Fixed-point Iteration Research Articles

Anderson acceleration of Picard/Newton iteration based on grad-div stabilization for the Smagorinsky model

In this paper, we consider Anderson acceleration of one- and two-level Picard/Newton iterations based on the grad-div stabilization for the stationary Smagorinsky model at high Reynolds number. First, based on the grad-div stabilization, we propose the Anderson-accelerated Picard iteration, and then we introduce the Newton iteration at the end of iteration to accelerate convergence. Second, to reduce the computational cost, we consider a two-level algorithm, i.e., we use the previous Anderson acceleration of Picard/Newton iteration for the Smagorinsky model on a coarse mesh and then solve a generalized Stokes problem on fine mesh by the grad-div stabilized Picard iteration. The proposed algorithm not only improves convergence and reduces computational costs but also enhances the capability to simulate the fluid flow with high Reynolds number. Several numerical experiments have been conducted to demonstrate the numerical performance of the proposed algorithms.

Controllability results for Sobolev-type analytic resolvent nonlocal neutral functional integrodifferential inclusions with impulses

This article aims to explore the existence and controllability strategies for Sobolev-type nonlocal neutral functional integrodifferential inclusions with impulses via the resolvent operator. Initially, the article is divided into two main parts. The first part is dedicated to establishing the existence of mild solutions for the given system and introducing a new set of sufficient conditions for approximate controllability. These findings are derived under the assumption that the corresponding linear system is approximately controllable. The second part builds on these results by establishing the existence of an optimal control pair for the Lagrange problem (P) based on an appropriate set of sufficient conditions. The development of these results heavily relies on Bohnenblust–Karlin's fixed-point method, semigroup theory, multivalued analysis, and the resolvent operator. Finally, an example is provided to clarify the concepts discussed.

Optimal stopping of Gauss–Markov bridges

Abstract We solve the non-discounted, finite-horizon optimal stopping problem of a Gauss–Markov bridge by using a time-space transformation approach. The associated optimal stopping boundary is proved to be Lipschitz continuous on any closed interval that excludes the horizon, and it is characterized by the unique solution of an integral equation. A Picard iteration algorithm is discussed and implemented to exemplify the numerical computation and geometry of the optimal stopping boundary for some illustrative cases.

The Good, the Bad and the Picky: Consumer Heterogeneity and the Reversal of Product Ratings

We study the impact of consumer heterogeneity on online ratings. Consumers differ in their experience, which can affect both their choices and ratings. Thus, biases in average ratings can arise when the opinions of experienced and novice users are aggregated. We first build a two-period model to characterize the biases’ drivers and consequences. We test our theory combining data from IMDb and MovieLens, two well-known movie ratings platforms. We proxy users’ experience with the total number of ratings posted on the platforms. First, using external measures of quality, such as the Academy awards and nominations, we show that, on both platforms, experienced users, on average, rate movies of higher quality compared with novices. Moreover, they post more stringent ratings than novices for more than 98% of movies. Combined, these imply a compression in aggregate ratings, and thus a bias against high quality movies. We then propose a simple, fixed-point algorithm to debias ratings. Our debiased ratings demonstrate the presence of ranking reversals for more than 8% of comparisons in our sample. As a result, our debiased ratings better correlate with external measures of quality. This paper was accepted by Eric Anderson, marketing. Supplemental Material: The online appendices and data files are available at https://doi.org/10.1287/mnsc.2022.03281 .

