Variational Problem Research Articles

On triple-adaptive projection method for bilevel split variational inequalities with CFPP constraint of finite Bregman relatively demicontractions in Banach spaces

In a p-uniformly convex and uniformly smooth Banach space E, the CFPP and VIP are utilized to indicate a common fixed-point problem and a variational inequality problem, respectively. We devise and discuss triple-adaptive projection method with inertial effect for resolving bilevel split VIP (BSVIP) with CFPP constraint of finite Bregman relatively demicontractive mappings in E. The method exploits the strong pseudocontractivity of one mapping at the upper-level VIP and the pseudomonotonicity of another mapping at the lower-level SVIP. We establish the strong convergence outcome for the proposed method under certain suitable restrictions on the algorithm parameters without the prior knowledge of the operator norm or the coefficient of the underlying operator. In the end, an illustrated instance is invoked to bear up the practicability and performability of the proposed method.

Solutions for the Nonlinear Mixed Variational Inequality Problem in the System

Our paper proposes a system of nonlinear mixed variational inequality problems (SNMVIPs) on Banach spaces. Under suitable assumptions, using the K-Fan fixed point theorem and Minty techniques, we demonstrate that the solution set to the SNMVIP is nonempty, weakly compact, and unique. Additionally, we suggest a stability result for the SNMVIPs by perturbing the duality mappings. Furthermore, we present an optimal control problem that is governed by the SNMVIPs and show that it can be solved.

A semi-analytical spectral element model for guided wave propagation in composite laminated conical shells

The conical shell is a typical non-uniform waveguide with its circumferential radius varies linearly. This work aims to provide physical insights for the composite laminated conical shell in terms of waves. A semi-analytical spectral element model is presented, which adopts the Fourier series in the circumferential direction and the higher-order shape functions in the axial direction. The first-order shear deformation theory and Hamilton principle are employed to derive the energy functions of the conical shell, and they are discretized using the spectral elements to achieve a weak-form variational problem. The dispersion equation and the group velocity expression are derived via the piecewise approximation concept and Bloch's periodic theorem. The accuracy of the developed spectral element is verified by comparison with published results. The deformation evolution mechanisms of the composite conical shell are revealed, and the wave propagation characteristics of uniform and ply drop-off composite conical shells are investigated. The results demonstrated that wavenumber distribution at 0–5 kHz is spatially dependent, and wavenumber becomes cut off and bifurcate near the cone vertex. When the frequency exceeds 2 kHz, the deformation of conical shell changes from rigid to flexible, and the structure tends to be a uniform waveguide. Compared to the geometric curvature effect, the asymmetric lay-up leads to more significant mode coupling.

A reliable numerical algorithm mixed with hypergeometric function for analyzing fractional variational problems

A reliable numerical algorithm mixed with hypergeometric function for analyzing fractional variational problems

Improved Equations of the Lagrange Top and Examples of Analytical Solutions

Equations of a heavy rotating body with one fixed point can be deduced starting from a variational problem with holonomic constraints. When applying this formalism to the particular case of a Lagrange top, in the formulation with a diagonal inertia tensor the potential energy has a more complicated form as compared with that assumed in the literature on dynamics of a rigid body. This implies the corresponding improvements in equations of motion. Therefore, we revised this case, presenting several examples of analytical solutions to the improved equations. The case of precession without nutation has a surprisingly rich relationship between the rotation and precession rates, which is discussed in detail.

