Inverse Inequalities Research Articles

Isogeometric Methodfor Least Squares Problem

In the framework of this work, we used the isogeometric method to solve the least squares problem in one dimension, on a curve of R d , d = 1, 2, including a semicircle. For this purpose, we presented the isogeometric method and the tools necessary for the description of this method, namely, the b splines basis, the parameterization of the R d , d = 1, 2 curve. We formulated the least squares problem which is a minimization problem. This problem was solved by using the Discontinuous Galerkin (DG) and the b splines basis as the approximation basis. The numerical method was validated by evaluating the error. For this purpose, an inverse inequality was therefore used.

On interpolation spaces of piecewise polynomials on mixed meshes

We consider fractional Sobolev spaces Hθ, θ ∈ (0,1), on 2D domains and H1-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shaperegular triangles and quadrilaterals. We prove that the norm obtained from interpolating between the discrete space equipped with the L2-norm on the one hand and the H1-norm on the other hand is equivalent to the corresponding continuous interpolation Sobolev norm, and the norm-equivalence constants are independent of meshsize and polynomial degree. This characterization of the Sobolev norm is then used to show an inverse inequality between H1 and Hθ.

On mappings with the inverse Poletsky inequality on Riemannian surfaces

On mappings with the inverse Poletsky inequality on Riemannian surfaces

Anisotropic error estimator for the Stokes–Biot system

This paper presents an a posteriori error analysis for the problem defining the interaction between a free fluid and poroelastic structure approximated by finite element methods on anisotropic meshes in Rd, d=2 or 3. Korn’s inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to nonconforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for conforming and nonconforming cases. With the obtained finite element solutions, local error indicators and a global estimator are generated, demonstrating reliability and efficiency. The efficiency is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the reliability, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media.

On the Linear Dilatation of the Mappings Satisfying an Inverse Poletsky Modular Inequality in Metric Spaces

AbstractWe study the linear dilatation of the mappings satisfying an inverse Poletsky inequality in metric spaces. We also show that under certain conditions such mappings are quasiregular.

Is pay for performance promoting inverse inequality in Irish general practice?

In 2020, the Chronic Disease Management (CDM) programme was introduced in Ireland. This programme resources GPs to review public (GMS) patients, diagnosed with eight named chronic diseases, twice yearly according to a structured protocol. This pay for performance initiative has been widely adopted by GPs. However, it is hypothesised that private patients (PPs) receive a poorer standard of care, as they may be reluctant to attend due to the cost involved. To assess whether the management of eight chronic diseases named in the CDM programme is to the same standard among both PPs and GMS patients. A retrospective audit of GP practices in the Midwest of Ireland. Data relating to 25 GMS patients and 25 PPs, matched by age, gender, and clinical condition, is collected from each practice. Patients have at least 1 of the eight named chronic diseases. Parameters include vaccination status (influenza, pneumococcal, COVID); body mass index; blood pressure; smoking status; renal function; HbA1c; lipid profile; brain natriuretic peptide (BNP) in patients with heart failure; and lung function tests in patients with COPD or asthma. COVID vaccination status acts as a control because it is freely available for both PPs and GMS patients. Preliminary results from 2 GP practices show large consistent disparities in management between PPs and GMS patients in most parameters. Limiting Pay for Performance to the care of GMS patients only, based on age or income, promotes inverse inequality. We argue that CDM care should be offered to all patients.

On the Equicontinuity of Generalized Quasiconformal Mappings by Prime Ends

We study mappings satisfying the inverse Poletsky type inequality with respect to a non-conformal modulus. It was prove that such mappings, with some additional conditions, have a continuous extension to the boundary in terms of prime ends and form equicontinuous families of mappings.

Optimal matching parameters of the inverse Hilbert-type integral inequality with quasihomogeneous kernels and their applications

UDC 517.9 By using the inverse Hölder inequality and the weight function method, we establish the inverse Hilbert-type integral inequality. In the case of a quasihomogeneous kernel, we obtain the necessary and sufficient conditions for the optimal matching parameters. Finally, their applications in the operator theory are discussed.

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy

Abstract In this paper, we study boundary controllability for the linear extension problem of a wave equation with space-dependent coefficients and having an internal degeneracy. For this purpose, we mainly focus on the well-posedness and the boundary null controllability of a relaxed version of the original problem, namely, to some degenerate transmission problem. The key ingredient is to derive direct and inverse inequalities for the associated homogeneous degenerate adjoint problem. By these inequalities, we deduce that the transmission problem has a unique solution by transposition and this solution is null controllable. Moreover, we give an explicit formula of the controllability time.

The projection algorithm for inverse quasi-variational inequalities with applications to traffic assignment and network equilibrium control problems

We consider a first-order dynamical system for solving inverse quasi-variational inequalities in finite dimensional spaces. Through an explicit time discrete version of the proposed dynamical system, we investigate the linear convergence of a projection algorithm under suitable conditions of parameters. Moreover, we investigate the application of inverse quasi-variational inequalities in traffic assignment problem and network equilibrium control problem. The numerical experiments for these practical problems confirm the linear convergence of the theoretical part. In particular, the obtained results provide a positive answer to an open question posted by S. Dey and S. Reich in Optimization, DOI: 10.1080/02331934.2023.2173525 (2023).

