Search Results (410)

In this study, a Gaussian process model is utilized to study the Fredholm integral equations of the first kind (FIEFKs). Based on the H–$H_k$ formulation, two cases of FIEFKs are under consideration with respect to the right-hand term: discrete data and analytical expressions. In the former case, explicit approximate solutions with minimum norm are obtained via a Gaussian process model. In the latter case, the exact solutions with minimum norm in operator forms are given, which can also be numerically solved via Gaussian process interpolation. The interpolation points are selected sequentially by minimizing the posterior variance of the right-hand term, i.e., minimizing the maximum uncertainty. Compared with uniform interpolation points, the approximate solutions converge faster at sequential points. In particular, for solvable degenerate kernel equations, the exact solutions with minimum norm can be easily obtained using our proposed sequential method. Finally, the efficacy and feasibility of the proposed method are demonstrated through illustrative examples provided in this paper. Full article

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and $C(K, X)$ is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators $T:C(K, X)\to Y$ with representing measures $m:\Sigma \to L(X,Y)$, where $L(X,Y)$ is the Banach space of all operators $T:X\to Y$ and $\Sigma$ is the $\sigma$-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

For the splitting iterative scheme to solve the system of linear equations, an equivalent form in terms of descent and residual vectors is formulated. We propose an extrapolation technique using the new formulation, such that a new splitting iterative scheme (NSIS) can be simply generated from the original one by inserting an acceleration parameter preceding the descent vector. The spectral radius of the NSIS is proven to be smaller than the original one, and so has a faster convergence speed. The orthogonality of consecutive residual vectors is coined into the second NSIS, from which a stepwise varying orthogonalization factor can be derived explicitly. Multiplying the descent vector by the factor, the second NSIS is proven to be absolutely convergent. The modification is based on the maximal reduction of residual vector norm. Two-parameter and three-parameter NSIS are investigated, wherein the optimal value of one parameter is obtained by using the maximization technique. The splitting iterative schemes are unified to have the same iterative form, but endowed with different governing equations for the descent vector. Some examples are examined to exhibit the performance of the proposed extrapolation techniques used in the NSIS. Full article

Matrices have an important role in modern engineering problems like artificial intelligence, biomedicine, machine learning, etc. The present paper proposes new algorithms to solve linear problems involving finite matrices as well as operators in infinite dimensions. It is well known that the power method to find an eigenvalue and an eigenvector of a matrix requires the existence of a dominant eigenvalue. This article proposes an iterative method to find eigenvalues of matrices without a dominant eigenvalue. This algorithm is based on a procedure involving averages of the mapping and the independent variable. The second contribution is the computation of an eigenvector associated with a known eigenvalue of linear operators or matrices. Then, a novel numerical method for solving a linear system of equations is studied. The algorithm is especially suitable for cases where the iteration matrix has a norm equal to one or the standard iterative method based on fixed point approximation converges very slowly. These procedures are applied to the resolution of Fredholm integral equations of the first kind with an arbitrary kernel by means of orthogonal polynomials, and in a particular case where the kernel is separable. Regarding the latter case, this paper studies the properties of the associated Fredholm operator. Full article

This study presents an image inpainting model based on an energy functional that incorporates the $L^p$ norm of the fractional Laplacian operator as a regularization term and the $H^{-1}$ norm as a fidelity term. Using the properties of the fractional Laplacian operator, the $L^p$ norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional $L^2$ norm with the $H^{-1}$ norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks. Full article

This paper presents a high-order structure-preserving difference scheme for the nonlinear space fractional sine-Gordon equation with damping, employing the triangular scalar auxiliary variable approach. The original equation is reformulated into an equivalent system that satisfies a modified energy conservation or dissipation law, significantly reducing the computational complexity of nonlinear terms. Temporal discretization is achieved using a second-order difference method, while spatial discretization utilizes a simple and easily implementable discrete approximation for the fractional Laplacian operator. The boundedness and convergence of the proposed numerical scheme under the maximum norm are rigorously analyzed, demonstrating its adherence to discrete energy conservation or dissipation laws. Numerical experiments validate the scheme’s effectiveness, structure-preserving properties, and capability for long-time simulations for both one- and two-dimensional problems. Additionally, the impact of the parameter $\epsilon$ on error dynamics is investigated. Full article

