Search

We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschitz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the step size unlike the GBM tamed method. [ABSTRACT FROM AUTHOR]

First-order optimization methods have long been established as effective methods for solving large-scale unconstrained optimization problems. Over time, these methods have been extended to address the task of finding zeros of single-valued monotone maps within the framework of finite-dimensional Hilbert spaces. Notably, Abubakar et al. (Comput Appl Math 39(2):129, 2020) recently introduced a projection method that incorporates a convex combination of two distinct positive spectral coefficients to find the zeros of a single-valued monotone map. Motivated by their work, this paper generalises the Abubakar et al. technique to a more general setting of infinite-dimensional Hilbert spaces. Under mild assumptions, we establish weak convergence of the generated sequence of iterates. Furthermore, the extended method does not involve computing the resolvent of the monotone operator. [ABSTRACT FROM AUTHOR]

We consider a class of stationary processes exhibiting both long-range dependence and heavy tails. Separate limit theorems for sums and for extremes have been established recently in the literature with novel objects appearing in the limits. In this article, we establish the joint sum-and-max limit theorems for this class of processes. In the finite-variance case, the limit consists of two independent components: a fractional Brownian motion arising from the sum and a long-range dependent random sup measure arising from the maximum. In the infinite-variance case, we obtain in the limit two dependent components: a stable process and a random sup measure whose dependence structure is described through the local time and range of a stable subordinator. For establishing the limit theorem in the latter case, we also develop a joint convergence result for the local time and range of subordinators, which may be of independent interest. [ABSTRACT FROM AUTHOR]

A control reconstruction problem for dynamic deterministic affine-control systems is considered. This problem consists of constructing piecewise constant approximations of an unknown control generating an observed trajectory from discrete inaccurate measurements of this trajectory. It is assumed that the controls are constrained by known nonconvex geometric constraints. In this case, sliding modes may appear. To describe the impact of sliding modes on the dynamics of the system, the theory of generalized controls is used. The notion of normal control is introduced. It is a control that generates an observed trajectory and is defined uniquely. The aim of reconstruction is to find piecewise constant approximations of the normal control that satisfy given nonconvex geometric constraints. The convergence of approximations is understood in the sense of weak convergence in the space superscript 퐿 2 . A solution to the control reconstruction problem is proposed. [ABSTRACT FROM AUTHOR]

We consider a nonlinear, hyperbolic population balance equation that incorporates both aggregation and collisional breakage events simultaneously. Our approach revolves around the development of a novel time-explicit finite volume scheme. Under a suitable time-step stability condition, we prove the convergence of the approximate solution for any non-uniform mesh. A first-order convergence is obtained by a thorough error analysis of the proposed scheme for a suitable choice of kernels. Finally, we compute some numerical test examples to explore the behavior of the solution in steady-state conditions as well as the occurrence of gelation phenomena. [ABSTRACT FROM AUTHOR]

A simple, transparent, two-dimensional, nonlinear model of cell reorientation is constructed in this paper. The cells are attached to a substrate by "focal adhesions" that transmit the deformation of the substrate to the "stress fibers" in the cell. When the substrate is subjected to a deformation, say an in-plane bi-axial deformation with stretches λ 1 and λ 2 , the stress fibers deform with it and change their length and orientation. In addition, the focal adhesions can detach from the substrate and reattach to it at new nearby locations, and this process of detachment and reattachment can happen many times. In this scenario the (varying) fiber angle Θ in the reference configuration plays the role of an internal variable. In addition to the elastic energy of the stress fibers, the energy associated with the focal adhesions is accounted for by a wiggly energy ϵ a cos Θ / ϵ , 0 < ϵ ≪ 1 . Each local minimum of this energy corresponds to a particular configuration of the focal adhesions. The small amplitude ϵ a indicates that the energy barrier between two neighboring configurations is relatively small, and the small distance 2 π ϵ between the local minima indicates that a focal adhesion does not have to move very far before it reattaches. The evolution of this system is studied using a gradient flow kinetic law, which is homogenized for ϵ → 0 using results from weak convergence. The results determine (a) a region of the λ 1 , λ 2 -plane in which the (referential) fiber orientation remains stuck at the angle Θ and does not evolve, and (b) the evolution of the orientation when the stretches move out of this region as the fibers seek to minimize energy. [ABSTRACT FROM AUTHOR]

