Approximation Of Fixed Points Research Articles

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces.

How to optimize container withholding decisions for reuse in the hinterland?

This study investigates how a hinterland consignee (importer) makes decisions regarding the storage of empty containers for reuse by a shipper (exporter). The system is modeled as a double-ended queue with non-zero matching times, limited truck resources, and both consignee and shipper having fixed withholding capacities. The consignee’s withholding threshold is strategically set to minimize overall transport and detention costs. We derive closed-form expressions for performance measures in the case of a single storage facility at the shipper, utilizing a matrix-based approach. We extend this methodology numerically to the general case. Additionally, we present an accurate fixed-point approximation facilitating the determination of performance measures and optimal threshold. Our findings show the importance of withholding decisions in import and balanced areas for cost reduction. In export areas, a policy of full reuse proves nearly optimal. Analyzing dynamic state-dependent consignee decisions via a Markov decision process, we establish that the optimal policy involves withholding thresholds increasing with stored quantity at the shipper. While optimal, state-dependent thresholds yield limited cost savings compared to a fixed threshold, suggesting minimal impact from consignee-shipper information sharing. Additionally, we examine the influence of variability in matching and production times, observing decreased costs with reduced variability, particularly in export areas, but with minor impacts on withholding decisions.

Results of coupled iterative fixed point approximation

Let X be a Banach space and F a subset of X. In this paper, we showed weak and strong convergence theorems for a Krasnoselskij-type iterative method to approximate coupled solution of nonexpansive operator E: FxF-> F, where F is nonempty, depending on two variables. Furthermore, we obtained a fixed point for a (b, \theta, L)-almost contraction operator E: X +X-> X using the Krasnoselskij iteration {xn}n=0infty.

Fixed point approximation via a new faster iteration process in Banach spaces with an application

Fixed point approximation via a new faster iteration process in Banach spaces with an application

A new fixed point approximation method for solving third-order BVPs based on Green's function

A new fixed point approximation method for solving third-order BVPs based on Green's function

Fixed-Point Approximation of Operators Satisfying (RCSC)—Condition in CAT(0) Spaces

In this research article, we have proved strong and Δ-convergence results for mapping satisfying (RCSC) condition via M-iteration process in CAT(0) spaces. Numerical examples are provided to show the superiority of our results over other existing results and to illustrate the faster convergence of the M iterative scheme as compared to many well-known iterative schemes. In this process, many results are improved in the current literature of CAT(0) spaces.

End-to-end Privacy Preserving Training and Inference for Air Pollution Forecasting with Data from Rival Fleets

Privacy-preserving machine learning (PPML) promises to train machine learning (ML) models by combining data spread across multiple data silos. Theoretically, secure multiparty computation (MPC) allows multiple data owners to train models on their joint data without revealing the data to each other. However, the prior implementations of this secure training using MPC have three limitations: they have only been evaluated on CNNs, and LSTMs have been ignored; fixed point approximations have affected training accuracies compared to training in floating point; and due to significant latency overheads of secure training via MPC, its relevance for practical tasks with streaming data remains unclear. The motivation of this work is to report our experience of addressing the practical problem of secure training and inference of models for urban sensing problems, e.g., traffic congestion estimation, or air pollution monitoring in large cities, where data can be contributed by rival fleet companies while balancing the privacy-accuracy trade-offs using MPC-based techniques.Our first contribution is to design a custom ML model for this task that can be efficiently trained with MPC within a desirable latency. In particular, we design a GCN-LSTM and securely train it on time-series sensor data for accurate forecasting, within 7 minutes per epoch. As our second contribution, we build an end-to-end system of private training and inference that provably matches the training accuracy of cleartext ML training. This work is the first to securely train a model with LSTM cells. Third, this trained model is kept secret-shared between the fleet companies and allows clients to make sensitive queries to this model while carefully handling potentially invalid queries. Our custom protocols allow clients to query predictions from privately trained models in milliseconds, all the while maintaining accuracy and cryptographic security.

