Power-law Convergence Research Articles

Horocycle averages on closed manifolds and transfer operators

We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$ discrete-time hyperbolic dynamics ($r>1$), obtaining anisotropic Banach or Hilbert spaces on which the resolvent of the generator of weighted transfer operators for the flow is quasi-compact. We apply this to study the ergodic integrals of the horocycle flows $h_\rho$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov flow with orientable strong-stable distribution, we establish power-law convergence of the ergodic average. We thereby implement the program of Giulietti-Liverani in the "real-life setting" of geodesic flows in variable negative curvature, where nontrivial resonances exist.

Quantum and classical annealing in a continuous space with multiple local minima

The protocol of quantum annealing is applied to an optimization problem with a one-dimensional continuous degree of freedom, a variant of the problem proposed by Shinomoto and Kabashima. The energy landscape has a number of local minima, and the classical approach of simulated annealing is predicted to have a logarithmically slow convergence to the global minimum. We show by extensive numerical analyses that quantum annealing yields a power-law convergence, thus an exponential improvement over simulated annealing. The power is larger, and thus the convergence is faster, than a prediction by an existing phenomenological theory for this problem. Performance of simulated annealing is shown to be enhanced by introducing quasiglobal searches across energy barriers, leading to a power-law convergence but with a smaller power than in the quantum case and thus a slower convergence classically even with quasiglobal search processes. We also reveal how diabatic quantum dynamics, quantum tunneling in particular, steers the systems toward the global minimum by a meticulous choice of annealing schedule. This latter result explicitly contrasts the role of tunneling in quantum annealing against the classical counterpart of stochastic optimization by simulated annealing.

On Accompanying Measures and Asymptotic Expansions in the B. V. Gnedenko Limit Theorem

We propose a sequence of accompanying laws in the B.,V. Gnedenko limit theorem for maxima of independent random variables with distributions lying in the Gumbel max domain of attraction. We show that this sequence provides a power-law convergence rate, whereas the Gumbel distribution provides only the logarithmic rate. As examples, we consider in detail the classes of Weibull and log-Weibull type distributions. For the entire Gumbel max domain of attraction, we propose a scale of classes of distributions that includes these two classes as a starting point.

The infinite convergence order of near minimal cubature formulas on classes of periodic functions

ABSTRACT The paper aims to estimate the error of a cubature formula acting on an arbitrary function from the Sobolev space on a multidimensional cube. The norm of the error in the dual space of the Sobolev class is represented as a positive definite quadratic form in the weights of the cubature formula. Given a finite smoothness s of the Sobolev space, we establish a power-law convergence order to zero of error functionals norms for near minimal cubature formulas and cubature formulas of high trigonometric degree. If the smoothness s of the Sobolev space tends to infinity then the power-law convergence order tends to infinity as well.

Optimal Truncations for Multivariate Fourier and Chebyshev Series: Mysteries of the Hyperbolic Cross: Part I: Bivariate Case

The key to most successful applications of Chebyshev and Fourier spectral methods in high space dimension are a combination of a Smolyak sparse grid together with so-called “hyperbolic cross” truncation. It is easy to find counterexamples for which the hyperbolic cross truncation is far from optimal. An important question is: what characteristics of a function make it “crossy”, that is, suitable for the hyperbolic crosss truncation? We have not been able to find a complete answer to this question. However, by combining low-rank SVD approximation, Poisson summation and imbricate series, hyperbolic coordinates and numerical experimentation, we are, to borrow from Fermi, “confused at a higher level”. For rank-one (separable) functions, which are the product of two univaraiate functions, we show that the hyperbolic cross truncation is indeed the best if the functions have weak singularities on the domain or boundaries so that the spectral series has a finite order of power-law convergence. For functions smooth on the domain, and therefore blessed with exponentially convergent spectral series, we have failed to find any reasonable examples where the hyperbolic cross truncation is best.

Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases

We present a systematic computation of the heat conductivity of the Markov jump process modeling the energy exchanges in an array of locally confined hard spheres at the conduction threshold. Based on a variational formula (Sasada 2016 (arXiv:1611.08866)), explicit upper bounds on the conductivity are derived, which exhibit a rapid power-law convergence towards an asymptotic value. We thereby conclude that the ratio of the heat conductivity to the energy exchange frequency deviates from its static contribution by a small negative correction, its dynamic contribution, evaluated to be in dimensionless units. This prediction is corroborated by kinetic Monte Carlo simulations which were substantially improved compared to earlier results.

History effects in the sedimentation of light aerosols in turbulence: The case of marine snow

We analyze the effect of the Basset history force on the sedimentation of nearly neutrally buoyant particles, exemplified by marine snow, in a three-dimensional turbulent flow. Particles are characterized by Stokes numbers much smaller than unity, and still water settling velocities, measured in units of the Kolmogorov velocity, of order one. The presence of the history force in the Maxey-Riley equation leads to individual trajectories which differ strongly from the dynamics of both inertial particles without this force, and ideal settling tracers. When considering, however, a large ensemble of particles, the statistical properties of all three dynamics become more similar. The main effect of the history force is a rather slow, power-law type convergence to an asymptotic settling velocity of the center of mass, which is found numerically to be the settling velocity in still fluid. The spatial extension of the ensemble grows diffusively after an initial ballistic growth lasting up to ca. one large eddy turnover time. We demonstrate that the settling of the center of mass for such light aggregates is best approximated by the settling dynamics in still fluid found with the history force, on top of which fluctuations appear which follow very closely those of the turbulent velocity field.

Fourier phase retrieval with a single mask by Douglas-Rachford algorithms.

The Fourier-domain Douglas-Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns without oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization.

Random sequential adsorption on imprecise lattice.

We report a surprising result, established by numerical simulations and analytical arguments for a one-dimensional lattice model of random sequential adsorption, that even an arbitrarily small imprecision in the lattice-site localization changes the convergence to jamming from fast, exponential, to slow, power-law, with, for some parameter values, a discontinuous jump in the jamming coverage value. This finding has implications for irreversible deposition on patterned substrates with pre-made landing sites for particle attachment. We also consider a general problem of the particle (depositing object) size not an exact multiple of the lattice spacing, and the lattice sites themselves imprecise, broadened into allowed-deposition intervals. Regions of exponential vs. power-law convergence to jamming are identified, and certain conclusions regarding the jamming coverage are argued for analytically and confirmed numerically.

General theory of asymmetric steric interactions in electrostatic double layers.

We study the mean-field Poisson-Boltzmann equation in the context of dense ionic liquids where steric effects become important. We generalise lattice gas theory by introducing a Flory-Huggins entropy for ions of differing volumes and then compare the effective free energy density to other existing lattice gas approximations, not based on the Flory-Huggins Ansatz. Within the methodology presented we also invoke more realistic equations of state, such as the Carnahan-Starling approximation, that are not based on the lattice gas approximation and lead to thermodynamic functions and properties that differ strongly from the lattice gas case. We solve the Carnahan-Starling model in the high density limit, and demonstrate a slow, power-law convergence at high potentials. We elucidate how equivalent convex free energy functions can be constructed that describe steric effects in a manner which is more convenient for numerical minimisation.

Convergence rates for loop-erased random walk and other Loewner curves

We estimate convergence rates for curves generated by Loewner's differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a geometric measure of regularity for Loewner curves parameterized by capacity. It is analogous to Warschawski's boundary structure modulus and closely related to annuli crossings. The main application we have in mind is that of a random discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the lattice size scaling limit. We carry out the approach in the case of loop-erased random walk (LERW) in a simply connected domain. Under mild assumptions of boundary regularity, we obtain an explicit power-law rate for the convergence of the LERW path toward the radial SLE$_2$ path in the supremum norm, the curves being parameterized by capacity. On the deterministic side, we show that the tip structure modulus gives a sufficient geometric condition for a Loewner curve to be H\"{o}lder continuous in the capacity parameterization, assuming its driving term is H\"{o}lder continuous. We also briefly discuss the case when the curves are a priori known to be H\"{o}lder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.

Caputo standard α-family of maps: fractional difference vs. fractional.