Let X be a linear diffusion taking values in (ℓ , r) and consider the standard Euler scheme to compute an approximation to E [ g (X T) 1 { T < ζ } ] for a given function g and a deterministic T , where ζ = inf { t ≥ 0 : X t ∉ (ℓ , r) } . It is well known since Gobet (Stoch. Process. Appl. 87:167–197, 2000) that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to 1 / N with N being the number of discretisations. We introduce a drift-implicit Euler method to bring the convergence rate back to 1 / N , i.e., the optimal rate in the absence of killing, using the theory of recurrent transformations developed in Çetin (Ann. Appl. Probab. 28:3102–3151, 2018). Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions. [ABSTRACT FROM AUTHOR]

The empirical likelihood (EL) method possesses desirable qualities such as automatically determining confidence regions and circumventing the need for variance estimation. As an extension, a quantile-based EL (QEL) method is considered, which results in a simpler form. In this paper, we explore the framework of the QEL method. Firstly, we explore the weak convergence of the −2 log empirical likelihood ratio for ROC curves. We also introduce a novel statistic for testing the entire ROC curve and the equality of two distributions. To validate our approach, we conduct simulation studies and analyze real data from hepatitis C patients, comparing our method with existing ones. [ABSTRACT FROM AUTHOR]

We consider the problem of detecting distributional changes in a sequence of high dimensional data. Our approach combines two separate statistics stemming from L p norms whose behavior is similar under H 0 but potentially different under H A , leading to a testing procedure that that is flexible against a variety of alternatives. We establish the asymptotic distribution of our proposed test statistics separately in cases of weakly dependent and strongly dependent coordinates as min { N , d } → ∞ , where N denotes sample size and d is the dimension, and establish consistency of testing and estimation procedures in high dimensions under one-change alternative settings. Computational studies in single and multiple change point scenarios demonstrate our method can outperform other nonparametric approaches in the literature for certain alternatives in high dimensions. We illustrate our approach through an application to Twitter data concerning the mentions of U.S. governors. [ABSTRACT FROM AUTHOR]

We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift. [ABSTRACT FROM AUTHOR]

Arriaza et al (Metrika 82:99–124, 2019) introduced the right and left shape functions, which enjoy interesting properties in terms of describing the global form of a distribution. This paper proposes and studies nonparametric estimators of those functions. The estimators involve nonparametric estimation of the quantile and density functions. Pointwise and uniform consistency are proved under general regularity assumptions, as well as the limit in law. Simulations are included to study the practical performance of the proposed estimators. The analysis of a real data set illustrates the methodology. [ABSTRACT FROM AUTHOR]

We derive a Cramér–von Mises test for testing a class of time dependent coefficients Coditional Heteroscedastic AutoRegressive Non Linear (CHARN) models. The test statistic is based on the log-likelihood ratio process whose weak convergence in a suitable Fréchet space is studied under the null hypothesis and under the sequence of local alternatives considered. This study makes use of the locally asymptotically normal (LAN) result previously established. Using the Karhunen–Loève expansion of the limiting process of the log-likelihood ratio process, the asymptotic null distribution and the power of the test statistic are accurately approximated. These results are applied to change-point analysis. An empirical study is done for evaluating the performance of the methodology proposed. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze four inertial algorithms for solving fixed point problems of Bregman quasi-nonexpansive mappings in the framework of reflexive Banach spaces. Our first two algorithms, we propose inertial-like methods based on Mann-type and Halpern-type iterations, and in the others, we propose relaxed inertial-like methods based on Mann-type and Halpern-type iterations. The weak and strong convergence of the algorithms are established under some appropriate conditions on the parameters. As an application, we utilize our main results to find a zero of the sum of Bregman inverse strongly monotone mappings and maximal monotone operators in real reflexive Banach spaces. Also, we provide several numerical experiments to show the convergence behaviour of our algorithms in both finite-dimensional and infinite-dimensional spaces. Finally, we further utilize our algorithms to numerically solve the data classification problems of lung cancer. [ABSTRACT FROM AUTHOR]

