General Convergence Theorem Research Articles

Linear repetitivity beyond abelian groups

Abstract We show that linearly repetitive weighted Delone sets in groups of polynomial growth have a uniquely ergodic hull. This result applies in particular to the linearly repetitive weighted Delone sets in homogeneous Lie groups constructed in the companion paper [S. Beckus, T. Hartnick and F. Pogorzelski. Symbolic substitution beyond Abelian groups. Preprint, 2021, arXiv:2109.15210] using symbolic substitution methods. More generally, using the quasi-tiling method of Ornstein and Weiss, we establish unique ergodicity of hulls of weighted Delone sets in amenable unimodular locally compact second countable groups under a new repetitivity condition which we call tempered repetitivity. For this purpose, we establish a general sub-additive convergence theorem, which also has applications concerning the existence of Banach densities and uniform approximation of the spectral distribution function of finite hopping range operators on Cayley graphs.

A qualitative investigation of a system of third-order difference equations with multiplicative reciprocal terms

In this paper, we study the system of third-order difference equations \begin{equation*} x_{n+1}=a+\frac{a_{1}}{y_{n}}+\frac{a_{2}}{y_{n-1}}+\frac{a_{3}}{y_{n-2}}% ,\quad y_{n+1}=b+\frac{b_{1}}{x_{n}}+\frac{b_{2}}{x_{n-1}}+\frac{b_{3}}{% x_{n-2}},\quad n\in \mathbb{N}_{0}, \end{equation*}% where the parameters $a$, $a_{i}$, $b$, $b_{i}$, $i=1,2,3$, and the initial values $x_{-j}$, $y_{-j}$, $j=0,1,2$, are positive real numbers. We first prove a general convergence theorem. By applying this convergence theorem to the system, we show that positive equilibrium is a global attractor. We also study the local asymptotic stability of the equilibrium and show that it is globally asymptotically stable. Finally, we study the invariant set of solutions.

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

The consistent boundary element method for potential and elasticity: Part III — Topologically challenging numerical assessments for 2D problems

The collocation boundary element method is consistently outlined in a companion (Part I) paper for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. Simple code schemes are proposed in a second companion (Part II) paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high – actually only machine-precision dependent – computational accuracy of all results of interest independently of a problem’s geometry and topology. We present here numerical results and convergence assessments – including numerical illustration of convergence Theorem 1 of the companion paper I – for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source–field distances — maybe with approximations due to a coarse mesh discretization but never introducing unduly singularities. We dare say the present results cannot be replicated by any code implementation other than the ones of our own, which resort to no ad hoc means beyond the problem’s correct mathematics. In fact, we present the objective and controllable tools of separately assessing a mesh discretization for a given practical problem in terms of precision, accuracy and liability to round-off-errors. We also evaluate the distance threshold for a source point to be considered sufficiently close to demand the proposed exact mathematical treatment, as, for large distances, only Gauss–Legendre quadrature is required, be it in terms of quadrature adaptivity of even a fast multipole scheme.

The consistent boundary element method for potential and elasticity: Part I — Formulation and convergence theorem

The collocation boundary element method is outlined for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana’s identity, as already proposed in the literature. However, this is done in a consistent way that sheds light on some conceptual and implementation-oriented aspects that should be – but are missing – in the textbooks on the subject. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. A rather recreative example of a simple truss element is brought in an Appendix. It enables the demystification of apparently convoluted concepts that might be thought only pertaining to highly elaborated mechanics and mathematics for singularities related to infinity and to ungraspable linear algebra. Simple code schemes based on the outlined concepts are proposed in a companion paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high computational accuracy of all results of interest independently of a problem’s geometry or topology. Numerical results are shown in another companion paper and convergence features are assessed for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source-field distances. Summing up, whether a problem we are attempting to simulate consists of smooth fields or not, we are able to reproduce it — maybe with numerical approximations due to a coarse mesh discretization but never introducing unduly singularities.