Parsimony and parameter estimation for mixtures of multivariate leptokurtic-normal distributions

Mixtures of multivariate leptokurtic-normal distributions have been recently introduced in the clustering literature based on mixtures of elliptical heavy-tailed distributions. They have the advantage of having parameters directly related to the moments of practical interest. We derive two estimation procedures for these mixtures. The first one is based on the majorization-minimization algorithm, while the second is based on a fixed point approximation. Moreover, we introduce parsimonious forms of the considered mixtures and we use the illustrated estimation procedures to fit them. We use simulated and real data sets to investigate various aspects of the proposed models and algorithms.

A queuing model for ventilator capacity management during the COVID-19 pandemic.

We applied a queuing model to inform ventilator capacity planning during the first wave of the COVID-19 epidemic in the province of British Columbia (BC), Canada. The core of our framework is a multi-class Erlang loss model that represents ventilator use by both COVID-19 and non-COVID-19 patients. Input for the model includes COVID-19 case projections, and our analysis incorporates projections with different levels of transmission due to public health measures and social distancing. We incorporated data from the BC Intensive Care Unit Database to calibrate and validate the model. Using discrete event simulation, we projected ventilator access, including when capacity would be reached and how many patients would be unable to access a ventilator. Simulation results were compared with three numerical approximation methods, namely pointwise stationary approximation, modified offered load, and fixed point approximation. Using this comparison, we developed a hybrid optimization approach to efficiently identify required ventilator capacity to meet access targets. Model projections demonstrate that public health measures and social distancing potentially averted up to 50 deaths per day in BC, by ensuring that ventilator capacity was not reached during the first wave of COVID-19. Without these measures, an additional 173 ventilators would have been required to ensure that at least 95% of patients can access a ventilator immediately. Our model enables policy makers to estimate critical care utilization based on epidemic projections with different transmission levels, thereby providing a tool to quantify the interplay between public health measures, necessary critical care resources, and patient access indicators.

A numerical scheme for the optimal control of groundwater pollution

AbstractA scheme is proposed for the numerical study of optimal control problems related to groundwater quality management in the case of nonpoint‐source pollution. The corresponding state equation is a system of coupled nonlinear partial differential equations. The approximation is given by a finite volume scheme based on a two‐point flux approximation with upwind mobilities embedded in an iterative fixed point approximation. The analysis of the convergence of the scheme allows to establish existence and uniqueness results under reasonable assumptions. Numerical illustrations of the performance of the algorithm are given in realistic situations.

Computational comparative analysis of fixed point approximations of generalized $ \alpha $-nonexpansive mappings in hyperbolic spaces

<abstract><p>In this article, we use the Picard-Thakur hybrid iterative scheme to approximate the fixed points of generalized $ \alpha $-nonexpansive mappings. For generalized $ \alpha $-nonexpansive mappings in hyperbolic spaces, we show several weak and strong convergence results. It is proved numerically and graphically that the Picard-Thakur hybrid iterative scheme converges more faster than other well-known hybrid iterative methods for generalized $ \alpha $-nonexpansive mappings. We also present an application to Fredholm integral equation.</p></abstract>

On fixed-point approximations for a class of nonlinear mappings based on the JK iterative scheme with application

<abstract><p>The goal of this manuscript is to introduce the JK iterative scheme for the numerical reckoning of fixed points in generalized contraction mappings. Also, weak and strong convergence results are investigated under this scheme in the setting of Banach spaces. Moreover, two numerical examples are given to illustrate that the JK iterative scheme is more effective than some other iterative schemes in the literature. Ultimately, as an application, the JK iterative scheme is applied to solve a discrete composite functional differential equation of the Volterra-Stieljes type.</p></abstract>

Generalization of Fixed Point Approximation of Contraction and Suzuki Generalized Non-Expansive Mappings in Banach Domain

By principal motivation from the results of the new iterative scheme that produces faster results than K-iteration. In this article, we study generalized results by a new iteration scheme to approximate fixed points of generalized contraction and Suzuki non-expansive mappings. We establish strong convergence results of generalized contraction mappings of closed convex Banach space and also deduce data dependent results. Furthermore, we prove some weak and strong convergence theorems in the sense of generalized Suzuki non-expansive mapping by applying condition (C).

