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Recently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM in terms of dual objective. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak–Łojasiewicz (PŁ) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two scenarios. [ABSTRACT FROM AUTHOR]

Let X, Y be continuous random variables with unknown distributions. The aim of this paper is to study the problem of estimating the probability θ : = P (X < Y) based on independent random samples from the distributions of X ′ , Y ′ , ζ and η , where X ′ = X + ζ , Y ′ = Y + η and X, Y, ζ , η are mutually independent random variables. In this context, ζ , η are referred to as measurement errors. We apply the ridge-parameter regularization method to derive a nonparametric estimator for θ depending on two parameters. Our estimator is shown to be consistent with respect to mean squared error if the characteristic functions of ζ , η only vanish on Lebesgue measure zero sets. Under some further assumptions on the densities of X, Y, ζ and η , we obtain some upper and lower bounds on the convergence rate of the estimator. A numerical example is also given to illustrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

In this work, we investigate a contraction-type method for solving monotone variational inclusion problems in real Hilbert spaces. We obtain strong convergence theorems for two algorithms with a self-adaptive step size for solving monotone variational inclusions. The advantage of our algorithms is that we do not require a cocoercivity assumption nor do we need to know the Lipschitz-type constant of the single-valued operator. Moreover, a convergence rate is derived in the case where one of the operators is maximally and strongly monotone, and the other is monotone and Lipschitz continuous. The performance of our proposed methods is illustrated by numerical experiments regarding signal recovery. Our results improve and extend some known results, and our experiments show that our proposed algorithms are efficient and outperform other algorithms which are available in the literature. [ABSTRACT FROM AUTHOR]

Stochastic approximation (SA) type methods have been well studied for solving single-stage stochastic variational inequalities (SVIs). This paper proposes a dynamic stochastic projection method (DSPM) for solving multistage SVIs. In particular, we investigate an inexact single-stage SVI and present an inexact stochastic projection method (ISPM) for solving it. Then we give the DSPM to a three-stage SVI by applying the ISPM to each stage. We show that the DSPM can achieve an O (1 ϵ 2) convergence rate regarding to the total number of required scenarios for the three-stage SVI. We also extend the DSPM to the multistage SVI when the number of stages is larger than three. The numerical experiments illustrate the effectiveness and efficiency of the DSPM. [ABSTRACT FROM AUTHOR]

We aim to solve the linearly constrained convex optimization problem whose objective function is the sum of a differentiable function and a non-differentiable function. We first propose an inertial continuous primal-dual dynamical system with variable mass for linearly constrained convex optimization problems with differentiable objective functions. The dynamical system is composed of a second-order differential equation with variable mass for the primal variable and a first-order differential equation for the dual variable. The fast convergence properties of the proposed dynamical system are proved by constructing a proper energy function. We then extend the results to the case where the objective function is non-differentiable, and a new accelerated primal-dual algorithm is presented. When both variable mass and time scaling satisfy certain conditions, it is proved that our new algorithm owns fast convergence rates for the objective function residual and the feasibility violation. Some preliminary numerical results on the ℓ 1 – ℓ 2 minimization problem demonstrate the validity of our algorithm. [ABSTRACT FROM AUTHOR]

This paper investigates the fast chemical diffusion limit from a parabolic-parabolic Keller-Segel system to the corresponding parabolic-elliptic Keller-Segel system by constructing approximate solutions with an appropriate order via an asymptotic expansion. Nonlinear stability of the precise initial layer is characterized with an exact convergence rate by using basic energy method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of state constrained linear parabolic optimal control problems. Instead of treating the inequality state constraints directly, we reformulate the problem as an equality-constrained optimization problem, and then apply the augmented Lagrangian method (ALM) to solve it. We prove the convergence of the ALM without any existence or regularity assumptions on the corresponding Lagrange multipliers, which is an essential complement to the classical theoretical results for the ALM because restrictive regularity assumptions are usually required to guarantee the existence of the Lagrange multipliers associated with the state constraints. In addition, under an appropriate choice of penalty parameter sequence, we can obtain a super-linear non-ergodic convergence rate for the ALM. Computationally, we apply a semi-smooth Newton (SSN) method to solve the ALM subproblems and design an efficient preconditioned conjugate gradient method for solving the Newton systems. Some numerical results are given to illustrate the effectiveness and efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

