Inverse Inequalities Research Articles

A proximal neurodynamic model for solving inverse mixed variational inequalities

This paper proposes a proximal neurodynamic model (PNDM) for solving inverse mixed variational inequalities (IMVIs) based on the proximal operator. It is shown that the PNDM has a unique continuous solution under the condition of Lipschitz continuity (L-continuity). It is also shown that the equilibrium point of the proposed PNDM is asymptotically stable or exponentially stable under some mild conditions. Finally, three numerical examples are presented to illustrate effectiveness of the proposed PNDM.

Reverses Hardy-type inequalities via Jensen integral inequality

The integral inequalities concerning the inverse Hardy inequalities have been studied by a large number of authors during this century, of these articles have appeared, the work of Sulaiman in 2012, followed by Banyat Sroysang who gave an extension to these inequalities in 2013. In 2020 B. Benaissa presented a generalization of inverse Hardy inequalities. In this article, we establish a new generalization of these inequalities by introducing a weight function and a second parameter. The results will be proved using the Hölder inequality and the Jensen integral inequality. Several the reverses weighted Hardy’s type inequalities and the reverses Hardy’s type inequalities were derived from the main results.

On nonhomeomorphic mappings with the inverse Poletsky inequality

The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of the Poletsky type. It is proved that mappings of this type are logarithmically Hölder-continuous under the condition that the function Q responsible for a distortion of the modulus of the families of curves is integrable. A continuous extension of the indicated mappings to the boundary is obtained. In addition, the conditions under which the families of mentioned mappings are equicontinuous inside the domain and at its boundary are studied.

Socioeconomic Position and the Cascade From SARS-CoV-2 Testing to COVID-19 Mortality: Population-Based Analysis of Swiss Surveillance Data

Background: The inverse care law states that disadvantaged populations need more health care than advantaged populations but receive less. Gaps in COVID-19 related health care and infection control are not well understood at present. We examined inequalities in health in the care cascade from testing for SARS-CoV-2 to COVID-19-related hospitalisation, ICU admission and death in Switzerland, a wealthy country strongly affected by the pandemic.Methods: We analysed surveillance data reported to the Federal Office of Public Health from March 1, 2020 to April 16, 2021 and 2018 population data. We geocoded residential addresses of notifications to determine the Swiss neighbourhood index of socioeconomic position (Swiss-SEP). The index describes 1.27 million small neighbourhoods of about 50 households, based on rent per square meter, education and occupation of household heads, and crowding. We used negative binomial regression models to calculate incidence rate ratios (IRR) with 95% credible intervals [CrI] of the association between ten groups of the Swiss-SEP index defined by deciles (1=lowest, 10=highest) and outcomes. Models were adjusted for sex, age, canton and wave of the epidemic (before or after June 8, 2020). We used three different denominators: the general population, the number of tests, and the number of positive tests.Findings: Analyses were based on 4,129,636 tests, 609,782 positive tests, 26,143 hospitalisations, 2,432 ICU admissions, 9,383 deaths, and 8,221,406 residents. Comparing the highest with the lowest Swiss-SEP group, and using the general population as the denominator, more tests were performed among people living in neighbourhoods of highest socioeconomic position (adjusted IRR 1.18; 95%CrI 1.02-1.36). Among tested people, test positivity was lower (adjusted IRR 0.75; 0.69-0.81) in neighbourhoods of highest socioeconomic position. Among people testing positive, the corresponding IRRs for hospitalisation, ICU admission and death were 0.68 (0.62-0.74), 0.54 (0.43-0.70) and 0.86 (0.76-0.99). The associations between neighbourhood socioeconomic position and outcomes were stronger in younger age groups, and there was heterogeneity between areas.Interpretation: The inverse care law and socioeconomic inequalities are evident in Switzerland throughout the care cascade. People living in neighbourhoods of low socioeconomic position were less likely to be tested but more likely to test positive, be hospitalised, or die. It is essential to continue to monitor testing for SARS-CoV-2, access and uptake of COVID-19 vaccination and outcomes of COVID-19. Governments and health care systems should address this pandemic of inequality by taking measures to reduce health inequalities in response to the SARS-CoV-2 pandemic.Funding Information: Federal Office of Public Health, Swiss National Science Foundation (grant 189498), European Union Horizon 2020 (grant 101003688).Declaration of Interests: The authors declare no competing interests.Ethics Approval Statement: We conducted this research using surveillance data according to the Swiss law on communicable diseases (EpG, SR 818.101). No ethical approval was required.

Unconditionally optimal error estimates of BDF2 Galerkin method for semilinear parabolic equation

AbstractIn this paper, a 2‐step backward differentiation formula (BDF2) Galerkin method is investigated for semilinear parabolic equation. More precisely, the second‐order time‐stepping scheme is used for time discretization and the piecewise linear continuous Galerkin method is employed for spatial discretization, respectively. Optimal error estimates in L2 and H1‐norms are obtained without any restriction on the time‐step size, while previous works always require certain conditions on time step‐size. The key to our analysis is to derive a uniform boundness of the numerical solution in energy norm so as to avoid the inverse inequality used in the usual convergence analysis of the finite element methods. Numerical experiments are carried out to confirm the theoretical analysis.

