Hilbert Space Setting Research Articles

Tikhonov regularized second-order plus first-order primal-dual dynamical systems with asymptotically vanishing damping for linear equality constrained convex optimization problems

In this paper, in the setting of Hilbert spaces, we consider a Tikhonov regularized second-order plus first-order primal-dual dynamical system with asymptotically vanishing damping for a linear equality constrained convex optimization problem. It is shown that convergence properties of the proposed inertial dynamical system depend upon the choice of the Tikhonov regularization parameter. When the Tikhonov regularization parameter decays rapidly to zero, we derive the fast convergence rates of the primal-dual gap, the objective residual, the feasibility violation and the gradient norm of the objective function along the generated trajectory. When the Tikhonov regularization parameter decreases slowly to zero, we prove the strong convergence of the primal trajectory of the Tikhonov regularized dynamical system to the minimal norm solution of the linear equality constrained convex optimization problem. Numerical experiments are performed to illustrate the efficiency of our approach.

A note on a Halmos problem

AbstractWe address the existence of non‐trivial closed invariant subspaces of operators on Banach spaces whenever their square have or, more generally, whether there exists a polynomial with such that the lattice of invariant subspaces of is non‐trivial. In the Hilbert space setting, the ‐problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the Invariant Subspace Problem.

An existence result for integro-differential degenerate sweeping process with convex sets.

In this paper, we study the well-posedness in the sense of existence and uniqueness of a solution of integrally perturbed degenerate sweeping processes, involving convex sets in Hilbert spaces. The degenerate sweeping process is perturbed by a sum of a singlevalued map satisfying a Lipschitz condition and an integral forcing term. The integral perturbation depends on two time-variables, by using a semi-discretization method. Unlike the previous works, the Cauchy’s criterion of the approximate solutions is obtained without any new Gronwall’s like inequality.

RELAXED DOUBLE INERTIA AND VISCOSITY ALGORITHMS FOR THE MULTIPLE-SETS SPLIT FEASIBILITY PROBLEM WITH MULTIPLE OUTPUT SETS

In this paper, we investigate a multiple-sets split feasibility problem with multiple output sets in infinite-dimensional Hilbert spaces. To address this problem, we propose relaxed inertial self-adaptive algorithms that do not use the least squares method and prove strong convergence results for the sequences generated by these algorithms. Finally, we validate the performances of the proposed algorithms by using numerical examples.

An accelerated inexact Newton-type regularizing algorithm for ill-posed operator equations

We propose and analyze a new iterative regularization approach, called IN-SETPG, for efficiently solving nonlinear ill-posed operator equations in the Hilbert-space setting. IN-SETPG consists of an outer iteration and an inner iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method, while the inner iteration is performed by a sequential subspace optimization method based on the two-point gradient iteration. The key idea behind IN-SETPG is that, unlike the standard Landweber method, it uses multiple search directions per iteration in combination with an adaptive step size in order to reduce the total number of iterations. The regularization property of IN-SETPG has been established, i.e., the iterate converges to a solution of the nonlinear problem with exact data when the noise level tends to zero. Various numerical experiments are presented to demonstrate that, compared with the original inexact Newton iteration, IN-SETPG can achieve better reconstruction results and remarkable acceleration.

A Hilbert‐space variant of Geršgorin's circle theorem

A Hilbert‐space variant of Geršgorin's circle theorem

Strong convergence of a viscosity proximal point algorithm with new error sequences

In the paper, we consider a viscosity proximal point algorithm with over-relaxed factor for solving inclusion problem in the setting of Hilbert space. Moreover, we establish its strong convergences under two different error criteria, which generalize some existing conclusions.

Strong and Weak Convergence Theorems for the Split Feasibility Problem of (β,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature.

Two novel two-step inertial algorithms for a class of bilevel variational inequalities

This paper introduces two novel two-step inertial and self-adaptive step sizes algorithms for solving a strongly monotone variational inequality over the solution set of a split convex feasibility problem with multiple output sets in real Hilbert spaces. We establish strong convergence properties of the proposed method under mild conditions and employ it to solve monotone variational inequalities over the solution set of the split feasibility problem. The method is further applied to the image classification problem via the support vector machine learning. Numerical results are given to demonstrate the accelerating behaviors of our method over other related methods in the literature.

Decay of operator semigroups, infinite-time admissibility, and related resolvent estimates

We study decay rates for bounded C0-semigroups from the perspective of Lp-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the resolvent behavior in the open right half-plane. A similar characterization based on Lp-infinite-time admissibility is provided for multiplication semigroups on Lq-spaces with 1≤q≤p<∞. For polynomially stable C0-semigroups on Hilbert spaces, we also give a sufficient condition for L2-infinite-time admissibility.