Numerical models based on seepage-deformation coupling governing equations are often used to simulate soil hydrodynamics and deformation in unsaturated porous media. Among them, Picard iteration method with pressure head as the main variable is widely used because of its simplicity and ability to deal with partial saturation conditions. It is well known that the method is prone to convergence failure under some unfavorable flow conditions and is also computationally time-consuming. In this study, the soil–water characteristic curve (SWCC) of unsaturated soil described by the exponential function is used to linearize the coupling equations to overcome the repeated assembly of nonlinear ordinary differential equations. The finite element method with six-node triangular element is used to discretely linearize the coupling governing equations. Further, the classical Gauss–Seidel iterative method (GS) can be used to solve the linear equations generated from the linearized coupling equations. However, the convergence rate of GS seriously restricts the ill-condition of the linear equations, especially when the condition number of linear equations is much larger than 1.0. Thus, we propose an improved Gauss–Seidel iterative methods MP(m)-GSCMGI by combining multistep preconditioning and cascadic multigrid. The applicability of the proposed methods in simulating variably saturated flow and deformation in unsaturated porous media is verified by numerical examples. The results show that the proposed improved methods have faster convergence rate and computational efficiency than the conventional Picard and GS. The hybrid improved method MP(m)-GSCMGI can achieve more robust convergence and economical simulation. [ABSTRACT FROM AUTHOR]

In 2015, Mao (J. Comput. Appl. Math., 290, 370–384, 2015) proposed the truncated Euler-Maruyama (EM) method for stochastic differential equations (SDEs) under the local Lipschitz condition plus the Khasminskii-type condition. Adapting the truncation idea from Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016), lots of modified truncated EM methods are proposed (see, e.g., Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) and the references therein). These truncated-type EM methods Mao (J. Comput. Appl. Math., 290, 370–384, 2015) and Mao (Appl. Numer. Math., 296, 362–375, 2016) and Guo et al. (Appl. Numer. Math., 115, 235–251, 2017,) and Lan and Xia (J. Comput. Appl. Math., 334, 1–17, 2018) and Li et al. (IMA J. Numer. Anal., 39(2), 847–892, 2019) construct the numerical solutions by defining an appropriate truncation projection, then applying the truncation projection to the numerical solutions before substituting them into the coefficients in each iteration. In this paper, we develop a new class of explicit schemes for superlinear stochastic differential equations with piecewise continuous arguments (SDEPCAs), which are defined by directly truncating the coefficients. Our method has a more simple structure and is easier to implement. We not only show the explicit schemes converge strongly to SDEPCAs but also demonstrate the convergence rate is optimal 1/2. A numerical example is provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper develops a second-order explicit predictor–corrector numerical approach for solving a mathematical model on the dynamics of cytokine expressions and human immune cell activation in response to the bacterium staphylococcus aureus (S. aureus). The proposed algorithm is at least zero-stable and second-order accurate. Mathematical modeling works that analyze the human body in response to some antigens have predicted concentrations of a broad range of cells and cytokines. This study deals with a coupled cellular-cytokine model which predicts cytokine expressions in response to gram-positive bacteria S. aureus. Tumor necrosis factor alpha, interleukin 6, interleukin 8 and interleukin 10 are included to assess the relationship between cytokine release from macrophages and the concentration of the S. aureus antigen. Ordinary differential equations are used to model cytokine levels while the cellular responses are modeled by partial differential equations. Interactions between both components provide a more robust and complete systems of immune activation. In the numerical simulations, a low concentration of S. aureus is used to measure cellular activation and cytokine expressions. Numerical experiments indicate how the human immune system responds to infections from different pathogens. Furthermore, numerical examples suggest that the new technique is faster and more efficient than a large class of statistical and numerical schemes discussed in the literature for systems of nonlinear equations and can serve as a robust tool for the integration of general systems of initial-boundary value problems. [ABSTRACT FROM AUTHOR]

AbstractLow-rank approximation is a popular strategy to tackle the “big n problem” associated with large-scale Gaussian process regressions. Basis functions for developing low-rank structures are crucial and should be carefully specified. Predictive processes simplify the problem by inducing basis functions with a covariance function and a set of knots. The existing literature suggests certain practical implementations of knot selection and covariance estimation; however, theoretical foundations explaining the influence of these two factors on predictive processes are lacking. In this paper, the asymptotic prediction performance of the predictive process and Gaussian process predictions are derived and the impacts of the selected knots and estimated covariance are studied. The use of support points as knots, which best represent data locations, is advocated. Extensive simulation studies demonstrate the superiority of support points and verify our theoretical results. Real data of precipitation and ozone are used as examples, and the efficiency of our method over other widely used low-rank approximation methods is verified. [ABSTRACT FROM AUTHOR]

