Asymptotic Regularity Research Articles

Quantitative asymptotic regularity of the VAM iteration with error terms for $m$-accretive operators in Banach spaces

In this paper, we obtain, by using proof mining methods, quantitative results on the asymptotic regularity of the viscosity approximation method (VAM) with error terms for m -accretive operators in Banach spaces. For concrete instances of the parameter sequences, linear rates are computed by applying a lemma due to Sabach and Shtern.

Fixed point theorems concerning mappings that exhibit asymptotic regularity in b-metric spaces

In this article, we establish fixed point outcomes for mappings that are asymptotically regular within the context of b -metric spaces. These findings broaden and enhance the familiar outcomes found in existing literature. Additionally, we present corollaries to demonstrate that our results are more encompassing compared to the established findings in the literature.

Asymptotic regularity results for a viscosity version of Halpern-type iterations

Asymptotic regularity results for a viscosity version of Halpern-type iterations

Nonstationary asymptotical regularization method with convex constraints for linear ill-posed problems

We investigate the method of nonstationary asymptotical regularization for solving linear ill-posed problems in Hilbert spaces. This method introduces the convex constraints that are proper lower semicontinuous and allowed to be non-smooth, therefore can be used for sparsity and discontinuity reconstruction. Under some suitable conditions , the convergence and regularity of the proposed method are established. Under the discretion of Runge–Kutta method, different iteration modes can be deduced for numerical implementation. The numerical results of iteration modes under one-stage explicit Euler, one-stage implicit Euler and two-stage explicit Runge–Kutta are presented to illustrate the efficiency of the proposed method.

A new second-order dynamical method for solving linear inverse problems in Hilbert spaces

A new second-order dynamic method (SODM) is proposed for solving ill-posed linear inverse problems in Hilbert spaces. The SODM can be viewed as a combination of Tikhonov regularization and second-order asymptotical regularization methods. As a result, a double-regularization-parameter strategy is adopted. The regularization properties of SODM are demonstrated under both a priori and a posteriori stopping rules. In the context of time discretization, we propose several iterative schemes with different choices of damping parameters. A truncated discrepancy principle is employed as the stop criterion. Finally, numerical experiments are performed to show the efficiency of the SODM: on the whole, compared with the classical Tikhonov method and the first-order dynamical-system method, the SODM leads to more-accurate approximate solutions while requiring fewer-iterative numbers.

Stochastic linear regularization methods: random discrepancy principle and applications

The a posteriori stopping rule plays a significant role in the design of efficient stochastic algorithms for various tasks in computational mathematics, such as inverse problems, optimization, and machine learning. Through the lens of classical regularization theory, this paper describes a novel analysis of Morozov’s discrepancy principle for the stochastic generalized Landweber iteration and its continuous analog of generalized stochastic asymptotical regularization. Unlike existing results relating to convergence in probability, we prove the strong convergence of the regularization error using tools from stochastic analysis, namely the theory of martingales. Numerical experiments are conducted to verify the convergence of the discrepancy principle and demonstrate two new capabilities of stochastic generalized Landweber iteration, which should also be valid for other stochastic/statistical approaches: improved accuracy by selecting the optimal path and the identification of multi-solutions by clustering samples of obtained approximate solutions.

Asymptotic regularity of invariant chains of edge ideals

Asymptotic regularity of invariant chains of edge ideals

Duality for asymptotic invariants of graded families

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants.We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals.In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.

Strong Convergence for the Alternating Halpern–Mann Iteration in CAT(0) Spaces

In this paper we consider, in the general context of CAT(0) spaces, an iterative schema which alternates between Halpern and Krasnoselskii–Mann style iterations. We prove, under suitable conditions, the strong convergence of this algorithm, benefiting from ideas from the proof mining program. We give quantitative information in the form of effective rates of asymptotic regularity and of metastability (in the sense of Tao). Motivated by these results, we are also able to obtain strongly convergent versions of the forward-backward and the Douglas–Rachford algorithms. Our results generalize recent work by Boţ, Csetnek, and Meier and Cheval and Leuştean.

Common stationary point of multivalued asymptotically regular mappings

We establish a relationship between asymptotic regularity and common stationary points of multivalued mappings on a metric space. As a consequence of our results, we obtain a new common fixed point result for two asymptotically regular single-valued mappings. Our work significantly improves and complements comparable results in the literature.