Local behavior of mappings of metric spaces with branching

The local behavior of mappings with the inverse Poletsky inequality between metric spaces is studied. The case where one of the spaces satisfies the condition of weak sphericalization, is similar to the Riemannian sphere (extended Euclidean space), and is locally linearly connected under a mapping is considered. It is proved that the equicontinuity of the corresponding families of mappings of two domains, one of which is a domain with a weakly flat boundary, and another one is a fixed domain with a compact closure, the corresponding weight in the main inequality being supposed to be integrable.

Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings

The goal of this paper is further to study a kind of generalized vector inverse quasi-variational inequality problems and to obtain error bounds in terms of the residual gap function, the regularized gap function, and the global gap function by utilizing the relaxed monotonicity and Hausdorff Lipschitz continuity. These error bounds provide effective estimated distances between an arbitrary feasible point and the solution set of generalized vector inverse quasi-variational inequality problems.

Averaged null controllability for some hyperbolic equations depending on a parameter

The aim of this paper is to prove the averaged null controllability property for a wave equation with an unknown velocity of propagation parameter and for parameter-dependent vibrating plate equation under the effect of a boundary control. The choice of the Hilbert uniqueness method seems to be the best-adapted method to our theory, where the key point to prove the desired result is an averaged inverse inequality (averaged observability inequality). Consequently, we'll prove the desired property for a large time enough and we'll design a single control chosen independently of the parameter value transferring the average of the state to the origin.

Simple and robust equilibrated fluxa posterioriestimates for singularly perturbed reaction–diffusion problems

We consider energy norma posteriorierror analysis of conforming finite element approximations of singularly perturbed reaction–diffusion problems on simplicial meshes in arbitrary space dimension. Using an equilibrated flux reconstruction, the proposed estimator gives a guaranteed global upper bound on the error without unknown constants, and local efficiency robust with respect to the mesh size and singular perturbation parameters. Whereas previous works on equilibrated flux estimators only considered lowest-order finite element approximations and achieved robustness through the use of boundary-layer adapted submeshes orviacombination with residual-based estimators, the present methodology applies in a simple way to arbitrary-order approximations and does not request any submesh or estimators combination. The equilibrated flux is obtainedvialocal reaction–diffusion problems with suitable weights (cut-off factors), and the guaranteed upper bound features the same weights. We prove that the inclusion of these weights is not only sufficient but also necessary for robustness of any flux equilibration estimate that does not employ submeshes or estimators combination, which shows that some of the flux equilibrations proposed in the past cannot be robust. To achieve the fully computable upper bound, we derive explicit bounds for some inverse inequality constants on a simplex, which may be of independent interest.

A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations

This paper is concerned with developing a medius error analysis for several nonconforming virtual element methods (VEMs) for the Poisson equation and the biharmonic equation in two dimensions, with the family of polygonal meshes satisfying a very general geometric assumption given in Brezzi et al. (2009) and Chen and Huang (2018). After some technical derivation, the inverse inequalities and norm equivalence are derived for some conforming VEMs. With the help of these results and following some ideas in Gudi (2010), we obtain medius error estimates for the nonconforming VEMs under discussion, which are optimal up to the regularity of the weak solution. Such estimates also imply that the nonconforming VEMs are convergent even if the exact solution only belongs to the admissible space while the right-hand side of the related equation has some additional regularity.

Intersecting Lorenz curves and aversion to inverse downside inequality

This paper defines and characterizes the concept of an increase in inverse downside inequality and show that, when the Lorenz curves of two income distributions intersect, how the change from one distribution to the other is judged by an inequality index exhibiting inverse downside inequality aversion often depends on the relative strengths of its aversion to inverse downside inequality and inequality aversion. For the class of linear inequality indices, of which the Gini coefficient is a member, a measure characterizing the strength of an index’s aversion to inverse downside inequality against its own inequality aversion is shown to determine the ranking by the index of two distributions whose Lorenz curves cross once. The precise condition under which the same result generalizes to the case of multiple-crossing Lorenz curves is also identified.

On nonhomeomorphic mappings with the inverse Poletsky inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

In this paper we study direct and inverse approximation inequalities in Lp(Rd), 1<p<∞, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function f via the fractional powers of the Dunkl Laplacian of approximants of f. Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier–Dunkl inequalities are derived.