Wireless Sensor Networks (WSNs) play a significant role in monitoring changes in the physical environment, relying on a subset of designated nodes known as the Connected Dominating Set (CDS) to facilitate communication and serve as the network's virtual backbone for routing and clustering. Despite the various algorithms for CDS construction, the challenges persist in terms of construction cost and scalability. In this context, this paper introduces the utilisation of the Hunger Games Search Optimisation Algorithm (HGSOA) for Connected Dominating Set Construction in Wireless Sensor Networks (HGSOA-CDS-WSN). The primary intention of this paper is “to develop a technique capable of constructing small-sized CDS with reduced construction costs, particularly suitable for uniformly or randomly distributed nodes”. The proposed HGSOA-CDS-WSN method is implemented in MATLAB based on various metrics. The results are compared with those of existing techniques, such as the Parallel Structure from Motion for UAV Pictures using Weighted CDS (PSfM-CDS-WSN), Polyhedra Associated with Locating-Dominating, Open Locating-Dominating, and Locating Total-Dominating Sets in Graphs (PAL-CDS-WSN), and Sustainable Maintenance of Connected Dominating Set by Solar Energy Harvesting for Internet of Things Networks (SHE-CDS-WSN). The proposed technique achieves lower delay by 13.24%, 25.64%, and 9.96%, and higher throughput by 18.34%, 24.55%, and 23.04% compared to the existing methods. This systematic evaluation underscores the efficacy and superiority of the proposed HGSOA-CDS-WSN technique in terms of connected dominating set size, construction time, energy efficiency, network stability, and overall performance, highlighting its potential to advance wireless sensor network applications. [ABSTRACT FROM AUTHOR]

ABSTRACT This article presents and analyzes a family of three‐term conjugate gradient projection methods with the inertial technique for solving large‐scale nonlinear pseudo‐monotone equations with convex constraints. The generated search direction exhibits good properties independent of line searches. The global convergence of the family is proved without the Lipschitz continuity of the underlying mapping. Furthermore, under the locally Lipschitz continuity assumption, we conduct a thorough analysis related to the asymptotic and non‐asymptotic global convergence rates in terms of iteration complexity. To our knowledge, this is the first iteration‐complexity analysis for inertial gradient‐type projection methods, in the literature, under such a assumption. Numerical experiments demonstrate the computational efficiency of the family, showing its superiority over three existing inertial methods. Finally, we apply the proposed family to solve practical problems such as ℓ2$$ {\ell}_2 $$‐regularized logistic regression, sparse signal restoration and image restoration problems, highlighting its effectiveness and potential for real‐world applications. [ABSTRACT FROM AUTHOR]

This paper focuses on developing and analyzing a new AR-iterative scheme designed for estimating fixed points of generalized non-expansive mappings of the Suzuki kind. The scheme’s stability and convergence properties are rigorously proven, highlighting its superior convergence rate for non-expansive mappings when compared to existing schemes in the literature. Additionally, an application of the AR-iterative scheme to differential equations is presented. These findings underscore the effectiveness and versatility of the proposed scheme in numerical analysis and computational mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two accelerated primal-dual methods tailored to address linearly constrained composite convex optimization problems, where the objective function is expressed as the sum of a possibly nondifferentiable function and a differentiable function with Lipschitz continuous gradient. The first method is the accelerated linearized augmented Lagrangian method (ALALM), which permits linearization to the differentiable function; the second method is the accelerated linearized proximal point algorithm (ALPPA), which enables linearization of both the differentiable function and the augmented term. By incorporating adaptive parameters, we demonstrate that ALALM achieves the O (1 / k 2) convergence rate and the linear convergence rate under the assumption of convexity and strong convexity, respectively. Additionally, we establish that ALPPA enjoys the O (1 / k) convergence rate in convex case and the O (1 / k 2) convergence rate in strongly convex case. We provide numerical results to validate the effectiveness of the proposed methods. • We extend a class of Nesterov's accelerated scheme for solving unconstrained optimization problems to handle composite convex optimization with linear constraints. • Propose an accelerated linearized augmented Lagrangian method (ALALM) with a linear convergence rate for the strongly convex case. • An accelerated linearized proximal point algorithm (ALPPA) is proposed for both convex and strongly convex cases. [ABSTRACT FROM AUTHOR]