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng's maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the difference between the deterministic and stochastic models of nonlocal diffusion. We use a nonlocal reaction-diffusion equation and a multi-dimensional jump Markov process to analyze these mathematical models. First, we demonstrate that the difference converges to 0 in probability with a supremum norm for a sizeable network. Next, we consider the rescaled difference and show that it converges to a stochastic process in distribution on the Skorokhod space. [ABSTRACT FROM AUTHOR]

In this paper, we find the common solution of a split variational inclusion problem and a fixed point problem of a finite collection of demicontractive mappings in real Hilbert spaces. We introduce two modified inertial-type iterative methods based on Mann and viscosity schemes to solve the considered problem. We prove weak and strong convergence results for the proposed methods under reasonable conditions on the control parameters. With the help of some numerical examples, we show that our proposed methods give better convergence results than some existing methods in the literature. [ABSTRACT FROM AUTHOR]

In 1643 P. de Fermat introduced the problem of finding a point in the plane such that the sum of its Euclidean distances to the three given points is minimal. Recently, Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023) extended this problem in Hilbert space setting and named it ‘generalized Fermat–Torricelli problem’. They introduced some iterative methods for approximating the solution of this problem. This paper aims to formulate some novel iterative methods with inertial effect for approximating the solution of this generalized Fermat–Torricelli problem in real Hilbert spaces. First, we prove the weak and strong convergence of the proposed methods with the mild conditions on the control parameters. Then, we establish the weak and strong convergence results for approximating the solution of a variant of the split feasibility problem with multiple output sets. Finally, we provide some numerical examples to justify the validity and efficiency of the proposed iterative methods. The results in this paper improve and generalize the work of Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023). [ABSTRACT FROM AUTHOR]

A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various feasibility problems such as common fixed point and convex feasibility problems. In 2001 Bauschke and Combettes introduced the notion of coherence and applied it to proving weak and strong convergence of many iterative methods. In 2019 Barshad, Reich and Zalas proposed a stronger variant of coherence which provides a more convenient sufficient convergence condition for such methods. In this paper we combine the ideas of both modular string averaging and coherence. Focusing on extending the above MSA procedure to an infinite sequence of operators with admissible controls, we establish strong coherence of its output operators. Various applications of these concepts are presented with respect to weak and strong convergence. They also provide important generalizations of known results, where the weak convergence of sequences of operators generated by the MSA procedure with intermittent controls was considered. [ABSTRACT FROM AUTHOR]

A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments. [ABSTRACT FROM AUTHOR]

It is well known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. Stone proved a continuous dependence of such diffusions on their speed measures. In this paper we establish the converse direction, i.e., we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures. [ABSTRACT FROM AUTHOR]

This paper studies transition probabilities from a Borel subset of a Polish space to a product of two Borel subsets of Polish spaces. For such transition probabilities it introduces and studies the property of semi-uniform Feller continuity. This paper provides several equivalent definitions of semi-uniform Feller continuity and establishes its preservation under integration. The motivation for this study came from the theory of Markov decision processes with incomplete information, and this paper provides the fundamental results useful for this theory. [ABSTRACT FROM AUTHOR]

In this paper, a general splitting method with alternated inertia and its relaxed version are proposed for solving the split equality problem in Hilbert spaces. The proposed methods combine both the relaxation and alternated inertial techniques to speed up the rate of convergence. Furthermore, the methods employ a simple self-adaptive stepsize, which does not require any prior information about the operator norm. Four options of inertial parameters and relaxation parameters are discussed. The weak convergence of the proposed algorithms is analyzed under mild conditions. Finally, two numerical experiments and an application in signal recovery problem are provided to demonstrate the advantages of the proposed algorithms compared to a recent related one. [ABSTRACT FROM AUTHOR]