European integration aims to achieve spatially sustainable development across the member states. However, the success of socio-economic integration is conditioned by structural features of the economies, which, hitherto, appear highly diversified across the EU countries. The paper focuses on the structural conditions of the process of income inequality convergence. It aims to identify differences in the convergence regarding the structural conditions of the economies. To fulfil the research tasks the paper classifies the 27 European member states according to their sectional employment structures using the Ward method. It then tests the appearance of beta convergence using FE panel models for the specified clusters of economies. It also considers structural change, measured by the NAV (norm of absolute value), as a determinant of income inequality convergence. The main research period covers 2009–2021. The findings of the paper confirm that income inequality convergence occurs within the groups of economies specified by different structural conditions. Importantly, the clustering according to the similarity of the employment structure overlaps with the division along the lines of the ‘new’ and ‘old’ member states, which proves the importance of historically shaped institutions for development. However, the observed convergence does not lead to improved social cohesion. Social policy, especially in the ‘new’ member states, is not able to offset the growth in market income inequality additionally stimulated by the structural changes. It can be concluded that an urgent need to design new solutions for social policy concerning structural transformation in employment in the EU emerges. Full article

This paper proposes a novel sparse twin parametric insensitive support vector regression (STPISVR) model, designed to enhance sparsity and improve generalization performance. Similar to twin parametric insensitive support vector regression (TPISVR), STPISVR constructs a pair of nonparallel parametric insensitive bound functions to indirectly determine the regression function. The optimization problems are reformulated as two sparse linear programming problems (LPPs), rather than traditional quadratic programming problems (QPPs). The two LPPs are originally derived from initial L1-norm regularization terms imposed on their respective dual variables, which are simplified to constants via the Karush–Kuhn–Tucker (KKT) conditions and consequently disappear. This simplification reduces model complexity, while the constraints constructed through the KKT conditions— particularly their geometric properties—effectively ensure sparsity. Moreover, a two-stage hybrid tuning strategy—combining grid search for coarse parameter space exploration and Bayesian optimization for fine-grained convergence—is proposed to precisely select the optimal parameters, reducing tuning time and improving accuracy compared to a singlemethod strategy. Experimental results on synthetic and benchmark datasets demonstrate that STPISVR significantly reduces the number of support vectors (SVs), thereby improving prediction speed and achieving a favorable trade-off among prediction accuracy, sparsity, and computational efficiency. Overall, STPISVR enhances generalization ability, promotes sparsity, and improves prediction efficiency, making it a competitive tool for regression tasks, especially in handling complex data structures. Full article

Photovoltaic (PV) models are hard to optimize due to their intrinsic complexity and changing operation conditions. Root mean square error (RMSE) is often given precedence in classic single-objective optimization methods, limiting them to address the intricate nature of PV model calibration. To bypass these limitations, this research proposes a novel multi-objective optimization framework balancing accuracy and robustness by considering both maximum error and the L2 norm as significant objective functions. Along with that, we introduce the Random Search Around Bests (RSAB) algorithm, which is a parameterless metaheuristic designed to be effective at exploring the solution space. The primary contributions of this work are as follows: (1) an extensive performance evaluation of the proposed framework; (2) an adaptable function to adjust dynamically the trade-off between robustness and error minimization; and (3) the elimination of manual tuning of the RSAB parameters. Rigorous testing across three PV models demonstrates RSAB’s superiority over 17 state-of-the-art algorithms. By overcoming significant issues such as premature convergence and local minima entrapment, the proposed procedure provides practitioners with a reliable tool to optimize PV systems. Hence, this research supports the overarching goals of sustainable energy technology advancements by offering an organized and flexible solution enhancing the accuracy and efficiency of PV modeling, furthering research in renewable energy. Full article

A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence and uniqueness of the $R_{\nu }$-generalized solution, its coercive and differential properties of biharmonic boundary value problems with a corner singularity. This paper establishes estimates of a function in the neighborhood of a singularity point in the norms of weighted Lebesgue spaces through its norms in weighted Sobolev spaces over the entire domain, with a minimum weight exponent. In addition, we obtain an estimate of the function norm in a boundary strip for the degeneration of a function on the entire boundary of the domain. These estimates will be useful not only for studying differential problems with singularity, but also in estimating the convergence rate of an approximate solution to an exact one in the weighted finite element method. Full article

