we give a ``theoretical'' explanation for that
The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids

The article of record as published may be located at http://dx.doi.org/10.1006/jcph.1998.6078The purpose of this paper is to introduce a new method formed by fusing the Lagrange-Galerkin and spectral element methods. The Lagrange-Galerkin method traces the characteristic curves of the solution and, consequently, is very well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations. Spectral element methods are essentially higher order finite element methods that exhibit spectral (exponential) convergence, provided that the solution is a smooth function. By combining these two methods, a numerical scheme can be constructed that resolves, with extremely high precision, the nonlinearities of the advection terms and the smooth regions of the flow generated by the diffusion terms. This paper describes the construction of the Lagrange-Galerkin spectral element method which permits the use of any grid type including unstructured grids. The only restriction at the moment is that the grid elements be quadrilaterals. The stability analysis of both methods demonstrates why these two methods are so powerful individually and how their fusion leads to an improved scheme. The Lagrange-Galerkin spectral element method is validated using the 1D and 2D advection and advection-diffusion equations. The results of the stability analysis and the numerical experiments demonstrate the utility of such an approach. (C) 1998 Academic Press
Adaptive Mesh Refinement Computation of Solidification Microstructures
  using Dynamic Data Structures

We study the evolution of solidification microstructures using a phase-field
model computed on an adaptive, finite element grid. We discuss the details of
our algorithm and show that it greatly reduces the computational cost of
solving the phase-field model at low undercooling. In particular we show that
the computational complexity of solving any phase-boundary problem scales with
the interface arclength when using an adapting mesh. Moreover, the use of
dynamic data structures allows us to simulate system sizes corresponding to
experimental conditions, which would otherwise require lattices greater that
$2^{17}\times 2^{17}$ elements. We examine the convergence properties of our
algorithm. We also present two dimensional, time-dependent calculations of
dendritic evolution, with and without surface tension anisotropy. We benchmark
our results for dendritic growth with microscopic solvability theory, finding
them to be in good agreement with theory for high undercoolings. At low
undercooling, however, we obtain higher values of velocity than solvability
theory at low undercooling, where transients dominate, in accord with a
heuristic criterion which we derive
A dissipative algorithm for wave-like equations in the characteristic
  formulation

We present a dissipative algorithm for solving nonlinear wave-like equations
when the initial data is specified on characteristic surfaces. The dissipative
properties built in this algorithm make it particularly useful when studying
the highly nonlinear regime where previous methods have failed to give a stable
evolution in three dimensions. The algorithm presented in this work is directly
applicable to hyperbolic systems proper of Electromagnetism, Yang-Mills and
General Relativity theories. We carry out an analysis of the stability of the
algorithm and test its properties with linear waves propagating on a Minkowski
background and the scattering off a Scwharszchild black hole in General
Relativity.Comment: 23 pages, 7 figure
Multiple Time Scales in Classical and Quantum–Classical Molecular Dynamics

<p>The existence of multiple time scales in molecular dynamics poses interesting and challenging questions from an analytical as well as a numerical point of view. In this paper, we consider simplified models with two essential time scales and describe how these two time scales interact. The discussion focuses on classical molecular dynamics (CMD) with fast bond stretching and bending modes and the so-called quantum–classical molecular dynamics (QCMD) model where the quantum part provides the highly oscillatory solution components. The analytic results on the averaging over fast degrees of motion will also shed new light on the appropriate implementation of multiple-time-stepping algorithms for CMD and QCMD. </p
Tests of Spurious Transport in Smoothed Particle Hydrodynamics

We have performed a series of systematic tests to evaluate the effects of
spurious transport in three-dimensional smoothed particle hydrodynamics (SPH)
calculations. Our tests investigate (i) particle diffusion, (ii) shock heating,
(iii) numerical viscosity, and (iv) angular momentum transport. The results are
useful for quantifying the accuracy of the SPH scheme, especially for problems
where shear flows or shocks are present, as well as for problems where true
hydrodynamic mixing is relevant.
  We examine the different forms of artificial viscosity (AV) which have been
proposed by Monaghan, by Hernquist & Katz, and by Balsara. For each form, our
tests suggest a single set of values for the AV parameters $\alpha$ and $\beta$
(coefficients of the linear and quadratic terms) which are appropriate in a
large number of situations. We also discuss how these parameters should be
adjusted depending on the goals of the particular application. We find that
both the Hernquist & Katz and Balsara forms introduce relatively small amounts
of numerical viscosity. Furthermore, both Monaghan's and Balsara's AV do well
at treating shocks and at limiting the amount of spurious mixing. For these
reasons, we endorse the Balsara AV for use in a broad range of applications.Comment: 49 pages, 26 figures as 30 postscript files, submitted to The Journal
  of Computational Physic
Parallelization of a Dynamic Monte Carlo Algorithm: a Partially
  Rejection-Free Conservative Approach

We experiment with a massively parallel implementation of an algorithm for
simulating the dynamics of metastable decay in kinetic Ising models. The
parallel scheme is directly applicable to a wide range of stochastic cellular
automata where the discrete events (updates) are Poisson arrivals. For high
performance, we utilize a continuous-time, asynchronous parallel version of the
n-fold way rejection-free algorithm. Each processing element carries an lxl
block of spins, and we employ the fast SHMEM-library routines on the Cray T3E
distributed-memory parallel architecture. Different processing elements have
different local simulated times. To ensure causality, the algorithm handles the
asynchrony in a conservative fashion. Despite relatively low utilization and an
intricate relationship between the average time increment and the size of the
spin blocks, we find that for sufficiently large l the algorithm outperforms
its corresponding parallel Metropolis (non-rejection-free) counterpart. As an
example application, we present results for metastable decay in a model
ferromagnetic or ferroelectric film, observed with a probe of area smaller than
the total system.Comment: 17 pages, 7 figures, RevTex; submitted to the Journal of
  Computational Physic
A new discrete velocity method for Navier-Stokes equations

The relation between Latttice Boltzmann Method, which has recently become
popular, and the Kinetic Schemes, which are routinely used in Computational
Fluid Dynamics, is explored. A new discrete velocity model for the numerical
solution of the Navier-Stokes equations for incompressible fluid flow is
presented by combining both the approaches. The new scheme can be interpreted
as a pseudo-compressibility method and, for a particular choice of parameters,
this interpretation carries over to the Lattice Boltzmann Method.Comment: 28 pages, 8 figure
Positivity of flux vector splitting schemes

Over the last ten years, robustness of schemes has raised an increasing interest among the CFD community. One mathematical aspect of scheme robustness is the positivity preserving property. At high Mach numbers, solving the conservative Euler equations can lead to negative densities or internal energy. Some schemes such as the flux vector splitting (FVS) schemes are known to avoid this drawback. In this study, a general method is detailed to analyze the positivity of FVS schemes. As an application, three classical FVS schemes (Van Leer's, Hänel's variant, and Steger and Warming's) are proved to be positively conservative under a CFL-like condition. Finally, it is proved that for any FVS scheme, there is an intrinsic incompatibility between the desirable property of positivity and the exact resolution of contact discontinuities
A New Technique for Sampling Multi-Modal Distributions

In this paper we demonstrate that multi-modal Probability Distribution
Functions (PDFs) may be efficiently sampled using an algorithm originally
developed for numerical integrations by Monte-Carlo methods. This algorithm can
be used to generate an input PDF which can be used as an independence sampler
in a Metropolis-Hastings chain to sample otherwise troublesome
distributions.Some examples in one two and five dimensions are worked out.Comment: One ps figure; submitted to "Journal of Computational Physics