We first establish weak convergence results regarding an inertial Krasnosel'skiĭ-Mann iterative method for approximating common fixed points of countable families of nonexpansive mappings in real Hilbert spaces with no extra assumptions on the considered countable families of nonexpansive mappings. The method of proof and the imposed conditions on the iterative parameters are different from those already available in the literature. We then present some applications to the Douglas–Rachford splitting method and image restoration problems, and compare the performance of our method with that of other popular inertial Krasnosel'skiĭ-Mann methods which can be found in the literature. [ABSTRACT FROM AUTHOR]

The multistage stochastic variational inequality is reformulated into a variational inequality with separable structure through introducing a new variable. The prediction–correction ADMM which was originally proposed in He et al. (J Comput Math 24:693–710, 2006) for solving deterministic variational inequalities in finite-dimensional spaces is adapted to solve the multistage stochastic variational inequality. Weak convergence of the sequence generated by that algorithm is proved under the conditions of monotonicity and Lipschitz continuity. When the sample space is a finite set, the corresponding multistage stochastic variational inequality is actually defined on a finite-dimensional Hilbert space and the strong convergence of the algorithm naturally holds true. Some numerical examples are given to show the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods. [ABSTRACT FROM AUTHOR]

In the paper we consider a problem for complex Ginzburg–Landau equations in a medium with locally periodic small obstacles. It is assumed that the obstacle surface can have different conductivity coefficients. We prove that the trajectory attractors of this system converge in a certain weak topology to the trajectory attractors of the homogenized Ginzburg–Landau equations with an additional potential (in the critical case), without an additional potential (in the subcritical case) in the medium without obstacles, or disappear (in the supercritical case). [ABSTRACT FROM AUTHOR]

Hadamard spaces have traditionally played important roles in geometry and geometric group theory. More recently, they have additionally turned out to be a suitable framework for convex analysis, optimization and non-linear probability theory. The attractiveness of these emerging subject fields stems, inter alia, from the fact that some of the new results have already found their applications both in mathematics and outside. Most remarkably, a gradient flow theorem in Hadamard spaces was used to attack a conjecture of Donaldson in Kähler geometry. Other areas of applications include metric geometry and minimization of submodular functions on modular lattices. There have been also applications into computational phylogenetics and image processing. We survey recent developments in Hadamard space analysis and optimization with the intention to advertise various open problems in the area. We also point out several fallacies in the existing proofs. [ABSTRACT FROM AUTHOR]

We come up with a new type of forward–backward–forward algorithms for monotone inclusion problems based on a self-adaptive technique to avoid the selection of Lipschitz assumption and also double inertial extrapolations to increase the convergence performance of our presented algorithm. We also prove its weak convergence theorem under mild hypothesis. Additionally, we provide numerical test in image deblurring and signal recovery as applications. The results show that our algorithm outperforms some known algorithms in the literature. [ABSTRACT FROM AUTHOR]

The two-dimensional system of Navier–Stokes equations in a medium with anisotropic variable viscosity and periodic small obstacles is considered. It is proved that the trajectory attractors of the system tend in a certain weak topology to the trajectory attractors of the homogenized system of Navier–Stokes equations with an additional potential in a medium without obstacles. [ABSTRACT FROM AUTHOR]

Numerical solutions of stochastic problems involve approximations of the random functions in their definitions by deterministic functions of time and/or space and finite numbers of random variables, referred to as finite dimensional (FD) random functions. Since the solutions of these problems are complex functionals of random functions, the resulting numerical solutions are informative only if target random functions and their FD representations have similar samples. We establish conditions under which extremes and other functionals of real-valued continuous processes X(t) can be estimated from samples of FD processes X d (t) . Four types of FD processes are developed for stationary and nonstationary processes X(t). Numerical examples are presented to illustrate the construction of these FD processes and assess their capabilities to characterize extremes of X(t). [ABSTRACT FROM AUTHOR]

