Asymptotic Regularity Research Articles

ON A DUAL CHARACTERIZATION OF THE ASYMPTOTIC CONE FOR THE SOLUTION SET OF A LINEAR OPTIMIZATION PROBLEM

The concept of the asymptotic cone is very useful in various branches of pure and applied mathematics, especially in optimization and variational inequalities. In recent years, many authors and researchers have studied asymptotic directions and asymptotically convergent algorithms for unbounded solution sets. In this paper, we consider the asymptotic cone of the solution set Ω of a linear optimization problem and investigate various results on its asymptotic cone, asymptotic regularity, the dual and polar cones of the asymptotic cone, the support function of the solution set, etc. Finally, we present a dual characterization of the asymptotic cone Ω∞ for the solution set of a linear optimization problem.

Convergence rates of stationary and non-stationary asymptotical regularization methods for statistical inverse problems in Banach spaces

We investigate stationary and non-stationary asymptotical regularization methods for statistical inverse problems in Banach spaces. The mean-squared errors (MSE) of both methods are estimated without the conventional assumption of the commutativity between the solution variance and the noise variance. Moreover, the a priori smoothness of solution is characterized in terms of real interpolation, which is strictly weaker than the classical settings. The sufficient and necessary conditions of optimal convergence rates of MSE are proven. Our results extend theoretical findings in Hilbert settings given in the recent paper by Lu et al. [SIAM/ASA J. Uncertain., 9 (2021), pp. 1488-1525]. Two examples of integral equations with white noise and inverse source problems in elliptic partial differential equations are demonstrated to show the usefulness of our theoretical results.

Stochastic asymptotical regularization for linear inverse problems

We introduce stochastic asymptotical regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. We demonstrate that SAR can quantify the uncertainty in error estimates for inverse problems. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an order-optimal regularization method for linear ill-posed problems provided that the terminating time of SAR is chosen according to the smoothness of the solution. This result is proven for both a priori and a posteriori stopping rules under general range-type source conditions. Furthermore, some converse results of SAR are verified. Two iterative schemes are developed for the numerical realization of SAR, and the convergence analyses of these two numerical schemes are also provided. A toy example and a real-world problem of biosensor tomography are studied to show the accuracy and the advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can provide the uncertainty quantification of the quantity of interest, which can in turn be used to reveal and explicate the hidden information about real-world problems, usually obscured by the incomplete mathematical modeling and the ascendence of complex-structured noise.

Generalized Cyclic p-Contractions and p-Contraction Pairs Some Properties of Asymptotic Regularity Best Proximity Points, Fixed Points

This paper studies a general p-contractive condition of a self-mapping T on X, where X ,d is either a metric space or a dislocated metric space, which combines the contribution to the upper-bound of dTx , Ty, where x and y are arbitrary elements in X of a weighted combination of the distances dx,y , dx,Tx,dy,Ty,dx,Ty,dy,Tx, dx,Tx−dy,Ty and dx,Ty−dy,Tx. The asymptotic regularity of the self-mapping T on X and the convergence of Cauchy sequences to a unique fixed point are also discussed if X,d is complete. Subsequently, T, S generalized cyclic p-contraction pairs are discussed on a pair of non-empty, in general, disjoint subsets of X. The proposed contraction involves a combination of several distances associated with the T, S-pair. Some properties demonstrated are: (a) the asymptotic convergence of the relevant sequences to best proximity points of both sets is proved; (b) the best proximity points are unique if the involved subsets are closed and convex, the metric is norm induced, or the metric space is a uniformly convex Banach space. It can be pointed out that both metric and a metric-like (or dislocated metric) possess the symmetry property since their respective distance values for any given pair of elements of the corresponding space are identical after exchanging the roles of both elements.