The infinite product of contraction semigroups on l^{1}({mathbb {N}}) and l^{infty }({mathbb {N}})

I provide an example of a family of commuting contraction semigroups (e^{t B_{n}})_{nin {mathbb {N}}} defined on l^{1}({mathbb {N}}) such that the product semigroup prod _{n=1}^{infty }e^{tB_{n}} exists and has bounded generator. The infinite product of the corresponding family of adjoint semigroups (e^{t B_{n}^{*}})_{nin {mathbb {N}}} defined on l^{infty }({mathbb {N}}) also exists and its generator is bounded. I give explicit formulae for these generators. The results follow from a general convergence theorem for such semigroups proved in Arendt et al. (J Funct Anal 160: 524–542, 1998).

A general approach to the study of the convergence of Picard iteration with an application to Halley's method for multiple zeros of analytic functions

In this paper, we define a new wide class of iteration functions and then we use it to prove a general convergence theorem that provides exact domain of initial approximations to guarantee the high Q-order of convergence of Picard iteration generated by this class of functions. As an application of this theorem, we prove some local convergence theorems about the famous Halley's method for simple and multiple zeros of analytic functions. All obtained results are supplied with a priori and a posteriori error estimates as well as with assessments of the asymptotic error constants.

Plug-and-Play gradient-based denoisers applied to CT image enhancement

Blur and noise corrupting Computed Tomography (CT) images can hide or distort small but important details, negatively affecting the consequent diagnosis. In this paper, we present a novel gradient-based Plug-and-Play (PnP) algorithm and we apply it to restore CT images. The plugged denoiser is implemented as a deep Convolutional Neural Network (CNN) trained on the gradient domain (and not on the image one, as in state-of-the-art works) and it induces an external prior onto the restoration model. We further consider a hybrid scheme which combines the gradient-based external denoiser with an internal one, obtained from the Total Variation functional. The proposed frameworks rely on the Half-Quadratic Splitting scheme and we prove a general fixed-point convergence theorem, under weak assumptions on both the denoisers. The experiments confirm the effectiveness of the proposed gradient-based approach in restoring blurred noisy CT images, both in simulated and real medical settings. The obtained performances outperform the achievements of many state-of-the-art methods.

First-Order Algorithms for a Class of Fractional Optimization Problems

We consider in this paper a class of single-ratio fractional minimization problems, in which the numerator part of the objective is the sum of a nonsmooth nonconvex function and a smooth nonconvex function while the denominator part is a nonsmooth convex function. In this work, we first derive its first-order necessary optimality condition, by using the first-order operators of the three functions involved. Then we develop first-order algorithms, namely, the proximity-gradient-subgradient algorithm (PGSA), PGSA with monotone line search (PGSA_ML) and PGSA with nonmonotone line search (PGSA_NL). It is shown that any accumulation point of the sequence generated by them is a critical point of the problem under mild assumptions. Moreover, we establish global convergence of the sequence generated by PGSA or PGSA_ML and analyze its convergence rate, by further assuming the local Lipschitz continuity of the nonsmooth function in the numerator part, the smoothness of the denominator part and the Kurdyka- Lojasiewicz property of the objective. The proposed algorithms are applied to the sparse generalized eigenvalue problem associated with a pair of symmetric positive semidefinite matrices and the corresponding convergence results are obtained according to their general convergence theorems. We perform some preliminary numerical experiments to demonstrate the efficiency of the proposed algorithms

Convergence of the Nelder-Mead method

AbstractWe develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.

On a general system of difference equations defined by homogeneous functions

AbstractThe aim of this paper is to study the following second order system of difference equations$$\begin{array}{} x_{n+1} = f(y_{n},y_{n-1}),\quad y_{n+1} = g(x_{n},x_{n-1}) \end{array}$$wheren∈ ℕ0, the initial valuesx−1,x0,y−1andy0are positive real numbers, the functionsf,g: (0, +∞)2→ (0, +∞) are continuous and homogeneous of degree zero. In this study, we establish results on local stability of the unique equilibrium point and to deal with the global attractivity, and so the global stability, some general convergence theorems are provided. Necessary and sufficient conditions on existence of prime period two solutions of our system are given. Also, a result on oscillatory solutions is proved. As applications of the obtained results, concrete models of systems of difference equations defined by homogeneous functions of degree zero are investigated. Our system generalize some existing works in the literature and our results can be applied to study new models of systems of difference equations. For interested readers, we left in the conclusion as open problems two more general systems of higher order defined by homogenous functions of degree zero.

