Pointwise Convergence Research Articles

Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes

AbstractWe study experimental convergence rates of three shock‐capturing schemes for hyperbolic systems of conservation laws: the second‐order central‐upwind (CU) scheme, the third‐order Rusanov‐Burstein‐Mirin (RBM), and the fifth‐order alternative weighted essentially non‐oscillatory (A‐WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A‐WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second‐order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A‐WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non‐oscillatory near the shocks.

Convergence of the heat kernel approximate solutions of the transport-diffusion equation in the half-space

A family of approximate solutions to transport-diffusion equation in the half-space R+d is constructed by using the heat kernel and the translation representing transport on each step of time discretization. The uniform convergence of these approximate solutions and that of their first derivatives with respect to the space variables as well as the point-wise convergence of their second derivatives with respect to the space variables are proved. We also show that the limit function satisfies the transport-diffusion equation in the half-space with homogenous boundary conditions. From the technical point of view it is essential to obtain for the third derivatives of approximate solutions with respect to the space variables a necessary estimate for the passage to the limit, taking into account the influence of the boundary conditions.

Local pointwise convergence of the 3D finite element

Local pointwise convergence of the 3D finite element

Convergence of spherical averages for actions of Fuchsian groups

We prove pointwise convergence of spherical averages for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen–Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.

Pointwise convergence of the solutions to wave equations with potentials

Pointwise convergence of the solutions to wave equations with potentials

The Polish topology of the isometry group of the infinite dimensional hyperbolic space

We consider the isometry group of the infinite-dimensional separable hyperbolic space with its Polish topology. This topology is given by pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite-dimensional groups like \mathcal{S}_\infty , the group of all permutations of a countable set on the other side. Our main results are All along the text, we lead a parallel study with the sibling group \mathrm{Isom}(\mathcal{H}) , where \mathcal{H} is a separable Hilbert space.

Pointwise convergence of sequential Schrödinger means

We study pointwise convergence of the fractional Schrödinger means along sequences t_{n} that converge to zero. Our main result is that bounds on the maximal function sup_{n} |e^{it_{n}(-Delta )^{alpha /2}} f| can be deduced from those on sup_{0< tle 1} |e^{it(-Delta )^{alpha /2}} f|, when {t_{n}} is contained in the Lorentz space ell ^{r,infty}. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou and Seeger, and Li, Wang, and Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence.

Embedding theorems for function spaces

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space C(X,Y) of all continuous functions from a topological space X into a uniform space Y with the topology of uniform convergence on a family of subsets of X and with the (weak) set-open topology. We also investigated the following question: how the topological embedding of the space C(X,Y) is related to algebraic structures (such as topological groups, topological rings and topological vector spaces) on C(X,Y).

Maximal Operator, Cotlar’s Inequality and Pointwise Convergence for Singular Integral Operators in Dunkl Setting

We establish the maximal operator, Cotlar’s inequality and pointwise convergence in the Dunkl setting for the (nonconvolution type) Dunkl–Calderón–Zygmund operators introduced recently in Tan et al. (https://arxiv.org/abs/2204.01886). The fundamental geometry of the Dunkl setting involves two nonequivalent metrics: the Euclidean metric and the Dunkl metric deduced by finite reflection groups, and hence the classical methods do not apply directly. The key idea is to introduce truncated singular integrals and the maximal singular integrals by the Dunkl metric and the Euclidean metric. We show that these two kind of truncated singular integrals are dominated by the Hardy–Littlewood maximal function, which yields the Cotlar’s inequalities and hence the boundedness of maximal Dunkl–Calderón–Zygmund operators. Further, as applications, two equivalent pointwise convergences for Dunkl–Calderón–Zygmund operators are obtained.

Sharp pointwise convergence on the Schrödinger operator along one class of curves

Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson, which was essentially solved by Du-Guth-Li [10] and Du-Zhang [12]. In this note, we obtain sharp pointwise convergence on the Schrödinger operator along one class of curves.

