Inverse Inequalities Research Articles

An inverse inequality for a Bresse–Timoshenko system without second spectrum of frequency

In this paper, we consider a truncated version of the Timoshenko beam model and we prove an inverse inequality in order to obtain the controllability of the total system. The present system is new in the control and stabilization setting according to recent contributions due to Almeida Junior and Ramos (Z Angew Math Phys 68(145):31, 2017). The advantage here relies on mathematical results which do not depend on the classical condition between wave propagation velocities.

High-order extended finite element methods for solving interface problems

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the piecewise H1-norm and L2-norm are rigorously proved for both schemes. In particular, we have devised a new parameter-friendly DG-XFEM method, which means that no “sufficiently large” parameters are needed to ensure the optimal convergence of the scheme. To prove the stability of bilinear forms, we derive non-standard trace and inverse inequalities for high-order polynomials on curved sub-elements divided by the interface. This paper is adapted from the work originally post on arXiv.com by the same authors (arXiv:1604.06171). New ingredients are an optimal multigrid solver for the generated linear system and its analysis. This multigrid method converges uniformly with respect to the mesh size, and is independent of the location of the interface relative to the meshes, just like all the other estimates in this paper. Numerical examples are given to support the theoretical results.

Nonconforming finite element method for a generalized nonlinear Schrödinger equation

In this paper, an efficient nonconforming finite element method (FEM) is studied with EQ1rot element for a generalized nonlinear Schrödinger equation. First, we prove a novel result of the consistency error estimate with order O(h2) for EQ1rot element which leads to the superconvergent error estimate in broken H1-norm for semi-discrete scheme, while previous work only derive convergent results for this element. Second, a linearized backward Euler scheme is established and a time-discrete system is introduced to split the error into two parts, the temporal error and the spatial error. By using a rigorous analysis for the regularity of the time-discrete system and the proved characters of EQ1rot element, the optimal L2-error estimate is obtained without any time-step restrictions, which leads to the numerical solution can be bounded in L∞-norm by an inverse inequality unconditionally. Then, the supercloseness estimate is arrived at with the above achievements. Third, global superconvergence results are deduced through interpolated postprocessing technique. At last, numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Notes on a neural network approach to inverse variational inequalities

We consider a neural network approach to an inverse variational inequality which is assumed to have a non-empty set of solutions. In the case of gradient mappings, we prove that every trajectory of the network converges to the solution set in the case of convex potentials and if the solution set is singleton, the network is globally asymptotically stable at the equilibrium point. We also prove that if the network has a strongly convex potential, then the network is globally exponentially stable at the equilibrium point. Another purpose of this paper is to point out certain fatal mistakes in the paper by Zou et al. [A novel method to solve inverse variational inequality problems based on neural networks. Neurocomputing. 2016;173:1163–1168].

Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

AbstractThis paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L2 norm with the second‐order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results.

Gap functions and error bounds for vector inverse mixed quasi-variational inequality problems

This paper is devoted to investigating a vector inverse mixed quasi-variational inequality (VIMQVI). Our aim is to obtain error bounds for VIMQVI in terms of different gap functions, i.e., the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of VIMQVI. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang. Our results cover and extend similar results for these problems.

Modifiable Factors Explain Socioeconomic Inequalities in Children’s Dental Caries

The aim of this article was to quantify socioeconomic inequalities in dental caries experience among Australian children and to identify factors that explain area-level socioeconomic inequalities in children’s dental caries. We used data from the National Child Oral Health Survey conducted in Australia between 2012 and 2014 (n = 24,664). Absolute and relative indices of socioeconomic inequalities in the dental caries experience in primary and permanent dentition (decayed, missing, and filled surfaces [dmfs] and DMFS, respectively) were estimated. In the first stage, we conducted multilevel negative binomial regressions to test the association between area-level Index of Relative Socioeconomic Advantage and Disadvantage (IRSAD) and dental caries experience (dmfs for 5- to 8-y-olds and DMFS for 9- to 14-y-olds) after adjustment for water fluoridation status, sociodemographics, oral health behaviors, pattern of dental visits, and sugar consumption. In the second stage, we performed Blinder-Oaxaca and Neumark decomposition analyses to identify factors that explain most of the area-level socioeconomic inequalities in dental caries. Children had a mean dmfs of 3.14 and a mean DMFS of 0.98 surfaces. Children living in the most disadvantaged and intermediately disadvantaged areas had 1.96 (95% confidence interval, 1.69–2.27) and 1.45 (1.26–1.68) times higher mean dmfs and 1.53 (1.36–1.72) and 1.43 (1.27–1.60) times higher mean DMFS than those living in the most advantaged areas, respectively. Water fluoridation status (33.6%), sugar consumption (22.1%), parental educational level (14.2%), and dental visit patterns (12.7%) were the main factors explaining area-level socioeconomic inequalities in dental caries in permanent dentition. Among all the factors considered, the factors that contributed most in explaining inequalities in primary dental caries were dental visits (30.3%), sugar consumption (20.7%), household income (20.0%), and water fluoridation status (15.9%). The inverse area-level socioeconomic inequality in dental caries was mainly explained by modifiable risk factors, such as lack of fluoridated water, high sugar consumption, and an unfavorable pattern of dental visits.