Regularized dynamics for monotone inverse variational inequalities in hilbert spaces

Regularized dynamics for monotone inverse variational inequalities in hilbert spaces

Notes on random optimal control equilibrium problem via stochastic inverse variational inequalities

The main objective of the paper is to analyze how policymakers influence the random oligopolistic market equilibrium problem. To this purpose, random optimal control equilibrium conditions are introduced. Since the random optimal regulatory tax is characterized by a stochastic inverse variational inequality, existence and well-posedness results on such an inequality are proved. At last a numerical example is discussed.

Controllability and stabilization of a degenerate/singular Schrödinger equation

The aim of this paper is to prove controllability and stabilization properties for a degenerate and singular Schrödinger equation with degeneracy and singularity occurring at the boundary of the spatial domain. We first address the boundary control problem. In particular, by combining multiplier techniques and compactness-uniqueness argument, we prove direct and inverse inequalities for the associated adjoint system. Consequently, via the Hilbert Uniqueness Method, we deduce exact boundary controllability for the control system under consideration in any time T>0. Moreover, we investigate the stabilization problem for this class of equations in the range of subcritical coefficients of the singular potential. By introducing a suitable linear boundary feedback, we prove that the solution decays exponentially in an appropriate energy space.

A second-order accurate and unconditionally energy stable numerical scheme for nonlinear sine-Gordon equation

In this work, a second-order finite difference method is proposed to solve a nonlinear sine-Gordon equation. The constructed implicit scheme is proved to be unconditionally energy stable. A linear iteration algorithm is used to solve this nonlinear numerical scheme, and we prove that this iteration algorithm is convergent with a negligible constraint for time step. By constructing a suitable high-precision numerical solution, and using the inverse inequality and refinement constraint Δt≤Ch, the error estimate in L∞(0,T;L∞) norm of the fully discrete scheme is obtained. Several numerical examples are given to confirm the sharpness of our theoretical analysis.

On direct and inverse Poletsky inequalities with a tangential dilatation

Abstract This article is devoted to the study of mappings defined in the plane domain. Under certain conditions, the upper estimate of the distortion of the modulus of families of paths is obtained. Similarly, the upper estimate of the modulus of the families of paths in the pre-image under the mapping is also obtained.

On inverse variational–hemivariational inequalities and its regularization

In this article, we propose the inverse variational-hemivariational inequalities problems (IVHIP) and its Tikhonov regularization theory. We establish the existence results for IVHIP and its regularized problems. Moreover, we also discuss the nonemptiness, boundedness and the asymptotic analysis of regularized solutions. Furthermore, we also investigate the existence results of the regularized solutions on the unbounded constrained through the coercive conditions. Several examples are given to support the new findings.

A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media

AbstractThis paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.

On homeomorphisms with integral constraints acting in a domain with Poincaré inequalities

We study homeomorphisms with one normalization condition that transform some domain of the Euclidean space onto a domain which is Ahlfors regular and supports Poincare inequality. We assume that, the homeomorphisms satisfy the weighted modulus Poletsky condition. If the majorant in this condition satisfies some relation written in terms of singular parameters, then the specified homeomorphisms satisfy the corresponding distortion estimate at the boundary points written in their terms. In more detail, the manuscript is written in the vein of research on mappings with bounded and finite distortion, which have been actively studied recently. Observe that, for mappings with direct and inverse Poletsky inequality, the estimates of the distortion under the mappings at the inner points of a domain may be considered known, since such estimates have been obtained by various authors over the last 10-15 years. We could, in particular, point to the results of V. Ryazanov, O. Martio, U. Srebro, E. Yakubov, M. Cristea, E. Sevost'yanov, S. Skvortsov, and O. Dovhopiatyi in this regard. It should be noted that, under certain additional assumptions, mappings with a direct Poletsky inequalities are included in the class of mappings with finite distortion by Iwaniec-Martin. At the same time, the above-mentioned estimates of distance distortion at the boundary points of the domain have not been sufficiently studied. Our manuscript is dedicated to obtaining such estimates. Of course, we require that the majorant in Poletsky's inequality satisfy certain requirements for obtaining them. The conditions written in terms of singular parameters are the most convenient and cover important partial cases. In particular, functions of finite and bounded mean oscillation by John-Nirenberg, as well as functions with the Lehto integral divergence condition can be described in terms of singular parameters, and this will be taken into account in our future research. In addition to singular parameters and the use of modulus techniques, the key point of our manuscript is spaces with Poincaré inequalities. Note that one of the characteristic features of such spaces is their fulfillment of Loewner-type inequalities, i.e., metric lower bounds of the modulus of families of curves through the diameter of the sets they join. This inequality is crucial for obtaining the result of the paper.

Local Hölder continuity of nonnegative weak solutions of inverse variation-inequality problems of non-divergence type

<abstract><p>Compared to the standard variational inequalities, inverse variational inequalities are more suitable for pricing American options with indefinite payoff. This paper investigated the initial-boundary value problem of inverse variational inequalities constituted by a class of non-divergence type parabolic operators. We established the existence and Hölder continuity of weak solutions. Since the comparison principle in the case of standard variational inequalities is no longer applicable, we constructed an integral inequality using differential inequalities to determine the global upper bound of the solution. By combining it with the continuous method, we obtained the existence of weak solutions. Additionally, by employing truncation factors, we obtained the lower bound of weak solutions in the cylindrical subdomain, thereby obtaining the Hölder continuity.</p></abstract>