Polynomial Rate Research Articles

Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner–Nordström black holes

We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner–Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis’ results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.

Polynomial decay of the energy of solutions of coupled wave equations with locally boundary fractional dissipation

In this paper, we investigate a system of coupled wave equations featuring boundary fractional damping applied to a portion of the domain. We first establish the well-posedness of the system, proving the existence and uniqueness of solutions through semi-group theory. While the system does not exhibit exponential stability, we demonstrate its strong stability. Furthermore, leveraging Arendt and Batty’s general criteria and certain geometric conditions, we prove a polynomial rate of energy decay for the solutions.

Polynomial and rational convergence rates for Laplace problems on planar domains

Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is non-convex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates and the theory of crowding of conformal maps.

Fourier transform and expanding maps on Cantor sets

abstract: We study the Fourier transforms $\widehat{\mu}(\xi)$ of non-atomic Gibbs measures $\mu$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $\widehat{\mu}(\xi)\to 0$ at a polynomial rate as $|\xi|\to\infty$.

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. Due to the singularity of the kernel, the definition of Caputo-Fabrizio appeared, which has a non-singular kernel and mathematical properties similar to the derivative of the integer order. This last definition attracted many mathematicians and researchers to use it in modeling phenomena and obtaining historical information about the development of the studied phenomena, but usually the analytical solution does not exist, which necessitated numerical methods to find an approximate solution. These approximate methods depend on finding an approximate formula for the fractional derivative, and then the problem is transformed into a system of algebraic equations that is easy to solve. In fact, all the numerical methods that have been used have a polynomial rate of convergence, which calls for thinking about a new method that is more effective and has a better rate of convergence. For this reason, in this paper, we propose an efficient numerical method to approximate the first order fractional derivative in the Caputo-Fabrizio sense. This method develops a new quadratic formula using Haar wavelet integration method. Error analysis of our proposed method gives an exponential convergence rate of . To check the effectiveness of the proposed method, we examine some examples with different fractional orders. The quantative results demonstrated the stability and efficiency of the proposed method.

Stability analysis of homogeneous cooperative positive differential systems with time-varying delays and its generalization

This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.

Exponential convergence of the weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

Variance-Reduced Accelerated First-Order Methods: Central Limit Theorems and Confidence Statements

In this paper, we consider a strongly convex stochastic optimization problem and propose three classes of variable sample-size stochastic first-order methods: (i) the standard stochastic gradient descent method, (ii) its accelerated variant, and (iii) the stochastic heavy-ball method. In each scheme, the exact gradients are approximated by averaging across an increasing batch size of sampled gradients. We prove that when the sample size increases at a geometric rate, the generated estimates converge in mean to the optimal solution at an analogous geometric rate for schemes (i)–(iii). Based on this result, we provide central limit statements, whereby it is shown that the rescaled estimation errors converge in distribution to a normal distribution with the associated covariance matrix dependent on the Hessian matrix, the covariance of the gradient noise, and the step length. If the sample size increases at a polynomial rate, we show that the estimation errors decay at a corresponding polynomial rate and establish the associated central limit theorems (CLTs). Under certain conditions, we discuss how both the algorithms and the associated limit theorems may be extended to constrained and nonsmooth regimes. Finally, we provide an avenue to construct confidence regions for the optimal solution based on the established CLTs and test the theoretical findings on a stochastic parameter estimation problem. Funding: This work was supported by the Office of Naval Research [Grant N00014-22-1-2589] and Basic Energy Sciences [Grant DE-SC0023303]. The work of J. Lei was partially supported by the National Natural Science Foundation of China [Grants 72271187 and 62088101]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0068 .

Fluid limits for QB-CSMA with polynomial rates, homogenization and reflection

Fluid limits for QB-CSMA with polynomial rates, homogenization and reflection

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

Convergence rates for sums-of-squares hierarchies with correlative sparsity

AbstractThis work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.

Diffusion Approximation for Symmetric Birth-and-Death Processes with Polynomial Rates

The symmetric birth and death stochastic process on the non-negative integers x ∈ Z + with polynomial rates x α , α ∈ [1, 2], x 6= 0, is studied. The process moves slowly and spends more time in the neighborhood of the state 0. We prove the convergence of the scaled process to a solution of stochastic differential equation without drift. Sticking phenomenon appears at the limiting process: trajectories, starting from any state, take finite time to reach 0 and remain there indefinitely.

A local polynomial moment approximation for compartmentalized biochemical systems

Compartmentalized biochemical reactions are a ubiquitous building block of biological systems. The interplay between chemical and compartmental dynamics can drive rich and complex dynamical behaviors that are difficult to analyze mathematically — especially in the presence of stochasticity. We have recently proposed an effective moment equation approach to study the statistical properties of compartmentalized biochemical systems. So far, however, this approach is limited to polynomial rate laws and moreover, it relies on suitable moment closure approximations, which can be difficult to find in practice. In this work we propose a systematic method to derive closed moment dynamics for compartmentalized biochemical systems. We show that for the considered class of systems, the moment equations involve expectations over functions that factorize into two parts, one depending on the molecular content of the compartments and one depending on the compartment number distribution. Our method exploits this structure and approximates each function with suitable polynomial expansions, leading to a closed system of moment equations. We demonstrate the method using three systems inspired by cell populations and organelle networks and study its accuracy across different dynamical regimes.