Generic properties of nonexpansive mappings on unbounded domains

We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families of metrics of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a Rakotch contraction on every bounded subset and that there is a bounded set which is mapped into itself by this mapping. In particular, we obtain that the typical nonexpansive mapping in this setting has a unique fixed point which can be reached by iterating the mapping. Nevertheless, it turns out that the typical mapping is not a Rakotch contraction on the whole space and that it has the maximal possible Lipschitz constant of one on a residual subset of its domain. By typical we mean that the complement of the set of mappings with this property is σ-ϕ-porous, that is, small in a metric sense. For a metric of pointwise convergence, we show that the set of Rakotch contractions is meagre.

An Explicit Expression of the Unit Step Function

In this paper, an analytical form of the Unit Step Function (or Heaviside Step Function) is presented. In particular, this piecewise – defined function, which constitutes a fundamental concept of Operational Calculus and is also involved in many other areas of applied and engineering mathematics, is explicitly performed in a very simple manner by the aid of purely algebraic representations. The novelty of this work, when compared with other analytical approximations to this discontinuous function, is that the proposed exact formula is not performed in terms of non – elementary special functions, e.g. Gamma function or Error function. In addition, this function does not contain any infinitesimal quantities and also is not the limit of a sequence of functions with a pointwise or uniform convergence. Hence, it may be much more appropriate and useful to the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

On statistical graph and pointwise convergence of sequences of set-valued functions defined on intuitionistic fuzzy normed spaces

On statistical graph and pointwise convergence of sequences of set-valued functions defined on intuitionistic fuzzy normed spaces

Equiconvergence for perturbed Jacobi polynomial expansions

We show asymptotic expansions of the eigenfunctions of certain perturbations of the Jacobi operator in a bounded interval, deducing equiconvergence results between expansions with respect to the associated orthonormal basis and expansions with respect to the cosine basis. Several results for pointwise convergence then follow.

Integral and Weighted Composition Operators on Fock-type Spaces

We study various structures of general Volterra-type integral and weighted composition operators acting between two Fock-type spaces mathcal {F}_{varphi }^p and mathcal {F}_{varphi }^q, where varphi is a radial function growing faster than the function zrightarrow |z|^2/2. The main results show that the unboundedness of the Laplacian of varphi provides interesting results on the topological and spectral structures of the operators in contrast to their actions on Fock spaces, where the Laplacian of the weight function is bounded. We further describe the invertible and unitary weighted composition operators. Finally, we show the spaces support no supercyclic weighted composition operator with respect to the pointwise convergence topology and hence with the weak and strong topologies.

Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability

Let $K$ be a compact space and let $C(K)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C(K)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. \noindent Moreover, the weak topology and that of pointwise convergence coincide on $[0,x]$ if and only if $\{x>0\}$ is scattered. Finally, the weak topology on $[0,x]$ is metrizable if and only if the topology of pointwise convergence on $[0,x]$ is such if and only if $\{x>0\}$ is countable.

Optimal pointwise convergence of the LDG method for singularly perturbed convection–diffusion problem

A local discontinuous Galerkin (LDG) method is considered for a one-dimensional singularly perturbed convection–diffusion problem with an exponential boundary layer. Based on the technique of discrete Green’s function, we establish optimal pointwise convergence (up to a logarithmic factor) of the LDG method on three typical families of layer-adapted meshes: Shishkin-type, Bakhvalov–Shishkin-type and Bakhvalov-type. Numerical experiments are also given.