Sharp error estimates for spline approximation: Explicit constants, n-widths, and eigenfunction convergence

In this paper, we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under [Formula: see text]-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.

Solving policy design problems: Alternating direction method of multipliers-based methods for structured inverse variational inequalities

Inverse variational inequalities have broad applications in various disciplines, and some of them have very appealing structures. There are several algorithms (e.g., proximal point algorithms and projection-type algorithms) for solving the inverse variational inequalities in general settings, while few of them have fully exploited the special structures. In this paper, we consider a class of inverse variational inequalities that has a separable structure and linear constraints, which has its root in spatial economic equilibrium problems. To design an efficient algorithm, we develop an alternating direction method of multipliers (ADMM) based method by utilizing the separable structure. Under some mild assumptions, we prove its global convergence. We propose an improved variant that makes the subproblems much easier and derive the convergence result under the same conditions. Finally, we present the preliminary numerical results to show the capability and efficiency of the proposed methods.

Nonlinear Separation Approach to Inverse Variational Inequalities in Real Linear Spaces

In this paper, we employ the image space analysis to study constrained inverse variational inequalities by means of a nonlinear separation approach. We introduce such a nonlinear functional, which is based on the known Gerstewitz functional, and show its property as a weak separation function and a regular weak separation function under different parameter sets. In contrast to known results, we do not assume any topology on the considered spaces. Then, an alternative theorem is established, which leads directly to a sufficient and necessary optimality condition of the constrained inverse variational inequality. Finally, a gap function and an error bound are obtained for the constrained inverse variational inequality.

Tikhonov type regularization methods for inverse mixed variational inequalities

In this paper, we focus on a Tikhonov type regularization method for inverse mixed variational inequalities. Existence of solutions of the inverse mixed variational inequalities and its regularized problem are established. We also present the relationships among three coercivity conditions for inverse mixed variational inequalities. Finally, the nonemptiness and boundedness for the solutions set of the regularized problem are derived under some mild conditions.

A high-order meshless Galerkin method for semilinear parabolic equations on spheres

We describe a novel meshless Galerkin method for numerically solving semilinear parabolic equations on spheres. The new approximation method is based upon a discretization in space using spherical basis functions in a Galerkin approximation. As our spatial approximation spaces are built with spherical basis functions, they can be of arbitrary order and do not require the construction of an underlying mesh. We will establish convergence of the meshless method by adapting, to the sphere, a convergence result due to Thomee and Wahlbin. To do this requires proving new approximation results, including a novel inverse or Nikolskii inequality for spherical basis functions. We also discuss how the integrals in the Galerkin method can accurately and more efficiently be computed using a recently developed quadrature rule. These new quadrature formulas also apply to Galerkin approximations of elliptic partial differential equations on the sphere. Finally, we provide several numerical examples.

逆向等周型不等式的研究进展

逆向等周型不等式的研究进展

Reverse Hilbert type inequalities

In this paper we establish a new inverse inequality of Hilbert type for a finite number of positive sequences of real numbers. The integral analogue of the inequality are also proved. The results of this paper reduce to that of B. G. Pachpatte.

An existence-uniqueness theorem and alternating contraction projection methods for inverse variational inequalities

Let C be a nonempty closed convex subset of a real Hilbert space mathcal{H} with inner product langle cdot , cdot rangle , and let f: mathcal{H}rightarrow mathcal{H} be a nonlinear operator. Consider the inverse variational inequality (in short, operatorname{IVI}(C,f)) problem of finding a point xi ^{*}in mathcal{H} such that \t\t\tf(ξ∗)∈C,〈ξ∗,v−f(ξ∗)〉≥0,∀v∈C.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ f\\bigl(\\xi ^{*}\\bigr)\\in C, \\quad \\bigl\\langle \\xi ^{*}, v-f \\bigl(\\xi ^{*}\\bigr)\\bigr\\rangle \\geq 0, \\quad \\forall v\\in C. $$\\end{document} In this paper, we prove that operatorname{IVI}(C,f) has a unique solution if f is Lipschitz continuous and strongly monotone, which essentially improves the relevant result in (Luo and Yang in Optim. Lett. 8:1261–1272, 2014). Based on this result, an iterative algorithm, named the alternating contraction projection method (ACPM), is proposed for solving Lipschitz continuous and strongly monotone inverse variational inequalities. The strong convergence of the ACPM is proved and the convergence rate estimate is obtained. Furthermore, for the case that the structure of C is very complex and the projection operator P_{C} is difficult to calculate, we introduce the alternating contraction relaxation projection method (ACRPM) and prove its strong convergence. Some numerical experiments are provided to show the practicability and effectiveness of our algorithms. Our results in this paper extend and improve the related existing results.