Cornering gravitational entropy

We present a new derivation of gravitational entropy functionals in higher-curvature theories of gravity using corner terms that are needed to ensure well-posedness of the variational principle in the presence of corners. This is accomplished by cutting open a manifold with a conical singularity into a wedge with boundaries intersecting at a corner. Notably, our observation provides a rigorous definition of the action of a conical singularity that does not require regularization. For Einstein gravity, we compute the Rényi entropy of gravitational states with either fixed-periodicity or fixed-area boundary conditions. The entropy functional for fixed-area states is equal to the corner term, whose extremization follows from the variation of the Einstein action of the wedge under transverse diffeomorphisms. For general Lovelock gravity the entropy functional of fixed-periodicity states is equal to the Jacobson-Myers (JM) functional, while fixed-area states generalize to fixed-JM-functional states, having a flat spectrum. Extremization of the JM functional is shown to coincide with the variation of the Lovelock action of the wedge. For arbitrary F(Riemann) gravity, under special periodic boundary conditions, we recover the Dong-Lewkowycz entropy for fixed-periodicity states. Since the variational problem in the presence of corners is not well-posed, we conjecture the generalization of fixed-area states does not exist for such theories without additional boundary conditions. Thus, our work suggests the existence of entropy functionals is tied to the existence of corner terms which make the Dirichlet variational problem well-posed.

Cross-modal semantic aligning and neighbor-aware completing for robust text–image person retrieval

Most existing text–image person re-identification (TIReID) methods are performed in an ideal environment where both image and text instances are fully intact and identity annotated. However, in real-world open environments, these ideal assumptions often cannot be satisfied. In this study, we are the first to explore how to enhance the robustness of the TIReID model to better adapt to open environments, focusing on two aspects: 1) unlabeled multi-modal data, and 2) incomplete multi-modal data. Here, unlabeled multi-modal data faces two key issues: intra-class variations and cross-modal aligning problems, while incomplete data leads to data semantic inconsistency problems and severe performance degradation. In order to tackle the above issues, we propose a novel Cross-modal Semantic Aligning and Neighbor-aware Completing (CANC) method for robust text–image person re-identification (RTIReID). Specifically, to tackle the intra-class variation problem, we propose intra-view prototype contrastive matching for image and text modalities, which helps the model learn intra-modal discriminative representations while increasing inter-class distance and suppressing intra-class variation. Furthermore, to resolve the cross-modal alignment problem, we propose the cross-view instance projection matching to establish the relationship between different modal representations. Additionally, regarding how to effectively handle incomplete samples, we introduce nearest neighbor consistent completion, which can restore high-quality completion features. We adopt CLIP-ViT and CLIP-Xformer as image and text encoders respectively, and Rank-k metrics (k=1,5,10) as primary evaluation metrics. Extensive experiments on several public datasets demonstrate that our method can effectively address TIReID in open environments, consistently surpassing several SOTA TIReID methods.

Reliability and Unreliability Aspects of Travel Time Analysis on the Stochastic Network Using the Target-Oriented Perspective

This study proposes a target-oriented method to study travelers’ route choice behavior under travel time variability, and discusses the resulted equilibrium flow patterns. Both travel time reliability and travel time unreliability are considered in this new method, and accordingly, there are two targets. The first one is target for travel time to ensure travel time reliability, and based on this target, another one is target for excess delay to mitigate travel time unreliability. In this model, travel time and excess delay (i.e., the random vector) are stochastically correlated with each other, which is modeled with the copula function based on Sklar’s Theorem, and the exact form of the copula is obtained by the proved comonotonicity relationship of this random vector. The target interaction, i.e., the complementarity relationship, is also modeled based on the utility functions, the meaning of which is that travelers have the will to make more targets achieved so as to obtain more utility. Furthermore, with this model, this paper formulates the user equilibrium as a variational inequality problem to study the long-term effect of the route choice behavior, and solves it with the method of successive average. Finally, numerical testings on the traffic network are conducted to show the convergence of the solution algorithm, and to illustrate the impact of targets on the equilibrium results. Results show that the flow change can be five times more than that with less risk-averse travelers.

Minimizers of L^2-critical inhomogeneous variational problems with a spatially decaying nonlinearity in bounded domains

We consider the minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $\Omega$ of $\mathbb{R}^{N}$ which contains $0$. We prove that there is a threshold $a^{*}> 0$ such that minimizers exist for $0< a< a^{*}$ and the minimizer does not exist for any $a> a^{*}$. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold $a^*$ depending on the value of $V(0)$, where $V(x)$ denotes the trapping potential. Moreover, under some suitable assumptions on $V(x)$, based on a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^*$, we prove local uniqueness of minimizers when $a$ is close enough to $a^*$.