In this paper, we introduce two algorithms for finding a common solution of the monotone inclusion problem and the fixed point problem for a relatively nonexpansive mapping in reflexive Banach spaces. The weak convergence results for both algorithms are established without the prior knowledge of the Lipschitz constant of the mapping. An application to the variational inequality problem is considered. Finally, some numerical experiments of the proposed algorithms including comparisons with other algorithms are provided. [ABSTRACT FROM AUTHOR]

In this paper, by combining the hyperplane projection technique and gradient projection method with Polyak’s stepsizes, we propose a novel projection algorithm to solve the split feasibility problem in Hilbert spaces. Unlike the existing projection methods, our method is designed such that the next iterate is closer to the solution set than the previous one. The weak and strong convergence of new algorithms under standard assumptions are established. We examine the performance of our method on the sparse signal recovery problem. The reported numerical results demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Projection methods have been one of the popular methods for solving variational inequality problems in the literature. Several projection methods have been studied for variational inequality problems and most of these methods require at least two evaluations of the cost operator and (or) two projections per iteration. One of the major interests of researchers in variational inequalities is to study projection methods with less computational complexity in terms of reduced number of evaluations of the cost operator and reduced number of projections onto feasible set per iteration especially when the cost operator is non-monotone. In this paper, we propose a simple projection method which requires only one evaluation of the cost operator and one projection onto the feasible set per iteration. We obtain weak convergence results when the cost operator is quasimonotone and Lipschitz continuous with self-adaptive line search strategy. We also give weak convergence analysis when the cost operator is non-monotone and Lipschitz continuous. Some numerical implementations using standard tests are given to illustrate our method and compare with other recently related methods. [ABSTRACT FROM AUTHOR]

Recently in Dvurečenskij and Zahiri (J Appl Log IfCoLog J Log Appl 8:2365–2399, 2021b, J Appl Log IfCoLog J Log Appl 8:2401–2433, 2021c), new algebras, called weak pseudo EMV-algebras (wPEMV-algebras in short), were introduced. The authors do not assume the existence of a top element —they generalize MV-algebras, pseudo MV-algebras, and pseudo EMV-algebras. A g-state is defined on a unital wPEMV-algebra M as a mapping from M into the positive half-line of reals such that it preserves a partial addition + , and in a fixed strong unit, it takes the value 1. They form a Bauer simplex, and extremal points are exactly g-states whose kernel is a maximal and normal ideal. We show that extremal g-states generate all g-states, and it can happen that in some unital wPEMV-algebra, even commutative, there is no g-state. We present some conditions for existence of g-states and establish an integral representation of g-states. In addition, we give a topological characterization of the spaces of g-states and extremal g-states, respectively. Moreover, discrete g-states are investigated. [ABSTRACT FROM AUTHOR]

We study the weak convergence of a greedy algorithm of approximation by a given set in a Banach space. It is proved that the greedy algorithm of approximation by a strongly norm-reducing set in a uniformly smooth Banach space with the WN-property weakly converges. In an arbitrary separable Banach space without the WN-property, we construct an example of a strongly norm-reducing set such that the greedy algorithm of approximation by this set does not weakly converge for some initial element. [ABSTRACT FROM AUTHOR]

U-statistics represent a fundamental class of statistics arising from modeling quantities of interest defined by multi-subject responses. U-statistics generalize the empirical mean of a random variable X to sums over every m-tuple of distinct observations of X. W. Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to : m (t) : = E [ φ (Y 1 , ... , Y m) | (X 1 , ... , X m) = t ] , for t ∈ X m. In this paper we are mainly interested in establishing weak convergence of conditional U-processes in a functional mixing data framework. More precisely, we investigate the weak convergence of the conditional empirical process indexed by a suitable class of functions and of conditional U-processes when the explicative variable is functional. We treat the weak convergence in both cases when the class of functions is bounded or unbounded satisfying some moment conditions. These results are proved under some standard structural conditions on the Vapnik-Chervonenkis classes of functions and some mild conditions on the model. The theoretical results established in this paper are (or will be) key tools for many further developments in functional data analysis. [ABSTRACT FROM AUTHOR]