We consider bounded linear operators from the view of functional reproducing property. For some bounded linear operators associated with orthogonal polynomials we define an inner product space associated with a kernel constructed with orthogonal polynomials, show that it is a functional reproducing kernel Hilbert space (FRKHS) associated with these bounded linear operators and give decay rate for the best FRKHS approximation with a K -functional associated with the FRKHS. On this basis, we provide a learning rate for kernel regularized regression whose hypothesis space is the defined FRKHS. As applications, we define some concrete FRKHSs associated with polynomial operators such as the Bernstein–Durrmeyer operators, the de la Vallée Poussin operators on both the unit sphere S d − 1 and the unit ball B d . We show that these polynomial operators have reproducing property with respect to the corresponding concrete FRKHSs and show the learning rate for the kernel regularized regression. In short, we provide a way of constructing FRKHS operators with Fourier multipliers and show a learning framework from the view of operator approximation. [ABSTRACT FROM AUTHOR]

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified. [ABSTRACT FROM AUTHOR]

In a Hilbert spaceH, we investigate the long time behavior of the trajectories of the following second-order differential system ¨ x(t)+... t p x(t) = 0; where a;b are two positive constants, p 2 [0;2], and the time scale parameter b is a positive function that tends to infinity as time approaches infinity. We based on Tikhonov regularization techniques and an appropriate Lyapunov energy function simultaneously prove fast convergence rates of the objective function to the global minimum, strong convergence of trajectories to the minimizer of the minimum norm, as well as convergence rates for the velocity and the gradient. Numerical examples are also given to illustrate these results. Finally, we extend our results to the non-smooth case. [ABSTRACT FROM AUTHOR]

We consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, is obtained. [ABSTRACT FROM AUTHOR]

This study aims to explore the problem of H∞ fuzzy control with an adjustable convergence rate for switched nonlinear systems with time-varying delays under the persistent dwell time (PDT) switching. Compared to the widely studied dwell time (DT) switching or average dwell time (ADT) switching in existing literature, PDT switching provides a more comprehensive consideration of the switching frequency and has a broader range of applicability. Subsequently, by combining the interval stability definition, T-S fuzzy model, PDT technique, and Lyapunov-Krasovskii (L-K) functional, a new H∞ fuzzy control criterion for adjusting the convergence rate of switched nonlinear systems with time-varying delays is proposed. This criterion enables the development of a novel method for constructing H∞ fuzzy controllers, which can regulate the system’s convergence rate and achieve the specified H∞ performance. Combining the above methods, an algorithm is introduced to precisely control the convergence rate of the target system. Finally, the effectiveness of this method is validated through a control example of a single-link robot arm. [ABSTRACT FROM AUTHOR]

This paper investigates numerical solutions of pseudomontone variational inequality problems in real Hilbert spaces. Two new algorithms based on the celebrated extragradient method are introduced. Under suitable conditions, it is proved that the two algorithms achieve weak convergence and linear convergence rates. The results extend and improve upon related works in the literature on the associated extragradient algorithms. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of parabolic-elliptic chemotaxis models with density-suppressed motility and general logistic source in an n-dimensional smooth bounded domain. With some conditions on the density-suppressed motility function, we show the convergence rate of solutions is exponential as time tends to infinity for such kind of models. [ABSTRACT FROM AUTHOR]

The paper discusses the Galton–Watson stochastic branching system. We deal only with the noncritical case. Our task in the work is to improve our recent result which explicitly calculated the famous constant in the theory of subcritical Galton–Watson branching systems, announced by Kolmogorov in 1938. We demonstrate the rate of convergence to the Kolmogorov constant. This improvement contributes to determining the speed of approximation rate in a number of classical limit theorems of the theory of Galton–Watson branching systems. [ABSTRACT FROM AUTHOR]