Two novel algorithms for solving variational inequality problems governed by fixed point problems and their applications

Two novel algorithms for solving variational inequality problems governed by fixed point problems and their applications

Stability of hybrid time integration scheme for Lord–Shulman thermopiezoelectricity

Based on the existing initial boundary value and variational problems of Lord–Shulman thermopiezoelectricity a transient analysis of the behavior of piezoelectric materials is performed numerically. For space discretization of the variational problem the finite element method is used and for time discretization a hybrid time integration scheme is constructed on the basis of Newmark scheme for hyperbolic equation and generalized trapezoidal rule for parabolic equations. The unconditional stability of the developed time integration scheme is proved for some specific values of the scheme parameters by making use of energy balance law for the obtained time-discretized variational problem. Finally, the applicability of the constructed numerical scheme is demonstrated by the results of the numerical experiment which are compared with the ones available in literature.

A Legendre Spectral-Element Method to Incorporate Topography for 2.5D Direct-Current-Resistivity Forward Modeling

An effective and accurate solver for the direct-current-resistivity forward-modeling problem has become a cutting-edge research topic. However, computational limitations arise due to the substantial amount of data involved, hindering the widespread use of three-dimensional forward modeling, which is otherwise considered the most effective approach for identifying geo-electrical anomalies. An efficient compromise, or potentially an alternative, is found in two-and-a-half-dimensional (2.5D) modeling, which employs a three-dimensional current source within a two-dimensional subsurface medium. Consequently, a Legendre spectral-element algorithm is developed specifically for 2.5D direct-current-resistivity forward modeling, taking into account the presence of topography. This numerical algorithm can combine the complex geometric flexibility of the finite-element method with the high precision of the spectral method. To solve the wavenumber-domain electrical potential variational problem, which is converted into the two-dimensional Helmholtz equation with mixed boundary conditions, the Gauss–Lobatto–Legendre (GLL) quadrature is employed in all discrete quadrilateral spectral elements, ensuring identical Legendre polynomial interpolation and quadrature points. The Legendre spectral-element method is applied to solve a two-dimensional Helmholtz equation and a resistivity half-space model. Numerical experiments demonstrate that the proposed approach yields highly accurate numerical results, even with a coarse mesh. Additionally, the Legendre spectral-element algorithm is employed to simulate the apparent resistivity distortions caused by surface topographical variations in the direct-current resistivity Wenner-alpha array. These numerical results affirm the substantial impact of topographical variations on the apparent resistivity data obtained in the field. Consequently, when interpreting field data, it is crucial to consider topographic effects to the extent they can be simulated. Moreover, our numerical method can be extended and implemented for a more accurate computation of three-dimensional direct-current-resistivity forward modeling.

Hardy inequalities for large fermionic systems

Given 0<s<\frac{d}{2} with s\leq 1 , we are interested in the large N -behavior of the optimal constant \kappa_{N} in the Hardy inequality \sum_{n=1}^{N} (-\Delta_{n})^{s} \geq \kappa_{N} \sum_{n<m}|X_{n}-X_{m}|^{-2s} , when restricted to antisymmetric functions. We show that N^{1-\frac{2s}{d}}\kappa_{N} has a positive, finite limit given by a certain variational problem, thereby generalizing a result of Lieb and Yau related to the Chandrasekhar theory of gravitational collapse.