Complementation and Lebesgue‐type decompositions of linear operators and relations

AbstractIn this paper, a new general approach is developed to construct and study Lebesgue‐type decompositions of linear operators or relations in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue‐type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of . The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue‐type decomposition for pairs of quadratic forms. The approach built in this paper uses so‐called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.

Rational approximation of operator semigroups via the [formula omitted]-calculus

We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the B- functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the B-calculus thus making the paper essentially self-contained.

Strong convergence theorem for split null point, split variational inequality and fixed point problems for a non-expansive semigroup

The primary goal of this paper is to find a common solution of a split null point problem, split variational inequality problem and fixed point problem for a non-expansive semigroup with the help of new type of iterative algorithm. We construct that the sequences induced by the proposed iterative method to solve the above stated problems in the setting of real Hilbert spaces and obtain a strong convergence theorem. We also deduce certain consequences from the main convergence result. At the end, we provide a numerical experiment to demonstrate the convergence analysis of proposed iterative method. The methodology and conclusion described in this work extend and unify previously published findings in this field.

A Modified Form of Inertial Viscosity Projection Methods for Variational Inequality and Fixed Point Problems

This paper aims to introduce an iterative algorithm based on an inertial technique that uses the minimum number of projections onto a nonempty, closed, and convex set. We show that the algorithm generates a sequence that converges strongly to the common solution of a variational inequality involving inverse strongly monotone mapping and fixed point problems for a countable family of nonexpansive mappings in the setting of real Hilbert space. Numerical experiments are also presented to discuss the advantages of using our algorithm over earlier established algorithms. Moreover, we solve a real-life signal recovery problem via a minimization problem to demonstrate our algorithm’s practicality.

Nonlinear Tikhonov regularization in Hilbert scales for inverse learning

In this paper, we study Tikhonov regularization scheme in Hilbert scales for a nonlinear statistical inverse problem with general noise. The regularizing norm in this scheme is stronger than the norm in the Hilbert space. We focus on developing a theoretical analysis for this scheme based on conditional stability estimates. We utilize the concept of the distance function to establish high probability estimates of the direct and reconstruction errors in the Reproducing Kernel Hilbert space setting. Furthermore, explicit rates of convergence in terms of sample size are established for the oversmoothing case and the regular case over the regularity class defined through an appropriate source condition. Our results improve upon and generalize previous results obtained in related settings.

A random elastic traffic equilibrium problem via stochastic quasi-variational inequalities

The aim of the paper is to introduce a random elastic traffic equilibrium problem in a Hilbert space setting. The equilibrium condition is expressed by a random extension of the elastic Wardrop principle. Its characterization with a stochastic quasi-variational inequality is proved. Under suitable assumptions, the existence of a random equilibrium distribution is established. Furthermore, a numerical scheme to compute the random elastic traffic equilibrium distribution is presented. Finally a numerical example is discussed.

A framework for fluid motion estimation using a constraint-based refinement approach

Physics-based optical flow models have been successful in capturing the deformities in fluid motion arising from digital imagery. However, a common theoretical framework analyzing several physics-based models is missing. In this regard, we formulate a general framework for fluid motion estimation using a constraint-based refinement approach. We demonstrate that for a particular choice of constraint, our results closely approximate the classical continuity equation-based method for fluid flow. This closeness is theoretically justified by augmented Lagrangian method in a novel way. The convergence of Uzawa iterates is shown using a modified bounded constraint algorithm. The mathematical well-posedness is studied in a Hilbert space setting. Further, we observe a surprising connection to the Cauchy-Riemann operator that diagonalizes the system leading to a diffusive phenomenon involving the divergence and the curl of the flow. Several numerical experiments are performed and the results are shown on different datasets. Additionally, we demonstrate that a flow-driven refinement process involving the curl of the flow outperforms the classical physics-based optical flow method without any additional assumptions on the image data.

Convergence results on the general inertial Mann–Halpern and general inertial Mann algorithms

In this paper, we prove strong convergence theorem of the general inertial Mann–Halpern algorithm for nonexpansive mappings in the setting of Hilbert spaces. We also prove weak convergence theorem of the general inertial Mann algorithm for k-strict pseudo-contractive mappings in the setting of Hilbert spaces. These convergence results extend and generalize some existing results in the literature. Finally, we provide examples to verify our main results.

On Bilevel Monotone Inclusion and Variational Inequality Problems

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted.

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.