A new approach of superconvergence analysis of a low order nonconforming MFEM for reaction\u2013diffusion equation

In this paper, a low order nonconforming mixed finite element method (MFEM) is studied with EQ_{1}^{mathrm{rot}} element and zero order Raviart–Thomas (R–T) element for a class of nonlinear reaction–diffusion equations. On the one hand, a priori bound is proved using Lyapunov functional, which leads to the global existence and uniqueness of the approximation solutions. Further, the superclose estimates of order O(h^{2}) for original variable u in broken H^{1} norm and the flux boldsymbol{p}=nabla u in (L^{2})^{2} norm are obtained respectively for a semi-discrete scheme. On the other hand, a linearized Crank–Nicolson fully-discrete scheme is established and the superclose estimates of order O(h^{2}+tau^{2}) are also obtained unconditionally by making full use of the special characters of the above mixed finite elements (MFEs) and two different splitting techniques, which are used to deal with the consistency errors and nonlinear terms, respectively. These approaches circumvent the restrictive condition on a time step size arising as an inverse inequality used to prove a posteriori bounds in L^{infty} norm, which is necessary for nonlinear problems for the conventional finite element analysis. What is more, the corresponding global superconvergent results are derived through interpolated postprocessing techniques. Finally, numerical results are provided to confirm the theoretical analysis. Here h is the subdivision parameter and τ is the time step.

A posteriori error analysis of an augmented fully mixed formulation for the nonisothermal Oldroyd–Stokes problem

In this article, we consider an augmented fully mixed variational formulation that has been recently proposed for the nonisothermal Oldroyd–Stokes problem, and develop an a posteriori error analysis for the 2‐D and 3‐D versions of the associated mixed finite element scheme. More precisely, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem on arbitrary (convex or nonconvex) polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the bilinear forms of the continuous formulation, suitable assumptions on the domain and the data, stable Helmholtz decompositions, and the local approximation properties of the Clément and Raviart–Thomas operators. On the other hand, inverse inequalities, the localization technique based on bubble functions, and known results from previous works are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the a posteriori error estimators and illustrating the performance of the associated adaptive algorithms are reported.

Physical Parameters of a Reactor-Stellarator with Small Ripples of the Helical Magnetic Field

The paper describes the calculation data on the physical parameters of a reactor-stellarator, where the nonuniformities of the helical field are smaller than the toroidal magnetic field nonuniformities: eh et case, in the present calculations, these coefficients were assumed to be in proportion to the first power of the collision frequency, Di ∝ ν for νeff < 2ωE, and to Di ∝ ν−1 for the inverse inequality. Here, ωE is the rotation frequency of plasma in the radial electric field. As before, the plasma electrons corresponded to the mode of De ∝ ν−1. As initial parameters for numerical calculations, a reactor with R = 8 m, rp = 2 m, and B0 = 5 Т was taken. A numerical code was used to solve the set of equations that describes the plasma space−time behavior in the reactor-stellarator under the conditions of equal diffusion fluxes. The start of reactor operation in the mode of thermonuclear burning was provided by heating sources with a power of several tens of megawatts. Steady-state operating conditions of a self-sustained thermonuclear reaction were attained by maintaining the plasma density through DT fuel pellet injection into the plasma.

A Posteriori Error Analysis of a Mixed-Primal Finite Element Method for the Boussinesq Problem with Temperature-Dependent Viscosity

We have recently proposed a new finite element method for a more general Boussinesq model in 2D given by the case in which the viscosity of the fluid depends on its temperature. Our approach is based on a pseudostress–velocity–vorticity mixed formulation for the momentum equations, which is suitably augmented with Galerkin-type terms, coupled with the usual primal formulation for the energy equation, along with the introduction of the normal heat flux on the boundary as a Lagrange multiplier taking care of the fact that the prescribed temperature there becomes an essential condition. Then, fixed-point arguments using Banach and Brouwer theorems, in addition to other classical tools from functional and numerical analysis, provide sufficient conditions ensuring well-posedness of the resulting continuous and discrete sytems, together with the corresponding error estimates and associated rates of convergence. In the present work we complement these results with the derivation of a reliable and efficient residual-based a posteriori error estimator for the aforementioned augmented mixed-primal finite element method. Duality techniques, Helmholtz decompositions, and the approximation properties of the Raviart–Thomas and Clement interpolants are applied to obtain a reliable global error indicator. In turn, standard tools including the usual localization technique of bubble functions and inverse inequalities, and a regularity assumption originally utilized in the previous well-posedness and a priori error analyses, are employed to prove its efficiency. In both cases, reliability and efficiency, the estimates are shown at global level. Finally, a reliable fully local and computable a posteriori error estimator induced by the aforementioned one is deduced, and several numerical results illustrating its performance and validating the expected behaviour of the associated adaptive algorithm are reported.

On the lightness of the mappings satisfying generalized inverse modular inequalities

We study the lightness of some mappings satisfying generalized inverse modular inequalities and we apply the results to obtain conditions of lightness, openness and discreetness of some Sobolev mappings.