P-Parabolicity and a Generalized Bochner’s Method with Applications

The notions of function growth and sharp global integral estimates in p-harmonic geometry in Wei (Contemp Math 756:247–269, 2020) and Wei et al. (Sharp estimates on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}-harmonic functions with applications in biharmonic maps) naturally lead to a generalized uniformization theorem and a generalized Bochner’s method. These tools enable one to explore various geometric and variational problems in complete noncompact manifolds of arbitrary dimensions. In particular, we find the first set of nontrivial geometric quantities that are p-subharmonic and p-superharmonic functions (cf. Sect. 4.2). As further applications, we establish Liouville theorems for nonnegative A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document}-superharmonic functions (cf. Theorem 2.1) and for p-harmonic morphisms (cf. Theorem 4.3), Picard type theorems (cf. Sect. 4.3), existence theorems of harmonic maps (cf. Theorem 4.6), and a solution of the generalized Bernstein problem under p-parabolicity condition (without any volume growth condition, cf. Theorem 5.1) or under p-moderate volume growth (2.4) (cf. Corollary 5.1). These findings generalize and extend the work of Schoen–Simon–Yau under volume growth condition (0.1) which is due to (i)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {i}})$$\\end{document} Corollary 2.1(iv), which states that a manifold with p-moderate volume growth (2.4) must be p-parabolic (generalizing a result of Cheng and Yau (Commun Pure Appl 28(3):333–354, 1975) for the case p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document}, F(r)≡1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(r)\\equiv 1$$\\end{document}), (ii)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {ii}})$$\\end{document} Example 2.1 of a p-parabolic manifold with exponential volume growth, and (iii)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {iii}})$$\\end{document} volume growth condition (0.1) ⟹(implies)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overset{(\ ext {implies})}{\\Longrightarrow }$$\\end{document}p-moderate volume growth (2.4). Furthermore, we offer applications to p-harmonic maps and stable p-harmonic maps. Notably, these outcomes address an intriguing question posed by Kobayashi if p-subharmonic functions are valuable tools for studying p-harmonic maps or related topics (cf. in: Kobayashi, Midwest Geometry Conference, University of Oklahoma, Oklahoma, 2006. Private Communication, 1996), and answer in the affirmative. Other applications of the notions and estimates to biharmonic maps, isometric immersions, holomorphic functions on Kähler manifolds, generalized harmonic forms, and Yang–Mills fields on complete noncompact manifolds can be found in Wei et al. (Sharp estimates on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal A$$\\end{document}-harmonic functions with applications in biharmonic maps), Chen and Wei (Glasg Math J 51(3):579–592, 2009), Wei (Bull Transilv Univ Brasov Ser III 1(50):415–453, 2008), Wei (Growth estimates for generalized harmonic forms on noncompact manifolds with geometric applications, Contemp Math 756 American Mathematical Society, Providence, 247–269, 2020), Wei (Sci China Math 64(7):1649–1702, 2021) and Wei (Isolation phenomena for Yang–Mills fields on complete manifolds). We generate the work of Schoen–Simon–Yau under p-parabolic condition, in which the result can be used in other types of new manifolds we found, by an extrinsic average variational method we proposed (cf. Wei in Rom J Math Comput Sci 13(2):100–124, 2023; in: Wei, Connecting Poincare inequality with Sobolev inequality on Riemannian manifolds, Int Electron J Geom 17(1):290–305, 2024).

Adaptive synergetic control for electronic throttle valve system

AbstractThis study has developed adaptive synergetic control (ASC) algorithm to control the angular position of moving plate in the electronic throttle valve (ETV) system. This control approach is inspired by synergetic control theory. The adaptive controller has addressed the problem of variation in systems parameters. The control design includes two elements: the control law and adaptive law. The adaptive law is developed based on Lyupunov stability analysis of the controlled system, and it is responsible for estimating the potential uncertainties in the system. The effectiveness of the proposed adaptive synergetic control has been verified by numerical simulation using MATLAB/Simulink. The results showed that the ASC algorithm could give good tracking performance in the presence of uncertainty perturbations. In addition, a comparison study has been made to compare the tracking performance of ASC and that based on conventional synergetic control (CSC) for the ETV system. The simulated results showed that the performance of ASC outperforms that based on CSC. Moreover, the results showed that the estimation errors between the actual and estimated uncertainties are bounded and there is no drift in the developed adaptive law of ASC.