Exact H∞ optimization of dynamic vibration absorbers: Univariate-polynomial-based algorithm and operability analysis

H∞ optimization of dynamic vibration absorbers (DVAs) to minimize the maximum response amplitude of primary structures is a classic topic. The commonly used fixed-point method only provides approximate solutions requiring the primary structure is undamped. Instead, we perform exact optimization and investigate the less-reported parametric effects on optimization operability. To handle the known restrictions posed by grounded dampers, a typical DVA model mounted on a damped primary structure, with components connected to both the primary and the base, is considered. We explore three elaborated cases depending on the grounded dampers distributing in the primary structure and the DVA. Our findings reveal that the frequency responses of the primary structure with dual resonant peaks of equal height may not be the global optimum, and we establish a nontrivial necessary condition for operable exact optimization. Furthermore, we elucidate the effects of structural arrangements on the optimized results and provide design rules to maximize vibration suppression performance. The optimization follows the proposal of a so-called resultant-based algorithm that guarantees global optimum with high efficiency and a generalizable core by exclusively constructing univariate polynomial equations. This study contributes a systematic analysis framework alongside efficient calculation tools for exact optimization and DVA performance evaluation.

Deep Single Image Defocus Deblurring via Gaussian Kernel Mixture Learning.

This paper proposes an end-to-end deep learning approach for removing defocus blur from a single defocused image. Defocus blur is a common issue in digital photography that poses a challenge due to its spatially-varying and large blurring effect. The proposed approach addresses this challenge by employing a pixel-wise Gaussian kernel mixture (GKM) model to accurately yet compactly parameterize spatially-varying defocus point spread functions (PSFs), which is motivated by the isotropy in defocus PSFs. We further propose a grouped GKM (GGKM) model that decouples the coefficients in GKM, so as to improve the modeling accuracy with an economic manner. Afterward, a deep neural network called GGKMNet is then developed by unrolling a fixed-point iteration process of GGKM-based image deblurring, which avoids the efficiency issues in existing unrolling DNNs. Using a lightweight scale-recurrent architecture with a coarse-to-fine estimation scheme to predict the coefficients in GGKM, the GGKMNet can efficiently recover an all-in-focus image from a defocused one. Such advantages are demonstrated with extensive experiments on five benchmark datasets, where the GGKMNet outperforms existing defocus deblurring methods in restoration quality, as well as showing advantages in terms of model complexity and computational efficiency.

Decay character and the semilinear structurally damped σ-evolution equations with two dissipative terms

In this paper, we study the decay rates of the solution and its derivatives to the Cauchy problem for the linear damped σ-evolution equation with two dissipative dampingsutt+(−Δ)σu+(−Δ)δ1ut+(−Δ)δ2ut=0 that are expressed in terms of decay characters of the initial data. Furthermore, by using the decay rates of solutions to the linear part and the fixed-point method, we investigate the existence and decay rate of the global solution with small data of the corresponding semi-linear problems with the nonlinear term of power type |ut|p, for the supercritical case p>p⁎. For some special data spaces, the optimality of p⁎ is guaranteed by nonexistence results in the subcritical case p<p⁎ and the critical case p=p⁎.

Certain Novel Fixed-Point Theorems Applied to Fractional Differential Equations

In this paper, we introduce a new class of contractions in normed spaces, referred to as generalized enriched Kannan contractions. These contractions expand the familiar enriched Kannan contractions to three-point versions, broadening the scope of Kannan contractions. These mappings are typically discontinuous, except at the fixed points, where they exhibit continuity, similar to enriched Kannan mappings. However, through suitable examples, we demonstrate that these two classes of mappings are distinct from one another. We present new results for generalized enriched Kannan contractions. Additionally, by incorporating conditions of continuity and asymptotic regularity, we extend the class of operators to which fixed-point methods can be applied. Additionally, we derive two more results for generalized enriched Kannan contractions in normed spaces, without the requirement that they be Banach spaces. Finally, we use our main result to demonstrate the existence of solutions for a boundary value problem involving a fractional differential equation.