Asymptotic Expansion Regularization for Inverse Source Problems in Two-Dimensional Singularly Perturbed Nonlinear Parabolic PDEs

Asymptotic Expansion Regularization for Inverse Source Problems in Two-Dimensional Singularly Perturbed Nonlinear Parabolic PDEs

Convergence analysis of the Halpern iteration with adaptive anchoring parameters

We propose an adaptive way to choose the anchoring parameters for the Halpern iteration to find a fixed point of a nonexpansive mapping in a real Hilbert space. We prove strong convergence of this adaptive Halpern iteration and obtain the rate of asymptotic regularity at least O ( 1 / k ) O(1/k) , where k k is the number of iterations. Numerical experiments are also provided to show advantages and outperformance of our adaptive Halpern algorithm over the standard Halpern algorithm.

Rates of asymptotic regularity for the alternating Halpern–Mann iteration

In this paper we extend to UCW-hyperbolic spaces the quantitative asymptotic regularity results for the alternating Halpern–Mann iteration obtained by Dinis and the second author for CAT(0) spaces. These results are new even for uniformly convex normed spaces. Furthermore, for a particular choice of the parameter sequences, we compute linear rates of asymptotic regularity in W-hyperbolic spaces and quadratic rates of T- and U-asymptotic regularity in CAT(0) spaces.

Certain Fixed Point Results On 𝔄-Metric Space Using Banach Orbital Contraction and Asymptotic Regularity

ABSTRACT In this paper, we investigate the existence of φ-fixed point for Banach orbital contraction over 𝔄-metric space. Also a fixed point result has been established via asymptotic regularity property over such generalized metric space. Our fixed point theorems have also been applied to the fixed circle problem. Moreover, we give some new solutions to the open problem raised by Özgür and Taş on the geometric properties of φ-fixed points of self-mappings and the existence and uniqueness of φ-fixed circles and φ-fixed discs for various classes of self-mappings.

On Modified Halpern and Tikhonov–Mann Iterations

We show that the asymptotic regularity and the strong convergence of the modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by A. Cuntavenapit and B. Panyanak and the Tikhonov–Mann iteration introduced by H. Cheval and L. Leuştean as a generalization of an iteration due to Y. Yao et al. that has recently been studied by Boţ et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Boţ et al. together with a generalization from Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity and metastability due to K. Schade and U. Kohlenbach can be adapted and transformed into rates for the Tikhonov–Mann iteration corresponding to recent quantitative results on the latter of H. Cheval, L. Leuştean and B. Dinis, P. Pinto, respectively. A transformation in the converse direction is also possible. We also obtain rates of asymptotic regularity of order O(1/n) for both the modified Halpern (and so in particular for the Halpern iteration) and the Tikhonov–Mann iteration in a general geodesic setting for a special choice of scalars.

Asymptotic regularity for the generalized MHD‐Boussinesq equations

The three‐dimensional generalized MHD‐Boussinesq equations are studied in this paper. Well‐posedness of the strong solutions for the 3D generalized MHD‐Boussinesq equations is proved. Asymptotic regularity for the 3D generalized MHD‐Boussinesq equations is proved in , and furthermore, higher regularity for the 3D generalized MHD‐Boussinesq equations is proved in for .

Asymptotic invariants constructed from generic initial ideals

AbstractGeneric initial ideals (gins for short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they have been of interest in commutative algebra and indirectly in algebraic geometry. Recently, Mayes in a series of articles associated with gins of graded families of ideals geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes, and some asymptotic invariants of homogeneous ideals which are associated with the gins, for example, asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view.

Common fixed point of nonlinear contractive mappings

<abstract><p>The purpose of this paper is to study the existence of a common fixed point for a pair of mappings without assumption of the contractive coefficient being fixed and less than 1. By replacing the fixed contractive coefficient with a nonlinear contractive function, we establish a unique common fixed point theorem for a pair of asymptotically regular self-mappings with either orbital continuity or $ q $-continuity in a metric space. Moreover, by the asymptotical regularity of two approximate mappings, we prove that a pair of nonexpansive and continuous self-mappings, which are defined on a nonempty closed convex subset of a Banach space, have a common fixed point. Some examples are given to illustrate that our results are extensions of a recent result in the existing literature.</p></abstract>

On fixed points of generalized Kannan and Reich type contractive mappings

Kannan or Reich type strict contractive conditions do not ensure the existence of fixed points unless some strong conditions such as compactness of the space and continuity of the mapping are assumed. In this paper, our main aim is to investigate the existence of fixed point of generalized Kannan type contractive mappings in the setting of boundedly compact and T-orbitally compact metric spaces via orbital continuity. In addition to it, asymptotic regularity has been used to prove the Reich type fixed point theorem via altering distance functions. Supporting examples have been given to strengthen the hypotheses of our proved theorems.

Enhanced Differentiable Architecture Search Based on Asymptotic Regularization

Enhanced Differentiable Architecture Search Based on Asymptotic Regularization