Reduced Basis Greedy Selection Using Random Training Sets

Reduced bases have been introduced for the approximation of parametrized PDEs in applications where many online queries are required. Their numerical efficiency for such problems has been theoretically confirmed in Binev et al. (SIAM J. Math. Anal. 43 (2011) 1457–1472) and DeVore et al. (Constructive Approximation 37 (2013) 455–466), where it is shown that the reduced basis space Vn of dimension n, constructed by a certain greedy strategy, has approximation error similar to that of the optimal space associated to the Kolmogorov n-width of the solution manifold. The greedy construction of the reduced basis space is performed in an offline stage which requires at each step a maximization of the current error over the parameter space. For the purpose of numerical computation, this maximization is performed over a finite training set obtained through a discretization of the parameter domain. To guarantee a final approximation error ε for the space generated by the greedy algorithm requires in principle that the snapshots associated to this training set constitute an approximation net for the solution manifold with accuracy of order ε. Hence, the size of the training set is the ε covering number for M and this covering number typically behaves like exp(Cε−1/s) for some C &gt; 0 when the solution manifold has n-width decay O(n−s). Thus, the shear size of the training set prohibits implementation of the algorithm when ε is small. The main result of this paper shows that, if one is willing to accept results which hold with high probability, rather than with certainty, then for a large class of relevant problems one may replace the fine discretization by a random training set of size polynomial in ε−1. Our proof of this fact is established by using inverse inequalities for polynomials in high dimensions.

A posteriori error analysis of a mixed virtual element method for a nonlinear Brinkman model of porous media flow

In this paper we present an a posteriori error analysis of a mixed-VEM discretization for a nonlinear Brinkman model of porous media flow, which has been proposed by the authors in a previous work. Therein, the system is formulated in terms of a pseudostress tensor and the velocity gradient, whereas the velocity and the pressure of the fluid are computed via postprocessing formulas. Furthermore, the well-posedness of the associated augmented formulation along with a priori error bounds for the discrete scheme also were established. We now propose reliable and efficient residual-based a posteriori error estimates for a computable approximation of the virtual solution associated to the aforementioned problem. The resulting error estimator is fully computable from the degrees of freedom of the solutions and applies on very general polygonal meshes. For the analysis we make use of a global inf–sup condition, Helmholtz decomposition, local approximation properties of interpolation operators and inverse inequalities together with localization arguments based on bubble functions. Finally, we provide some numerical results confirming the properties of our estimator and illustrating the good performance of the associated adaptive algorithm.

Partially penalized IFE methods and convergence analysis for elasticity interface problems

In this paper, some partial penalized immersed finite element methods (PPIFEMs) are proposed and analyzed for solving elasticity interface problems. Through verifying the inverse trace inequality on the interface edges, the optimal convergence in the energy norm is derived. A new test function is constructed to obtain the discrete inf–sup condition of the penalty-free nonsymmetric PPIFEM and is utilized in the proof of the optimal convergence. Furthermore, the effect of the Lamé parameters on convergence is also studied. Various numerical examples and comparisons are provided to confirm the theoretical results.

Error bounds for mixed set-valued vector inverse quasi-variational inequalities

The purpose of this paper is to introduce and study the mixed set-valued vector inverse quasi-variational inequality problems (MSVIQVIPs) and to obtain error bounds for this kind of MSVIQVIP in terms of 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 mixed set-valued vector inverse quasi-variational inequality problem. The results presented in the paper improve and generalize some recent results.

A minimax approach for inverse variational inequalities

In this work, we characterize the existence of a solution for a certain variational inequality by means of a classical minimax theorem. In addition, we propose a numerical algorithm for the solution of an inverse problem associated with a variational inequality. To this end we state a collage-type result in this variational framework.

Robust admissible consensus of singular multi‐agent systems using sliding mode approach

AbstractThe admissible consensus problem of singular multi‐agent systems (SMASs) with mismatched norm‐bounded parametric uncertainties and unknown norm‐bounded disturbances is investigated by an integral sliding mode technique. Under a directed spanning tree network of the state transmission graph, the error states are defined. Then a linear matrix inequality (LMI)‐based sufficient condition is presented to guarantee regularity, free impulse, quadratic stability and H∞ performance of the sliding mode dynamics of the error states. Next, an integral sliding manifold parameter is designed to decrease the order of the LMI caused by the parametric uncertainties, and matrix inversion formulas and inequality techniques are applied to reduce the conservativeness of the LMI. Finally, the control laws, relying on the error states, the property of the state transmission graph and the upper norm bound of disturbances, are presented to ensure the state trajectories of the resulting closed‐loop system to achieve robust admissible consensus with H∞ performance. A simulation example is provided to show the effectiveness of the proposed method.

An inertial projection neural network for solving inverse variational inequalities

A novel inertial projection neural network (IPNN) is proposed for solving inverse variational inequalities (IVIs) in this paper. It is shown that the IPNN has a unique solution under the condition of Lipschitz continuity and that the solution trajectories of the IPNN converge to the equilibrium solution asymptotically if the corresponding operator is co-coercive. Finally, several examples are presented to illustrtae the effectiveness of the proposed IPNN.