In this paper, the author compares behaviors of systems which can be described by fractional differential and fractional difference equations using the fractional and fractional difference Caputo standard α-Families of maps as examples. The author shows that properties of fractional difference maps (systems with falling factorial-law memory) are similar to the properties of fractional maps (systems with power-law memory). The similarities (types of attractors, power-law convergence of trajectories, existence of cascade of bifurcations and intermittent cascade of bifurcations type trajectories, and dependence of properties on the memory parameter α) and differences in properties of falling factorial- and power-law memory maps are investigated.

Comparison of Fault Representation Methods in Finite Difference Simulations of Dynamic Rupture

Assessing accuracy of numerical methods for spontaneous rupture simulation is challenging because we lack analytical solutions for reference. Previous comparison of a boundary integral method (bi) and finite-difference method (called dfm) that explicitly incorporates the fault discontinuity at velocity nodes (traction- at-split-node scheme) shows that both converge to a common, grid-independent solution and exhibit nearly identical power-law convergence rates with respect to grid spacing Δ x . We use this solution as a reference for assessing two other proposed finite-difference methods, the thick fault (tf) and stress glut (sg) methods, both of which approximate the fault-jump conditions through inelastic increments to the stress components (inelastic-zone schemes). The tf solution fails to match the qualitative rupture behavior of the reference solution and has quantitative misfits in root- mean-square rupture time of ∼30% for the smallest computationally feasible Δ x (with ∼9 grid-point resolution of cohesive zone, denoted N c = 9). For sufficiently small values of Δ x , the sg method reproduces the qualitative features of the reference solution, but rupture velocity remains systematically low for sg relative to the reference solution, and sg lacks the well-defined power-law convergence seen for bi and dfm. The rupture-time error for sg, with N c ∼ 9, remains well above uncertainty in the reference solution, and the split-node method attains comparable accuracy with N c 1/4 as large (and computation timescales as ( N c ) 4 ). Thus, accuracy is highly sensitive to the formulation of the fault-jump conditions: The split-node method attains power-law convergence. The sg inelastic-zone method achieves solutions that are qualitatively meaningful and quantitatively reliable to within a few percent, but convergence is uncertain, and sg is computationally inefficient relative to the split-node approach. The tf inelastic-zone method does not achieve qualitatively meaningful solutions to the 3D test problem and is sufficiently computationally inefficient that it is not feasible to explore convergence quantitatively.

Longitudinal polarizability of long polymeric chains: Quasi-one-dimensional electrostatics as the origin of slow convergence

The longitudinal linear polarizability alpha(N) of a stereoregular oligomer of size N is proportional to N in the large-N limit, provided the system is nonconducting in that limit. It has long been known that the convergence of alpha(N)/N to the asymptotic alpha(infinity) value is slow. We show that the leading term in the difference between alpha(N)/N and alpha(infinity) is of the order of 1/N. The difference [alpha(N)-alpha(N-1)], as well as alpha(center)(N) (when computationally accessible), also converge to alpha(infinity), but faster, the leading term being of the order of 1/N(2). We also present evidence that in these cases the power law convergence behavior is due to quasi-one-dimensional electrostatics, with one exception. Specifically, in molecular systems the difference between alpha(N)/N and alpha(infinity) has not just one but two sources of the O(1/N) term, with one being due to the aforementioned Coulomb interactions, and the second due to the short ranged exponentially decaying perturbations on chain ends. The major role of electrostatics in the convergence of the remainders is demonstrated by means of a Clausius-Mossotti-type classical model. The conclusions derived from the model are also shown to be applicable in molecular systems, by means of test-case ab initio calculations on linear stacks of H(2) molecules, and on polyacetylene chains. The implications of the modern theory of polarization for extended systems are also discussed.