Monotone Inclusions, Acceleration, and Closed-Loop Control

We propose and analyze a new dynamical system with a closed-loop control law in a Hilbert space [Formula: see text], aiming to shed light on the acceleration phenomenon for monotone inclusion problems, which unifies a broad class of optimization, saddle point, and variational inequality (VI) problems under a single framework. Given an operator [Formula: see text] that is maximal monotone, we propose a closed-loop control system that is governed by the operator [Formula: see text], where a feedback law [Formula: see text] is tuned by the resolution of the algebraic equation [Formula: see text] for some [Formula: see text]. Our first contribution is to prove the existence and uniqueness of a global solution via the Cauchy–Lipschitz theorem. We present a simple Lyapunov function for establishing the weak convergence of trajectories via the Opial lemma and strong convergence results under additional conditions. We then prove a global ergodic convergence rate of [Formula: see text] in terms of a gap function and a global pointwise convergence rate of [Formula: see text] in terms of a residue function. Local linear convergence is established in terms of a distance function under an error bound condition. Further, we provide an algorithmic framework based on the implicit discretization of our system in a Euclidean setting, generalizing the large-step hybrid proximal extragradient framework. Even though the discrete-time analysis is a simplification and generalization of existing analyses for a bounded domain, it is largely motivated by the aforementioned continuous-time analysis, illustrating the fundamental role that the closed-loop control plays in acceleration in monotone inclusion. A highlight of our analysis is a new result concerning [Formula: see text]-order tensor algorithms for monotone inclusion problems, complementing the recent analysis for saddle point and VI problems. Funding: This work was supported in part by the Mathematical Data Science Program of the Office of Naval Research [Grant N00014-18-1-2764] and by the Vannevar Bush Faculty Fellowship Program [Grant N00014-21-1-2941].

Mean-square Approximation of Iterated Ito and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Ito Sdes and Semilinear Spdes (third Edition)

This is the third edition of the monograph (first edition 2020, second edition 2021) devoted to the problem of mean-square approximation of iterated Ito and Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. The mentioned problem is considered in the book as applied to the numerical integration of non-commutative Ito stochastic differential equations and semilinear stochastic partial differential equations with nonlinear non-commutative trace class noise. The book opens up a new direction in researching of iterated stochastic integrals. For the first time we use the generalized multiple Fourier series converging in the sense of norm in Hilbert space for the expansion of iterated Ito stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process (Chapter 1). Sections 1.11-1.13 (Chapter 1) are new and generalize the results of Chapter 1 obtained earlier by the author and are also closely related to the multiple Wiener stochastic integral introduced by Ito in 1951. The convergence with probability 1 as well as the convergence in the sense of n-th (n=2, 3,...) moment for the expansion of iterated Ito stochastic integrals have been proved (Chapter 1). Moreover, the rate of both types of convergence has been established. The main difference between the third and second editions of the book is that the third edition includes original material (Chapter 2, Sections 2.10-2.19) on a new approach to the series expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k with respect to components of the multidimensional Wiener process. The above approach allowed us to generalize some of the author's earlier results and also to make significant progress in solving the problem of series expansion of iterated Stratonovich stochastic integrals. In particular, for iterated Stratonovich stochastic integrals of the fifth and sixth multiplicity, series expansions based on multiple Fourier-Legendre series and multiple trigonometric Fourier series are obtained. In addition, expansions of iterated Stratonovich stochastic integrals of multiplicities 2 to 4 were generalized. These results (Chapter 2) adapt the results of Chapter 1 for iterated Stratonovich stochastic integrals. Two theorems on expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity k based on generalized iterated Fourier series with pointwise convergence are formulated and proved (Chapter 2). The results of Chapters 1 and 2 can be considered from the point of view of the Wong-Zakai approximation for the case of a multidimensional Wiener process and the Wiener process approximation based on its series expansion using Legendre polynomials and trigonometric functions. The integration order replacement technique for iterated Ito stochastic integrals has been introduced (Chapter 3). Exact expressions are obtained for the mean-square approximation error of iterated Ito stochastic integrals of arbitrary multiplicity k (Chapter 1) and iterated Stratonovich stochastic integrals of multiplicities 1 to 4 (Chapter 5). Furthermore, we provided a significant practical material (Chapter 5) devoted to the expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 from the unified Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4). These approximations were obtained using Legendre polynomials and trigonometric functions. The methods constructed in the book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of iterated stochastic integrals with respect to the finite-dimensional approximation of the Q-Wiener process (for integrals of multiplicity k) and with respect to the infinite-dimensional Q-Wiener process (for integrals of multiplicities 1 to 3).

Approximation of functions by wavelet expansions with dilation matrix

In this paper, we obtain the degree of approximation of a function f in Lp(1 ? p ? ?) norm under general conditions of the pointwise and uniform convergence of wavelet expansions associated with the multiresolution analysis with dilation matrix. Our results show that the degree has the exponential decay (faster than any polynomial) for the function f in Lp(R) on a finite interval (a, b).