Effective Rates for Iterations Involving Bregman Strongly Nonexpansive Operators

We develop the theory of Bregman strongly nonexpansive maps for uniformly Fréchet differentiable Bregman functions from a quantitative perspective. In that vein, we provide moduli witnessing quantitative versions of the central assumptions commonly used in this field on the underlying Bregman function and the Bregman strongly nonexpansive maps. In terms of these moduli, we then compute explicit and effective rates for the asymptotic regularity of Picard iterations of Bregman strongly nonexpansive maps and of the method of cyclic Bregman projections. Further, we also provide similar rates for the asymptotic regularity and metastability of a strongly convergent Halpern-type iteration of a family of such mappings and we use these new results to derive rates for various special instantiations like a Halpern-type proximal point algorithm for monotone operators in Banach spaces as well as Halpern-Mann- and Tikhonov-Mann-type methods.

Quasi-linear homogenization for large-inertia laminar transport across permeable membranes

Porous membranes are thin solid structures that allow the flow to pass through their tiny openings, called pores. Flow inertia may play a significant role in several filtration flows of natural and engineering interest. Here, we develop a predictive macroscopic model to describe solvent and solute flows past thin membranes for non-negligible inertia. We leverage homogenization theory to link the solvent velocity and solute concentration to the jumps of solvent stress and solute flux across the membrane. Within this framework, the membrane acts as a boundary separating two distinct fluid regions. These jump conditions rely on several coefficients, stemming from closure problems at the microscopic pore scale. Two approximations for the advective terms of Navier–Stokes and advection–diffusion equations are introduced to include inertia in the microscopic problem. The approximate inertial terms couple the micro- and macroscopic fields. Here, this coupling is solved numerically using an iterative fixed-point procedure. We compare the resulting models against full-scale simulations, with a good agreement both in terms of averaged values across the membrane and far-field values. Eventually, we develop a strategy based on unsupervised machine learning to improve the computational efficiency of the iterative procedure. The extension of homogenization towards weak-inertia flow configurations as well as the performed data-driven approximation may find application in preliminary analyses as well as optimization procedures towards the design of filtration systems, where inertia effects can be instrumental in broadening the spectrum of permeability and selectivity properties of these filters.

A fractal-fractional mathematical model for COVID-19 and tuberculosis using Atangana–Baleanu derivative

ABSTRACT This study aims to develop a compartmental epidemiological model for the co-infection of COVID-19 and tuberculosis, incorporating a Holling type II treatment rate for individuals with tuberculosis, COVID-19, and dual infections while considering incomplete treatment in some TB cases. The model analysis examines the sub-models for COVID-19, TB, and the combined co-infection model. Using the fixed-point method, the research investigates the existence and uniqueness of solutions for the proposed model. It also explores a stability analysis to evaluate Ulam-Hyer’s reliability. Furthermore, it discusses and validates Lagrange’s interpolation polynomial through a specific case study to numerically compare different fractal and fractional orders.

Improving the Performance and Efficiency of Convolutional Neural Networks (CNNs) on Image Classification Tasks via Fixed-Point Quantization and Structured Pruning

This study explores the combined use of fixed-point quantization and structured pruning to optimize the performance and efficiency of convolutional neural networks (CNNs) in image classification tasks. These techniques can be used to reduce model size and computational complexity, making CNNs more suitable for deployment in resource-constrained environments such as mobile devices and embedded systems. Fixed-point quantization methods can reduce the bit-width of weights and activations, thereby reducing the computational load and memory footprint. On the other hand, structured pruning systematically removes unimportant convolutional filters or channels, which further reduces the model size and increases the inference speed. An experimental evaluation was performed on the ImageNet dataset using the ResNet-50 architecture. The results show that the combined strategy of quantization and pruning reduces the model size by up to 75% and increases the inference speed by 50%, while maintaining a classification accuracy of 74.5%, compared to 76.4% for the baseline model. Considering the significant increase in efficiency, a slight decrease in accuracy is acceptable. The results show that the integrated approach effectively compresses and accelerates the CNN model without a significant drop in accuracy, making it ideal for real-time applications.