Pattern forming pulled fronts: bounds and universal convergence

We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading speed v * . We discuss a method that allows to derive bounds on the front velocity, and which, hence, can be used to prove for, among others, the Swift–Hohenberg equation, the extended Fisher–Kolmogorov equation and the cubic complex Ginzburg–Landau equation, that the dynamically relevant fronts are of the pulled type. In addition, we generalize the derivation of the universal power law convergence of the dynamics of uniformly translating pulled fronts to both coherent and incoherent pattern forming fronts. The analysis is based on a matching analysis of the dynamics in the leading edge of the front, to the behavior imposed by the nonlinear region behind it. Numerical simulations of fronts in the Swift–Hohenberg equation are in full accord with our analytical predictions.

An analytical solution for stratified tidal lagoons: Application to the validation of a numerical open boundary condition

An analytical solution is presented for the case of a stratified, tidally forced lagoon. This solution, especially its energy and mass balances, is useful for the validation of numerical shallow water models under stratified, tidally forced conditions. The utility of the analytical solution for validation is demonstrated for a simple finite difference numerical model. The energy, mass, and their balances from the numerical model are shown to converge to the analytical solution with a power law dependence as spatial and temporal resolution are increased. The power law convergence of the numerical results to the analytical solution validates both the finite difference scheme and the open boundary conditions used for the tidal forcing. The open boundary conditions work well because they are consistent with the characteristics of the analytical solution.

Parity, precision and spin inversion within the density matrix renormalization group

A way of implementing the density matrix renormalization group (DMRG) method which simplifies the use of real-space parity as a conserved quantum number is discussed. The use of parity in the infinite-system DMRG calculations is often necessary in order to calculate more than the lowest excitation gap in a system. In addition, the use of parity reduces the computational overhead by a factor of two. Using parity as a symmetry we give numerical evidence that the infinite-system DMRG method in some cases displays a power-law convergence with the number of states retained. In particular we show that the often-used measure of the error in a DMRG calculation, , bears only a marginal resemblance to the true error. Spin inversion is shown to be a very useful symmetry when performing calculations in the subspace, allowing for a distinction between even and odd multiplets even when using the finite-system method.

Weak selection and stability of localized distributions in Ostwald ripening

We support and generalize a weak selection rule predicted recently for the self-similar asymptotics of the distribution function (DF) in the zero-volume-fraction limit of Ostwald ripening (OR). An asymptotic perturbation theory is developed that, when combined with an exact invariance property of the system, yields the selection rule, predicts a power-law convergence towards the selected self-similar DF and agrees well with our numerical simulations for the interface- and diffusion-controlled OR.

Fermi-edge singularity in one-dimensional electron systems with long-range Coulomb interactions.

Effects of long-range Coulomb interactions on the Fermi-edge singularity in optical spectra are investigated theoretically for one-dimensional spin-1/2 fermion systems with the use of the Tomonaga-Luttinger bosonization technique. Low-energy excitation spectrum near the Fermi level shows that dispersion of the charge-density fluctuation remains gapless but is nonlinear when the electron-electron (e-e) Coulomb interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., an infinite force range). Temporal behavior of the current-current correlation function is calculated analytically for arbitrary force ranges, ${\ensuremath{\lambda}}_{\mathit{e}}$ and ${\ensuremath{\lambda}}_{\mathit{h}}$, of the e-e and the electron-hole (e-h) Coulomb interactions. (i) When both the e-e and the e-h interactions have large but finite force ranges (${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\infty} and ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\infty}), the correlation function yields a power-law decay only for a long-time regime, which is determined by the force ranges as tgmax[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]/${\mathit{v}}_{\mathit{F}}$. Corresponding optical spectrum near the Fermi edge (within an energy range of \ensuremath{\Elzxh}${\mathit{v}}_{\mathit{F}}$/max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]) exhibits the power-law divergence or the power-law convergence, which is an ordinary Fermi-edge singularity. (ii) When either the e-e or the e-h interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., ${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\rightarrow}\ensuremath{\infty} and/or ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\rightarrow}\ensuremath{\infty}), an exponent of the correlation function is dependent on time to lead the faster decay than that of any power laws. Then the optical spectra show no power-law dependence and always converge (become zero) at the Fermi edge, which is in striking contrast to the ordinary power-law singularity. Relations between the spectral shape and a measurement time (frequency resolution) are also clarified. \textcopyright{} 1996 The American Physical Society.