The system of automatic control of the substrate temperature in the installation for the production of film material spray pyrolysis method

A system of automatic control of the substrate temperature is proposed, which ensures the required accuracy of temperature maintenance and its minimum gradient over the surface when obtaining transparent conductive coatings by spray pyrolysis. During pyrolysis, aerosol droplets are transferred to a heated substrate; at the right temperature, they decompose, and then synthesize or melt on a hot substrate. The ideal temperature regime for evaporation of aerosol droplets will be the mode when a drop appears at the substrate immediately after evaporation of the solvent from it. Precise control of the substrate temperature and temperature gradient ensures this mode. The functional scheme of the proposed automatic control system consists of an automatic temperature controller, a power regulator, an infrared heater on halogen lamps, and a temperature sensor. The variational problem of minimizing power consumption when the infrared heater is turned on has been solved. This activation eliminates current emissions and thereby increases the service life of the heater. It is proposed to connect the heater to a current source in order to eliminate current emissions, thereby ensuring the optimal switching mode. The synthesis and analysis of the specified automatic control system has been performed, its time and frequency parameters have been calculated. Acceptable performance, zero static control error and zero overshoot, a significant margin of stability in phase and amplitude have been confirmed. The use of such an automatic temperature control system contributes to the creation of an optimal mode of evaporation of aerosol droplets, allows you to stabilize the substrate temperature and temperature gradient, thereby ensuring the stability of the characteristics of the resulting films. The investigated transparent conductive coatings are part of glasses with adjustable transparency, and are also used as layers of solar cells.

Widely linear RLS equalizer with variable forgetting factor and UD factorization for mud pulse telemetry

To address the problems of time variations and high interferences in continuous-wave mud pulse telemetry channels, a widely linear recursive least squares decision feedback equalizer with a variable forgetting factor was proposed, a smaller steady-state error was achieved, and the tracking performance of the equalizer in time-varying mud channels was enhanced. Furthermore, employing UD factorization in the recursive algorithm improves the computational stability and ensures robust performance. The simulation results indicate that the proposed equalizer enhances the convergence speed by 46 %. The experimental findings demonstrate that within a 3000 m mud pulse telemetry channel, the proposed equalizer reduces the steady-state mean squared error by 3 to 4 dB and increases the error-free transmission rate from 3 bps to 6 bps. These advancements significantly enhance the performance of continuous wave mud pulse telemetry systems.

Levitin–Polyak well-posedness of split multivalued variational inequalities

ABSTRACT We introduce and study the split multivalued variational inequality problem (SMVIP) and the parametric SMVIP. We examine, in particular, Levitin–Polyak well-posedness of SMVIPs and parametric SMVIPs in Hilbert spaces. We provide several examples to illustrate our theoretical results. We also discuss several important special cases.

Fiberwise Symmetrizations for Variational Problems on Fibred Manifolds

Fiberwise Symmetrizations for Variational Problems on Fibred Manifolds

A Shape-Preserving Variational Spline Approximation Problem for Hole Filling in Generalized Offset Surfaces

In the study of some real cases, it is possible to encounter well-defined geometric conditions, of an industrial or design type—for example, the case of a specific volume within each of several holes. In most of these cases, it is recommended to fulfil a function defined in a domain in which information is missing in one or more sub-domains (holes) of the global set, where the function data are not known. The problem of filling holes or completing a surface in three dimensions appears in many fields of computing, such as computer-aided geometric design (CAGD). A method to solve the shape-preserving variational spline approximation problem for hole filling in generalized offset surfaces is presented. The existence and uniqueness of the solution of the studied method are established, as well as the computation, and certain convergence results are analyzed. A graphic and numerical example complete this study to demonstrate the effectiveness of the presented method. This manuscript presents the resolution of a complicated problem due to the study of some criteria that can be traduced via an approximation problem related to generalized offset surfaces with holes and also the preservation of the shape of such surfaces.