SUBGEOMETRICALLY ERGODIC AUTOREGRESSIONS WITH AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY

In this paper, we consider subgeometric (specifically, polynomial) ergodicity of univariate nonlinear autoregressions with autoregressive conditional heteroskedasticity (ARCH). The notion of subgeometric ergodicity was introduced in the Markov chain literature in the 1980s, and it means that the transition probability measures converge to the stationary measure at a rate slower than geometric; this rate is also closely related to the convergence rate of $\beta $ -mixing coefficients. While the existing literature on subgeometrically ergodic autoregressions assumes a homoskedastic error term, this paper provides an extension to the case of conditionally heteroskedastic ARCH-type errors, considerably widening the scope of potential applications. Specifically, we consider suitably defined higher-order nonlinear autoregressions with possibly nonlinear ARCH errors and show that they are, under appropriate conditions, subgeometrically ergodic at a polynomial rate. An empirical example using energy sector volatility index data illustrates the use of subgeometrically ergodic AR–ARCH models.

Strong limit theorem for largest entry of large-dimensional random tensor

Suppose that [Formula: see text] are i.i.d. copies of random vector [Formula: see text]. Let [Formula: see text] then the random tensor product constructed by [Formula: see text] is defined by [Formula: see text] In this paper, we obtain the strong limit theorems of the largest entry of large-dimensional random tensor product [Formula: see text] under two high-dimensional settings the polynomial rate and the exponential rate. The conclusions are established under weaker moment condition than the exist papers and the relationship between [Formula: see text] and [Formula: see text] is more flexible.

Global output feedback control for uncertain strict-feedback nonlinear systems: A logic-based switching event-triggered approach

This paper addresses the global output-feedback stabilization for a class of uncertain nonlinear systems via a switching event-triggered approach. Typically, the system growth rates are allowed to be arbitrary unknown non-polynomial functions of system output. Note that such nonlinearities cannot be handled by generalizing the design rationales employed in polynomial rate cases. For this reason, an event-triggered mechanism together with a logic-based switching mechanism is proposed to determine not only when to sample but also when to switch the control parameter. With the help of the switching control parameter, an observer-based control scheme is developed to achieve global output-feedback stabilization. Particularly, the control parameter can be adaptively adjusted to a sufficiently large value, so the proposed control scheme has a stronger capability to deal with large uncertainties, inherent nonlinearities, and sampling errors of the system. For more efficient resource saving, an extended investigation is presented by constructing a dynamic event-triggered scheme. Finally, two simulation examples are given to illustrate the effectiveness of the switching event-triggered schemes.

Approximation of smooth functionals using deep ReLU networks

In recent years, deep neural networks have been employed to approximate nonlinear continuous functionals F defined on Lp([−1,1]s) for 1≤p≤∞. However, the existing theoretical analysis in the literature either is unsatisfactory due to the poor approximation results, or does not apply to the rectified linear unit (ReLU) activation function. This paper aims to investigate the approximation power of functional deep ReLU networks in two settings: F is continuous with restrictions on the modulus of continuity, and F has higher order Fréchet derivatives. A novel functional network structure is proposed to extract features of higher order smoothness harbored by the target functional F. Quantitative rates of approximation in terms of the depth, width and total number of weights of neural networks are derived for both settings. We give logarithmic rates when measuring the approximation error on the unit ball of a Hölder space. In addition, we establish nearly polynomial rates (i.e., rates of the form exp−a(logM)b with a>0,0<b<1) when measuring the approximation error on a space of analytic functions.

Analysis of the Whiplash gradient descent dynamics

AbstractIn this paper, we propose the Whiplash inertial gradient dynamics, a closed‐loop optimization method that utilizes gradient information. We introduce the symplectic asymptotic convergence analysis for the Whiplash system for convex functions. We also introduce relaxation sequences to explain the nonclassical nature of the algorithm and an exploring heuristic variant of the Whiplash algorithm to escape saddle points, deterministically. We provide a practical methodology for analyzing convergence rates using integral constraint bounds and a novel Lyapunov rate method. Our results demonstrate polynomial and exponential rates of convergence for quadratic costs.

Asymptotic Behaviour of a Viscoelastic Transmission Problem with a Tip Load

We consider a transmission problem for a string composed by two components: one of them is a viscoelastic material (with viscoelasticity of memory type), and the other is an elastic material (without dissipation effective over this component). Additionally, we consider that in one end is attached a tip load. The main result is that the model is exponentially stable if and only if the memory effect is effective over the string. When there is no memory effect, then there is a lack of exponential stability, but the tip load produces a polynomial rate of decay. That is, the tip load is not strong enough to stabilize exponentially the system, but produces a polynomial rate of decay.

Identifiability and inference of phylogenetic birth–death models

Recent theoretical work on phylogenetic birth–death models offers differing viewpoints on whether they can be estimated using lineage-through-time data. Louca and Pennell (2020) showed that the class of models with continuously differentiable rate functions is nonidentifiable: any such model is consistent with an infinite collection of alternative models, which are statistically indistinguishable regardless of how much data are collected. Legried and Terhorst (2022) qualified this grave result by showing that identifiability is restored if only piecewise constant rate functions are considered.Here, we contribute new theoretical results to this discussion, in both the positive and negative directions. Our main result is to prove that models based on piecewise polynomial rate functions of any order and with any (finite) number of pieces are statistically identifiable. In particular, this implies that spline-based models with an arbitrary number of knots are identifiable. The proof is simple and self-contained, relying mainly on basic algebra. We complement this positive result with a negative one, which shows that even when identifiability holds, rate function estimation is still a difficult problem. To illustrate this, we prove some rates-of-convergence results for hypothesis testing using birth–death models. These results are information-theoretic lower bounds which apply to all potential estimators.