Hilbert Space Setting Research Articles

Study on Orthogonal Sets for Birkhoff Orthogonality

We introduce the notion of orthogonal sets for Birkhoff orthogonality, which we will call Birkhoff orthogonal sets in this paper. As a generalization of orthogonal sets in Hilbert spaces, Birkhoff orthogonal sets are not necessarily linearly independent sets in finite-dimensional real normed spaces. We prove that the Birkhoff orthogonal set A={x1,x2,…,xn}(n≥3) containing n−3 right symmetric points is linearly independent in smooth normed spaces. In particular, we obtain similar results in strictly convex normed spaces when n=3 and in both smooth and strictly convex normed spaces when n=4. These obtained results can be applied to the mutually Birkhoff orthogonal sets studied in recently.

Homeier-like methods for regularization of nonlinear ill-posed equations in Hilbert space

Homeier-like methods have been extensively studied in the literature for solving non-linear operator equation of the form F(x)=y in the Banach or Hilbert space setting. Motivated by the work of George et al. (2023), we extend some class of Homeier-like methods to approximate ill-posed operator equation in Hilbert space. Two Homeier-like iterative schemes are proposed for approximating the nonlinear ill-posed operator equation F(x)=y in Hilbert space, where F is monotone. Lavrentiev regularization method and recurrence relation are employed for convergence analysis. The regularization parameter is chosen by apriori parameter choice strategy with new source condition as studied by George et al. (2023). We also supply a numerical example for comparison study of the proposed method with some of the Lavrentiev regularization methods in the literature.

A Modified Viscosity-Type Self-Adaptive Iterative Algorithm for Common Solution of Split Problems with Multiple Output Sets in Hilbert Spaces

A modified viscosity-type self-adaptive iterative algorithm is presented in this study, having a strong convergence theorem for estimating the common solution to the split generalized equilibrium problem along with the split common null point problem with multiple output sets, subject to some reasonable control sequence restrictions. The suggested algorithm and its immediate consequences are also discussed. The effectiveness of the proposed algorithm is finally demonstrated through analytical examples. The findings presented in this paper will help to consolidate, extend, and improve upon a number of recent findings in the literature.

An algorithmic approach to solving split common fixed point problems for families of demicontractive operators in Hilbert spaces

In this article, we propose a new iterative algorithm for solving split common fixed point problems for finite families of single-valued demicontractive mappings in the setting of real Hilbert spaces. In order to solve this problem, we introduce a Halpern method with with an Armijo-line search for approximating the solution of the aforementioned problems. We establish a strong convergence result of the proposed method and display some numerical examples to illustrate the performance of our main result. Our result complements and generalizes some related results in literature.

An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution set of the split variational inequality problem with multiple output sets in real Hilbert spaces and establish a new iterative algorithm for it. Our algorithms are accelerated by the inertial technique and eliminate the dependence on the norm of the transformation operators and the strongly monotone and Lipschitz continuous constants of the involved operator by employing a self-adaptive step size criterion. The strong convergence result is given under some mild conditions widely used in the convergence analysis. Two corollaries for the solutions of the split variational inequality problem and the split feasibility problem with multiple output sets are also obtained using our main result. Finally, some numerical experiments have been conducted to illustrate the effectiveness of the proposed algorithms and compare them with the related ones.

Orthogonal Polynomials and Operator Convergence in Hilbert Spaces: Norm-Attainability, Uniform Boundedness, and Compactness

This research paper investigates the convergence properties of operators constructed from orthogonal polynomials in the context of Hilbert spaces. The study establishes norm-attainability and explores the uniform boundedness of these operators, extending the analysis to include complex-valued orthogonal polynomials. Additionally, the paper uncovers connections between operator compactness and the convergence behaviors of orthogonal polynomial operators, revealing how sequences of these operators converge weakly to both identity and zero operators. These results advance our understanding of the intricate interplay betweenalgebraic and analytical properties in Hilbert spaces, contributing to fields such as functional analysis and approximation theory. The research sheds new light on the fundamental connections underlying the behavior of operators defined by orthogonal polynomials in diverse Hilbert space settings.

Inertial Iterative Algorithms for Split Variational Inclusion and Fixed Point Problems

This paper aims to present two inertial iterative algorithms for estimating the solution of split variational inclusion (SpVIsP) and its extended version for estimating the common solution of (SpVIsP) and fixed point problem (FPP) of a nonexpansive mapping in the setting of real Hilbert spaces. We establish the weak convergence of the proposed algorithms and strong convergence of the extended version without using the pre-estimated norm of a bounded linear operator. We also exhibit the reliability and behavior of the proposed algorithms using appropriate assumptions in a numerical example.

Mild solutions and controllability of fractional evolution inclusions of Clarke's subdifferential type with nonlocal conditions in Hilbert spaces

In this work, our aim is to ensure the existence of mild solutions and study the controllability of fractional evolution inclusions of Clarke's subdifferential type of order q∈(1,2) with nonlocal conditions in the setting of Hilbert spaces. Using fixed point techniques, fractional calculus, multivalued maps, cosine and sine function operators, we discuss the existence of mild solutions for the considered system. Moreover, under suitable conditions, we investigate the controllability of the proposed system. Finally, we present an illustrated example.

Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces

ABSTRACT In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method.

Solvability of the System of Extended Nonlinear Mixed Variational-Like Inequalities and Proximal Dynamical System

In this article, our goal is to study the system of extended nonlinear mixed variational-like inequalities (in short, SENMVLI) with a nonconvex functional in the setting of real Hilbert spaces and discuss the existence of solution of our considered problem. We propose a three-step iterative algorithm to calculate the approximate solutions of SENMVLI and investigate the convergence analysis as well as stability analysis of the proposed algorithm. Furthermore, we also study the proximal dynamical system for SENMVLI and prove that the trajectory of the solution of the extended proximal dynamical system converges globally exponentially to a unique solution of SENMVLI. Our suggested iterative algorithm and results have become the significant improvement, enhancement, and generalization of many previously known results in the literature.

New Algorithms for Solving the Split Common Zero Point Problem in Hilbert Space

We introduce new self-adaptive algorithms for solving the split common zero point problem with multiple output sets in Hilbert space. We also apply our main results to solving split feasibility problems with multiple output sets.

Accelerated Krasnoselski-Mann type algorithm for hierarchical fixed point and split monotone variational inclusion problems in Hilbert spaces

In this paper, a new accelerated extrapolation Krasnoselski-Mann type algorithm for finding common element in the solution set of the hierarchical fixed point and split monotone variational inclusion problems are introduced in the setting of a real Hilbert space. We then prove that a sequence generated by the algorithm converges strongly to such common element which also approximates solution of some fixed point problem of demimetric mapping in the space. Finally, some applications and numerical experiment are given to show effectiveness of the proposed algorithm over the recently known related results in the literature. The established results extend and generalize many recent ones announced in the literature.

On the Arrow–Hurwicz differential system for linearly constrained convex minimization

ABSTRACT In a real Hilbert space setting, we reconsider the classical Arrow–Hurwicz differential system in view of solving linearly constrained convex minimization problems. We investigate the asymptotic properties of the differential system and provide conditions for which its solutions converge towards a saddle point of the Lagrangian associated with the convex minimization problem. Our convergence analysis mainly relies on a ‘Lagrangian identity’ which naturally extends on the well-known descent property of the classical continuous steepest descent method. In addition, we present asymptotic estimates on the decay of the solutions and the primal-dual gap function measured in terms of the Lagrangian. These estimates are further refined to the ones of the classical damped harmonic oscillator provided that second-order information on the objective function of the convex minimization problem is available. Finally, we show that our results directly translate to the case of solving structured convex minimization problems. Numerical experiments further illustrate our theoretical findings.

Exact separation theorem for disjoint closed sets in Hilbert spaces

Exact separation theorem for disjoint closed sets in Hilbert spaces

Alternating projection‐based iterative learning control for discrete‐time systems with non‐uniform trial lengths

AbstractThis article develops a novel framework for iterative learning control (ILC) design of discrete‐time systems with non‐uniform trial lengths by using the method of alternating projections. In contrast to existing results for the non‐uniform trial length problem, this article uses the Hilbert space setting and hence the linear discrete‐time system dynamics with non‐uniform trial lengths can be represented by multiple affine subspaces (or linear varieties). Motivated by the successive projection design between two closed convex sets, the considered ILC problem can be transformed into alternating projections between multiple sets, then the Hilbert space setting is used to establish key systems theoretic properties. Moreover, an optimal ILC design is developed for systems with non‐uniform trial lengths, which is also extended to the case of input constraints. A numerical case study is given to illustrate the applicability of the new design.

Density Estimation in RKHS with Application to Korobov Spaces in High Dimensions

A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate probability density functions belonging to weighted Korobov spaces, for which a dimension-independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.

Controlled polyhedral sweeping processes: Existence, stability, and optimality conditions

This paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations.

Approximate oblique dual frames

In representations using frames, oblique duality appears in situations where the analysis and the synthesis has to be done in different subspaces. In some cases, we cannot obtain an explicit expression for the oblique duals and in others there exists only one oblique dual frame which has not the properties we need. Also, in practice the computations are not exact. To give a solution to these problems, in this work we introduce and investigate the notion of approximate oblique dual frames first in the setting of separable Hilbert spaces. We present several properties and provide different characterizations of approximate oblique dual frames. We focus then on approximate oblique dual frames in shift-invariant subspaces of L2(R) and give different conditions on the generators that assure their existence. The importance of approximate oblique dual frames from a numerical and computational point of view is illustrated with an example of frame sequences generated by B-splines, where the previous results are used to construct approximate oblique dual frames which have better attributes than the exact ones. We provide an expression for the approximation error and study its behaviour.

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.

Two new self-adaptive algorithms for solving split common fixed point problems with multiple output sets

In this paper, we study split common fixed point problems with multiple output sets in Hilbert spaces. For solving this problem, we improve Reich and Tuyen's algorithms [10] and propose two new self-adaptive algorithms which do not need any priori information on the norms of the transfer operators. Then, we establish weak convergence theorems for them. It can be seen from our numerical experiments that the proposed methods depict a very significant improvement in terms of the number of iterations and execution time.