Let G : H → H and K : H → H be monotone mappings that are either sequentially weakly continuous or continuous, where H is a real Hilbert space. In this work, we introduce two new iterative methods for approximating solutions of the Hammerstein equation u + G K u = 0 , if they exist. The first iterative method is shown to always converge weakly to an element in the solution set of the Hammerstein equation if this solution set is nonempty. The second iterative method is a modification of the first method to upgrade weak convergence to strong convergence. Convergence results are obtained without requiring the maps to be bounded. Numerical examples are provided to demonstrate the convergence of one of these methods. Comparisons with some existing methods show that the method is cost effective in terms of the number of iterations required to obtain a solution and the computational time. [ABSTRACT FROM AUTHOR]

We study the problem of the existence of a coordinate network of curves on a -smooth bordered surface consisting of geodesic lines covering it up to the set of zero Hausdorff measure and lines orthogonal to them. We additionally assume that the surface is symmetric with respect to some geodesic line. Finding this kind of network is reduced to finding a solution to the nonlinear conjugate Beltrami equation in the unit disk, which degenerates at the points at which the Jacobian of the mapping, carried out by the solution, turns to either zero or infinity. The need for this kind of parametrization is associated with the search for equilibrium forms of liquid droplets in a model that takes into account the presence of an intermediate layer when studying the equilibrium state of a two-phase medium. Using the two-fold method of successive approximations, we establish the presence of a mapping of the unit disk to itself satisfying the Beltrami equation. The constructed mapping is continuous on the closed disk. The real part of the map has the generalized first order derivatives square-integrable on the upper half of the disk and the generalized first order derivatives of the imaginary part are locally square-integrable on it, and the map has the Lusin property. The inverse map, defined almost everywhere on the disk, is almost everywhere injective in it and has the Lusin property. The map found defines an almost global semi-geodesic parametrization of the surface and is a natural generalization of the -quasi-conformal maps introduced by R. Finn and J. Serrin. [ABSTRACT FROM AUTHOR]

In this work, we propose and study a framework of generalized proximal point algorithms associated with a maximally monotone operator. We indicate sufficient conditions on the regularization and relaxation parameters of generalized proximal point algorithms for the equivalence of the boundedness of the sequence of iterations generated by this algorithm and the non-emptiness of the zero set of the maximally monotone operator, and for the weak and strong convergence of the algorithm. Our results cover or improve many results on generalized proximal point algorithms in our references. Improvements of our results are illustrated by comparing our results with related known ones. [ABSTRACT FROM AUTHOR]

The focus of this paper is to obtain weak and linear convergence analysis of the subgradient extragradient method with alternated inertial step for solving equilibrium problems in real Hilbert spaces. The proposed method uses self-adaptive step sizes. Weak convergence is established without Lipschitz constant of the bifunction as an input parameter. Linear convergence is obtained without the modulus of strong pseudomonotonicity and Lipschitz constant as input parameters. We report some priori and posteriori error estimates and some numerical experiments to illustrate the behavior of our proposed method with related methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider discounted penalty functions, also called Gerber-Shiu functions, in a Markovian shot-noise environment. At first, we exploit the underlying structure of piecewise-deterministic Markov processes (PDMPs) to show that these penalty functions solve certain partial integro-differential equations (PIDEs). Since these equations cannot be solved exactly, we develop a numerical scheme that allows us to determine an approximation of such functions. These numerical solutions can be identified with penalty functions of continuous-time Markov chains with finite state space. Finally, we show the convergence of the corresponding generators over suitable sets of functions to prove that these Markov chains converge weakly against the original PDMP. That gives us that the numerical approximations converge to the discounted penalty functions of the original Markovian shot-noise environment. [ABSTRACT FROM AUTHOR]

We study the forward-backward-forward splitting method with alternated inertial step in this paper. We consider under-relaxation and over-relaxation approaches to the proposed method and obtain weak convergence results in both cases in real Hilbert spaces. Linear convergence alongside priori and apriori error estimates are obtained under some standard assumptions. Finally, we give some numerical implementations and comparisons to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a modification of the forward–backward splitting method with an adaptive step size rule for inclusion problems in real Hilbert spaces. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish weak convergence of the proposed algorithm. Moreover, if the single-valued operator is cocoercivity, then the proposed algorithm strongly converges to the unique solution of the problem with an R-linear rate. Finally, we give several numerical experiments to illustrate the convergence of the proposed algorithm and also to compare them with others. [ABSTRACT FROM AUTHOR]