Fixed point approximation for a class of generalized nonexpansive multi-valued mappings in Banach spaces

In this paper, we propose a new iteration process, called multi-valued F-iteration process, for the approximation of fixed points. We introduce a new class of multi-valued generalized nonexpansive mappings satisfying a B_{gamma ,mu } property. Moreover, we establish certain weak and strong convergence theorems in uniformly convex Banach spaces. We also discuss the stability of the modified F-iteration process. Furthermore, a numerical example is presented to illustrate the superiority of our results.

Approximations of Fixed Point of Nonexpansive Mappings in Banach Spaces

n this paper, we used condition (E) to study approximation on the Banach space, demonstrating the convergence theorem as well as an example that supports the main theorem. Iterative fixed point approximation for nonlinear operators is a novel area of investigation. As a result, the literature contains a number of iterative techniques for overcoming such impediments and improving the rate of convergence.

Some convergence results using K iteration process in Banach Spaces

In this paper, we establish several strong and weak convergence results for a class of mappings which is essentially wider than the class of Suzuki mappings using [Formula: see text] iteration process in the context of uniformly convex Banach spaces. We also provide an example for support of our results. The obtained results are the extension and improvement of some recent results in the fixed point approximations theory.

Riemannian stochastic fixed point optimization algorithm

This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets, such as the intersection of many closed convex sets, the set of all minimizers of a nonsmooth convex function, and the intersection of sublevel sets of nonsmooth convex functions. We focus on adaptive learning rate optimization algorithms, which adapt step-sizes (referred to as learning rates in the machine learning field) to find optimal solutions quickly. We then propose a Riemannian stochastic fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with the adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm for nonsmooth convex and smooth nonconvex optimization. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem with a guaranteed convergence rate. This paper also provides numerical comparisons that demonstrate the effectiveness of the proposed algorithms with formulas based on the adaptive learning rate optimization algorithms, such as Adam and AMSGrad.

Generalized multiplicative nonlinear elastic matching distance measure and fixed point approximation

In this paper, inspired by the concept of nonlinear elastic matching, generalized metrics, and multiplicative triangle inequality, we propose a ‐multiplicative metric space in which we propose a generalized notion of a distance function, , with degree 2 is a distance of 3 points in generalizing the ordinary distance between 2 points and G‐metric between 3 points. We first discuss ‐multiplicative metric space mapping and later on the fundamental properties of ‐multiplicative metric are also discussed. Fundamental properties and several examples for the ‐multiplicative metric are also given. Moreover, topological properties on the ‐multiplicative metric space including the convergence of sequences and the continuity of mappings on the ‐multiplicative metric space are studied. Finally, we generalize some well‐known fixed point theorems including Banach contraction mapping as ‐multiplicative contraction mapping and proved some fixed point theorems on complete ‐multiplicative metric spaces. We prove the existence and uniqueness of a solution to a system of ‐multiplicative integral equations as an application. Our results generalize many pre‐existing results in literature.

Fixed Point Approximation of Monotone Nonexpansive Mappings in Hyperbolic Spaces

Fixed points of monotone α -nonexpansive and generalized β -nonexpansive mappings have been approximated in Banach space. Our purpose is to approximate the fixed points for the above mappings in hyperbolic space. We prove the existence and convergence results using some iteration processes.

Fixed Point Approximation for a Class of Generalized Nonexpansive Mappings in Hadamard Spaces

In this paper, we establish strong and Δ convergence results for mappings satisfying condition B γ , μ through a newly introduced iterative process called JA iteration process. A nonlinear Hadamard space is used the ground space for establishing our main results. A novel example is provided for the support of our main results and claims. The presented results are the good extension of the corresponding results present in the literature.