An augmented Lagrangian method for optimization problems with structured geometric constraints

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations

In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter. As a by-product, we obtain an approximate smooth solution, constructed from a sequence of reduced stationary PDEs with vanished high-order derivative terms. We prove that the accuracy of the constructed approximate solution can be in any order of this small-scale parameter in the whole domain, except a negligible transition layer. Furthermore, based on a simpler link equation between this approximate solution and the source function, we propose an efficient algorithm, called the asymptotic expansion regularization (AER), for solving nonlinear inverse source problems governed by the original PDE. The convergence-rate results of AER are proven, and the a posteriori error estimation of AER is also studied under some a priori assumptions of source functions. Various numerical examples are provided to demonstrate the efficiency of our new approach.

Asymptotic Regularity and Existence of Time-Dependent Attractors for Second-Order Undamped Evolution Equations with Memory

Our purpose in this article is to study the asymptotic behavior of undamped evolution equations with fading memory on time-dependent spaces. By means of the theory of processes on time-dependent spaces, asymptotic a priori estimate and the technique of operator decomposition and the existence and asymptotic regularity of time-dependent attractors are, respectively, established in the critical case. At the same time, we also obtain the asymptotic regularity of the solution.

On the asymptotical regularization with convex constraints for nonlinear ill-posed problems

We investigate the method of asymptotical regularization for solving nonlinear ill-posed problems F(x)=y in Hilbert spaces. A general uniformly convex functional has been embedded in the evolution equations which is allowed to be non-smooth, thus the algorithm can be applied for sparsity and discontinuity reconstruction. Assuming certain conditions concerning the nonlinear operator F and functional Θ, we establish the convergence and stability of the method.

Quadratic rates of asymptotic regularity for the Tikhonov–Mann iteration

In this paper, we compute quadratic rates of asymptotic regularity for the Tikhonov–Mann iteration in W-hyperbolic spaces. This iteration is an extension to a nonlinear setting of the modified Mann iteration defined recently by Bo, Csetnek and Meier in Hilbert spaces. Furthermore, we show that the Douglas–Rachford and forward-backward algorithms with Tikhonov regularization terms are special cases, in Hilbert spaces, of our Tikhonov–Mann iteration.

Powers of monomial ideals with characteristic-dependent Betti numbers

We explore the dependence of the Betti numbers of monomial ideals on the characteristic of the field. A first observation is that for a fixed prime p either the i-th Betti number of all high enough powers of a monomial ideal differs in characteristic 0 and in characteristic p or it is the same for all high enough powers. In our main results, we provide constructions and explicit examples of monomial ideals all of whose powers have some characteristic-dependent Betti numbers or whose asymptotic regularity depends on the field. We prove that, adding a monomial on new variables to a monomial ideal allows to spread the characteristic dependence to all powers. For any given prime p, this produces an edge ideal such that all its powers have some Betti numbers that are different over mathbb {Q} and over mathbb {Z}_p. Moreover, we show that, for every r ge 0 and i ge 3 there is a monomial ideal I such that some coefficient in a degree ge r of the Kodiyalam polynomials {mathfrak {P}}_3(I),ldots ,{mathfrak {P}}_{i+r}(I) depends on the characteristic. We also provide a summary of related results and speculate about the behavior of other combinatorially defined ideals.

An asymptotical regularization with convex constraints for inverse problems

We investigate the method of asymptotical regularization for the stable approximate solution of nonlinear ill-posed problems F(x) = y in Hilbert spaces. The method consists of two components, an outer Newton iteration and an inner scheme providing increments by solving a local coupling linearized evolution equations. In addition, a non-smooth uniformly convex functional has been embedded in the evolution equations which is allowed to be non-smooth, including L 1-liked and total variation-like penalty terms. We establish convergence properties of the method, derive stability estimates, and perform the convergence rate under the Hölder continuity of the inverse mapping. Furthermore, based on Runge–Kutta (RK) discretization, different kinds of iteration schemes can be developed for numerical realization. In our numerical experiments, four types iterative scheme, including Landweber type, one-stage explicit, implicit Euler and two-stage RK are presented to illustrate the performance of the proposed method.

Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity constrained and bilevel programming

Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The second part of the paper presents two applications of the obtained theory. On the one hand, we specify our findings for complementarity constrained optimization problems and recover recent results from the literature which demonstrates the power of the approach. Furthermore, we hint at potential extensions to or- and vanishing constrained optimization. On the other hand, we demonstrate the usefulness of asymptotic regularity in the context of bilevel optimization. More precisely, we justify a well-known stationarity system for affinely constrained bilevel optimization problems in a novel way. Afterwards, we suggest a solution algorithm for this class of bilevel optimization problems which combines a penalty method with ideas from DC-programming. After a brief convergence analysis, we present the results of some numerical experiments.

Asymptotic topological regularity of CAT(0) spaces

We study asymptotic topological regularity of \({\text {CAT}}(0)\) spaces. We prove that if a purely n-dimensional, proper, geodesically complete \({\text {CAT}}(0)\) space has small volume growth, then it is homeomorphic to the n-dimensional Euclidean space. We also discuss asymptotic geometry of proper, geodesically complete \({\text {CAT}}(0)\) spaces of small volume growth.

R.E. Bruck, proof mining and a rate of asymptotic regularity for ergodic averages in Banach spaces

R.E. Bruck, proof mining and a rate of asymptotic regularity for ergodic averages in Banach spaces

Quantitative inconsistent feasibility for averaged mappings

Bauschke and Moursi have recently obtained results that implicitly contain the fact that the composition of finitely many averaged mappings on a Hilbert space that have approximate fixed points also has approximate fixed points and thus is asymptotically regular. Using techniques of proof mining, we analyze their arguments to obtain effective uniform rates of asymptotic regularity.

Kannan-Type Contractions on New Extended b -Metric Spaces

This article is focused on the generalization of some fixed point theorems with Kannan-type contractions in the setting of new extended b -metric spaces. An idea of asymptotic regularity has been incorporated to achieve the new results. As an application, the existence of a solution of the Fredholm-type integral equation is presented.

On alpha -Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

We introduce the class of alpha -firmly nonexpansive and quasi alpha -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where alpha -firmly nonexpansive operators coincide with so-called alpha -averaged operators. For our more general setting, we show that alpha -averaged operators form a subset of alpha -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) alpha -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) alpha -firmly nonexpansive. Moreover, we will see that quasi alpha -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates x_{n+1}:=Tx_{n} belong to the fixed point set {{,mathrm{Fix},}}T whenever the operator T is nonexpansive and quasi alpha -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in {{,mathrm{Fix},}}T. Further, the projections P_{{{,mathrm{Fix},}}T}x_n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in L_p, p in (1,infty ) backslash {2} spaces on probability measure spaces.

A continuum limit for the PageRank algorithm

Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalisedgraph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit partial differential equation. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.

Remarks on asymptotic regularity in A-metric spaces

The purpose of this paper is to present a potential generalizations of celebrated theorems due to Kannan, Reich, Bianchini and Chatterjee in the set-up of an A-metric space. We unify these well-known fixed point theorems in A-metric space and give the simple proof of the result presented herein. For this purpose, we use concept of asymptotic regularity. Also, we give most general form of Kannan, Reich, Bianchini and Chatterjee Fixed Point Theorems in A-metric spaces, and prove these fixed point theorems using asymptotically regular sequences and maps rather than Picard iteration, respectively. Furthermore, we give some examples in support of our results.

On a weighted time-fractional asymptotical regularization method

In this paper we study a weighted time-fractional asymptotical regularization method for solving linear ill-posed problems Ax=y where only noisy data yδ with ‖yδ−y‖≤δ are available and A is a linear bounded operator. This regularization method can be considered as a preconditioned version for the newly-developed time-fractional asymptotical regularization method. Error analysis and numerical tests show that the method has acceleration effect on the original time-fractional asymptotical regularization method.