Consider a general branching process, a.k.a. Crump–Mode–Jagers process, generated by a perturbed random walk η1, ξ1 + η2, ξ1 + ξ2 + η3, ..., where (ξ1, η1), (ξ2, η2), ... are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by Nj(t) the number of the jth generation individuals with birth times ⩽ t. Assume that j = j(t) → ∞ and j(t) = o(ta) as t → ∞ for some explicitly given a > 0 (to be specified in the paper). The corresponding jth generation belongs to the set of intermediate generations. We provide sufficient conditions under which the finite-dimensional distributions of the process (N⌊j(t)u⌋(t))u > 0, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean. [ABSTRACT FROM AUTHOR]

This paper deals with the weak convergence of nonparametric estimators of the multidimensional and multidimensional-multivariate renewal functions on Skorohod topology spaces. It is an extension of Harel et al. (J Math Anal Appl 189:240–255, 1995) from the one-dimensional case to the multivariate and multidimensional case. The estimators are based on a sequence of non-negative independent and identically distributed (iid) random vectors. They are expressed as infinite sums of k-folds convolutions of the empirical distribution function. Their weak convergence study heavily rests on that of the empirical distribution function. [ABSTRACT FROM AUTHOR]

Consider the Krasnosel’skiĭ-Mann iteration in real Hilbert spaces. In this note, we suggest weaker assumptions on the involved two factors of an accelerated version, and we use them to prove its weak convergence and analyze its rate of convergence. Furthermore, we suggest a practical way of choosing adaptive acceleration factors and confirm its numerical superiority. [ABSTRACT FROM AUTHOR]

Consider a general branching process, a.k.a. Crump–Mode–Jagers process, generated by a perturbed random walk η1, ξ1 + η2, ξ1 + ξ2 + η3, ..., where (ξ1, η1), (ξ2, η2), ... are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by Nj(t) the number of the jth generation individuals with birth times ⩽ t. Assume that j = j(t) → ∞ and j(t) = o(ta) as t → ∞ for some explicitly given a > 0 (to be specified in the paper). The corresponding jth generation belongs to the set of intermediate generations. We provide sufficient conditions under which the finite-dimensional distributions of the process (N⌊j(t)u⌋(t))u > 0, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean. [ABSTRACT FROM AUTHOR]

In this work, we are interested in solving a convex minimization problem in real Hilbert spaces. We propose a new modified proximal algorithm using the inertial extrapolation and the linesearch technique. Its weak convergence theorems are established under mild conditions. Numerical experiments are presented to illustrate the performance of the proposed algorithm in image deblurring. [ABSTRACT FROM AUTHOR]

In this paper, we present a modification of the forward-backward splitting method with inertial extrapolation step and self-adaptive step sizes to solve variational inequalities in a quasi-monotone setting. Our proposed method involves one computation of the projection onto the feasible set and one evaluation of the operator per iteration, which is simpler than most methods available in the literature to solve similar problems. We first establish weak convergence result when the set of solutions of the Minty formulation of the variational inequality is nonempty in infinite dimensional Hilbert spaces under appropriate conditions. Next, we give linear convergence result when the operator is strongly pseudo-monotone. We also give numerical implementations of our proposed method and some comparisons with some other methods available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new splitting algorithm to find the zero of a monotone inclusion problem that features the sum of three maximal monotone operators and a Lipschitz continuous monotone operator in Hilbert spaces. We prove that the sequence of iterates generated by our proposed splitting algorithm converges weakly to the zero of the considered inclusion problem under mild conditions on the iterative parameters. Several splitting algorithms in the literature are recovered as special cases of our proposed algorithm. Another interesting feature of our algorithm is that one forward evaluation of the Lipschitz continuous monotone operator is utilized at each iteration. Numerical results are given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]