Fractional-order four-wing (FO 4-wing) systems are of significant importance due to their complex dynamics and wide-ranging applications in secure communications, encryption, and nonlinear circuit design, making their control and stabilization a critical area of study. In this research, a novel model-free finite-time flexible sliding mode control (FTF-SMC) strategy is developed for the stabilization of a particular category of hyperchaotic FO 4-wing systems, which are subject to unknown uncertainties and input saturation constraints. The proposed approach leverages fractional-order Lyapunov stability theory to design a flexible sliding mode controller capable of effectively addressing the chaotic dynamics of FO 4-wing systems and ensuring finite-time convergence. Initially, a dynamic sliding surface is formulated to accommodate system variations. Following this, a robust model-free control law is designed to counteract uncertainties and input saturation effects. The finite-time stability of both the sliding surface and the control scheme is rigorously proven. The control strategy eliminates the need for explicit system models by exploiting the norm-bounded characteristics of chaotic system states. To optimize the parameters of the model-free FTF-SMC, a deep reinforcement learning framework based on the adaptive dynamic programming (ADP) algorithm is employed. The ADP agent utilizes two neural networks (NNs)—action NN and critic NN—aiming to obtain the optimal policy by maximizing a predefined reward function. This ensures that the sliding motion satisfies the reachability condition within a finite time frame. The effectiveness of the proposed methodology is validated through comprehensive simulations, numerical case studies, and comparative analyses. Full article

We propose a novel space-time discretization method for a time-fractional dissipative wave equation. The approach employs a structured framework in which a fully discrete formulation is produced by combining virtual elements for spatial discretization and the Newmark predictor–corrector method for the temporal domain. The virtual element technique is regarded as a generalization of the finite element method for polygonal and polyhedral meshes within the Galerkin approximation framework. To discretize the time-fractional dissipation term, we utilize the Grünwald-Letnikov approximation in conjunction with the predictor–corrector scheme. The existence and uniqueness of the discrete solution are theoretically proved, together with the optimal convergence order achieved and an error analysis associated with the $H^1$-seminorm and the $L^2$-norm. Numerical experiments are presented to support the theoretical findings and demonstrate the effectiveness of the proposed method with both convex and non-convex polygonal meshes. Full article

In this paper, we present a compact finite difference method for solving the cubic–quintic Schrödinger equation with an additional anti-cubic nonlinearity. By applying a special treatment to the nonlinear terms, the proposed method preserves both mass and energy through provable conservation properties. Under suitable assumptions on the exact solution, we establish upper and lower bounds for the numerical solution in the infinity norm, and further prove that the errors are fourth-order accurate in space and second-order in time in both the 2-norm and infinity norm. A detailed description of the nonlinear system solver at each time step is provided. We validate the proposed method through numerical experiments that demonstrate its efficiency, including fourth-order convergence (when sufficiently small time steps are used) and machine-level accuracy in the relative errors of mass and energy. Full article

Joint inversion of gravity and gravity gradient data are of paramount importance in geophysical exploration, as the integration of these datasets enhances subsurface resolution and facilitates the accurate delineation of ore body shapes and boundaries. Conventional regularization methods, such as the L₂-norm, frequently yield excessively smooth solutions, which complicates the recovery of sharp boundaries. Furthermore, disparities in data units, magnitudes, and noise levels introduce additional complexities in selecting appropriate weighting functions and inversion parameters. To address these challenges, this study proposes a greedy inversion method based on cosine similarity, which identifies the most relevant cells and reduces the complexity involved in data weighting and parameter selection. Additionally, it incorporates prior information on density limits to achieve a high-resolution and sparse solution. To further enhance the stability and accuracy of the inversion process, a pruning mechanism is introduced to dynamically detect and remove erroneously selected cells, thereby suppressing error propagation. Synthetic model experiments demonstrate that incorporating the pruning mechanism significantly improves inversion accuracy. The method not only accurately resolves models of varying volumes while avoiding local convergence issues in the presence of major anomalies, but also exhibits strong robustness against noise, successfully delineating clear boundaries even when applied to complex composite models contaminated with 10% Gaussian noise. Finally, when applied to the joint inversion of measured gravity and gravity gradient tensor data from the Vinton salt dome, the results closely align with previous studies and actual geological observations. Full article

In this work, we employ a more thorough contractive condition to examine the stability and convergence behavior of an Jungck-type iterative scheme for a pair of non-self mappings in a Banach space. Our results show that this iterative scheme has a better rate of convergence as compared to all existing Jungck-type iterative schemes. The norm of a Banach space is symmetric with respect to the origin. Symmetry can significantly influence both the theoretical underpinnings and practical convergence behavior of iterative schemes. Furthermore, we show the convergence behaviour of various Jungck-type iterative schemes with an Jungck-KF iterative scheme through an example. We also prove the data-dependence result for our proposed iterative scheme for non-self-mapping. Additionally, we provide an application of the Jungck-KF iterative scheme related to Dynamic Market Equilibrium. Full article