Federated Primal Dual Fixed Point Algorithm

Federated Primal Dual Fixed Point Algorithm

Mesh adaptation and dynamic positioning for the efficient simulation of lifting hydrofoil flows

Accurate simulations of the flow around lifting hydrofoils are challenging, since they need to capture the flow near the foil surface precisely, represent the free surface, and take into account body motion and deformation. Therefore, these simulations are often computationally expensive.This paper studies numerical methods to limit the costs of hydrofoil simulation. Mesh adaptation is used to efficiently capture the free-surface flow, to resolve flow details around the foil surface, and to ensure the accuracy of mesh motion techniques, like overset meshing. For maximum precision of the boundary-layer flow, adaptation is started from dedicated body-aligned meshes.Hydrofoil flexibility is taken into account through a linear eigenmode-based reduced-order model of the structural response. This approach removes the need to couple directly the fluid and structure solvers and reduces the computational overhead for fluid–structure simulation. Equilibrium positions for flexible and rigid motion are determined with a fixed-point iteration based on quasi-Newton and approximate models for the forces respectively, which eliminate the need for costly time-accurate simulation.Test cases demonstrate that these methods work together, providing accurate simulations of realistic hydrofoils with reasonable computational costs. Mesh adaptation allows to target a specific numerical uncertainty through the refinement threshold parameter. Together with the body-aligned ‘sock’ meshes, adaptive refinement produces the same accuracy as non-adapted meshes for up to 10 times less CPU hours. Both fluid–structure interaction methods lead to computations which have the same convergence speeds as for non-moving bodies. Thus, foil flexibility can be simulated for a computational overhead below 40% and it leads to significantly better agreement with experiments.

On “A new three-step fixed point iteration scheme with strong convergence and applications”

&lt;p&gt;This note delves into the convergence analysis of several iterative methods and elucidates their behaviors. Furthermore, we demonstrate that the findings presented in “A new three-step fixed point iteration scheme with strong convergence and applications” are not entirely novel. In particular, some of the results either overlap with or restate previously established methods without introducing significant innovations.&lt;/p&gt;

Higher-order iterative decoupling for poroelasticity

For the iterative decoupling of elliptic–parabolic problems such as poroelasticity, we introduce time discretization schemes up to order five based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As the main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

How to Count Parrots: Comparing the Performance of Point and Transect Counts for Surveying Tasman Parakeets (Cyanoramphus cookii)

Obtaining precise estimates of population size and trends through time is important for the effective management and conservation of threatened species. For parrots (Psittaciformes: Psittacidae), obtaining such estimates can be challenging, particularly for cryptic species that occur in low densities in complex and/or fragmented habitats. We used a statistical resampling approach with the aim to compare the reliability and precision of counts for the critically endangered Tasman parakeet (Cyanoramphus cookii) that were taken using two methods on Norfolk Island (Pacific Ocean), namely, fixed-point counts and line transect counts. The detections obtained during fixed-point counts had better estimated precision (0.274) than line transect counts (0.476). The fixed-point method was also more efficient, yielding 1.338 parakeet detections per count compared to the 0.642 parakeet detections per count obtained by the line transect method. Although Tasman parakeets can be detected by either of these methods, our research demonstrates that the fixed-point method is more precise and reliable. These findings can help prioritise resources for the long-term monitoring of recovering populations of this species and similar island species.

Preconditioned inexact fixed point iteration method for solving tensor absolute value equation

Preconditioned inexact fixed point iteration method for solving tensor absolute value equation

THE FIXED POINT ITERATIONS METHOD FOR THE CONTRACTION MAPPING

This paper studies the fixed point iterations method that considers the contraction mapping and one of its applications for the parameter re-estimation of the normal (Gaussian) distribution, where the presence of outliers is considered. In order to study the observed method Banach’s fixed point theorem is presented, where it is shown that the contraction property is directly related to the Jacobian of the observed mapping. Furthermore, convergence analysis is conducted in order to estimate the convergence order of the observed model.

Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory

AbstractWe address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.