Adaptive output feedback regulation for a class of uncertain feedforward stochastic nonlinear systems

This paper addresses the global output feedback regulation problem for a class of uncertain feedforward stochastic nonlinear systems. Unlike the previous works, the drift’s growth rate depends not only on an unknown constant but also on an output polynomial function and an input function. Moreover, the diffusion’s growth rate depends on an unknown constant or an output polynomial function and an input function. By using the general stochastic convergence theorem, an output controller is constructed to guarantee the boundedness and stochastic convergence of the resulting closed-loop system.

Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros

In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.

Strong convergence for weighted sums of END random variables under the sub-linear expectations

In this article, we study strong limit theorems for weighted sums of extended negatively dependent (END, for short) random variables under the sub-linear expectations. We establish some general complete convergence theorems for weighted sums of END random variables under the sub-linear expectations. Our results partly generalize and improve the corresponding ones previously obtained by Cai (Metrika, 68:323-331, 2008), Huang and Wang (J. Inequal. Appl. 233, doi:10.1186/1029-242X-2012-233, 2012) and Wang et al. (RACSAM, 106:235-245, 2012) in the classical probability space to the sub-linear expectation space under weaker moment conditions.

General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

Finite difference approach to fourth-order linear boundary-value problems

Abstract Discrete approximations to the equation $$\begin{equation*}L_\textrm{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A^{\prime}(x)+H(x)) u^{(1)} + B(x) u = f, \quad x\in[0,1]\end{equation*}$$are considered. This is an extension of the Sturm–Liouville case $D(x)\equiv H(x)\equiv 0$ (Ben-Artzi et al. (2018) Discrete fourth-order Sturm–Liouville problems. IMA J. Numer. Anal., 38, 1485–1522) to the non-self-adjoint setting. The ‘natural’ boundary conditions in the Sturm–Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second-, third- and fourth-order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties of compactness and coercivity. It allows us to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.

Empirical risk minimization and complexity of dynamical models

A dynamical model consists of a continuous self-map $T:\mathcal{X}\to \mathcal{X}$ of a compact state space $\mathcal{X}$ and a continuous observation function $f:\mathcal{X}\to \mathbb{R}$. This paper considers the fitting of a parametrized family of dynamical models to an observed real-valued stochastic process using empirical risk minimization. The limiting behavior of the minimum risk parameters is studied in a general setting. We establish a general convergence theorem for minimum risk estimators and ergodic observations. We then study conditions under which empirical risk minimization can effectively separate signal from noise in an additive observational noise model. The key condition in the latter results is that the family of dynamical models has limited complexity, which is quantified through a notion of entropy for families of infinite sequences that connects covering number based entropies with topological entropy studied in dynamical systems. We establish close connections between entropy and limiting average mean widths for stationary processes, and discuss several examples of dynamical models.

Complete convergence theorem for negatively dependent random variables under sub-linear expectations

Under the condition that the Choquet integral exists, we study the complete convergence theorem for negatively dependent random variables under sub-linear expectation space. Two general complete convergence theorems under sub-linear expectation space are obtained, where the coefficient of weighted sum is the general function. This paper not only extends the complete convergence theorem in the traditional probability space to the sub-linear expectation space, but also extends the coefficient of weighted sum as a general function.

Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

Set-valued contractions of Leader type in quasi-triangular spaces are constructed, conditions guaranteeing the existence of nonempty sets of periodic points, fixed points and endpoints of such contractions are established, convergence of dynamic processes of these contractions are studied, uniqueness properties are derived, and single-valued cases are considered. Investigated dynamic systems are not necessarily continuous and spaces are not necessarily sequentially complete or Hausdorff. Obtained results suggest, in particular, strategies to new studies of functional Bellman equations and variable discounted Bellman equations in metric spaces and integral Volterra equations in locally convex spaces. Results in this direction are also presented in this paper. More precisely, without continuity of Bellman and Volterra appropriate operators, the sets of solutions of these equations (which are periodic points of these operators) are studied and new and general convergence, existence and uniqueness theorems concerning such equations are proved.

Convergence theorems for the Nelder-Mead method

We develop a matrix form of the Nelder-Mead method and after discussing the concept of convergence we prove a general convergence theorem. The new theorem is demonstrated in low dimensional spaces.