This paper proposes a novel estimator for support vector machines with matrix-valued covariates in a high-dimensional setting. We assume that the underlying parameter matrix lies in a low-dimensional subspace that is simultaneously low-rank and row (column) sparse. We formulate the problem as a regularized hinge loss minimization problem using the nuclear and group lasso norms as penalties to exploit the low-dimensional structure. Our primary focus is deriving the statistical convergence rate of the regularized estimator for the unknown parameter matrix. To validate our theoretical findings, we conducted numerical experiments on both simulated and real-world datasets, demonstrating the efficacy of the regularized support matrix machines framework. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to enhance the performance of the stochastic gradient descent (SGD) algorithm by incorporating a modified decay step size based on .... The proposed step size integrates a logarithmic term, leading to the selection of smaller values in the final iterations. Our analysis establishes a convergence rate of O ... for smooth non-convex functions without the Polyak-Łojasiewicz condition. To evaluate the effectiveness of our approach, we conducted numerical experiments on image classification tasks using the Fashion-MNIST and CIFAR10 datasets, and the results demonstrate significant improvements in accuracy, with enhancements of 0:5% and 1:4% observed, respectively, compared to the traditional p1 t step size. [ABSTRACT FROM AUTHOR]

The paper introduces two novel activation functions known as modExp and modExpm. The activation functions possess several desirable properties, such as being continuously differentiable, bounded, smooth, and non-monotonic. Our studies have shown that modExp and modExpm consistently outperform ReLU and other activation functions across a range of challenging datasets and complex models. Initially, the experiments involve training and classifying using a multi-layer perceptron (MLP) on benchmark data sets like the Diagnostic Wisconsin Breast Cancer and Iris Flower datasets. Both modExp and modExpm demonstrate impressive performance, with modExp achieving 94.15 and 95.56% and modExpm achieving 94.15 and 95.56% respectively, when compared to ReLU, ELU, Tanh, Mish, Softsign, Leaky ReLU, and TanhExp. In addition, a series of experiments were carried out on five different depths of deeper neural networks, ranging from five to eight layers, using MNIST datasets. The modExpm activation function demonstrated superior performance accuracy on various neural network configurations, achieving 95.56, 95.43, 94.72, 95.14, and 95.61% on wider 5 layers, slimmer 5 layers, 6 layers, 7 layers, and 8 layers respectively. The modExp activation function also performed well, achieving the second highest accuracy of 95.42, 94.33, 94.76, 95.06, and 95.37% on the same network configurations, outperforming ReLU, ELU, Tanh, Mish, Softsign, Leaky ReLU, and TanhExp. The results of the statistical feature measures show that both activation functions have the highest mean accuracy, the lowest standard deviation, the lowest Root Mean squared Error, the lowest variance, and the lowest Mean squared Error. According to the experiment, both functions converge more quickly than ReLU, which is a significant advantage in Neural network learning. [ABSTRACT FROM AUTHOR]

This study aims to explore the problem of $ H_{\infty} $ fuzzy control with an adjustable convergence rate for switched nonlinear systems with time-varying delays under the persistent dwell time (PDT) switching. Compared to the widely studied dwell time (DT) switching or average dwell time (ADT) switching in existing literature, PDT switching provides a more comprehensive consideration of the switching frequency and has a broader range of applicability. Subsequently, by combining the interval stability definition, T-S fuzzy model, PDT technique, and Lyapunov-Krasovskii (L-K) functional, a new $ H_{\infty} $ fuzzy control criterion for adjusting the convergence rate of switched nonlinear systems with time-varying delays is proposed. This criterion enables the development of a novel method for constructing $ H_{\infty} $ fuzzy controllers, which can regulate the system's convergence rate and achieve the specified $ H_{\infty} $ performance. Combining the above methods, an algorithm is introduced to precisely control the convergence rate of the target system. Finally, the effectiveness of this method is validated through a control example of a single-link robot arm.

One of the most significant challenges in adaptive filters is the slow convergence speed of the adaptive algorithm when dealing with highly correlated input signals. The adaptive affine projection algorithm (APA) which is a generalized version of the well-known normalized least mean square algorithm, improves convergence speed against correlated input signals, but with a high steady-state weight error and considerable computational complexity. In this paper, to enhance the convergence performance of APA and reduce its complexity, a selective projection order APA and its proportionate variant are proposed. In these algorithms, the projection order varies during convergence. Initially, a high projection order is employed in adaptation, gradually decreasing as the mean-square deviation (MSD) of the weight error decreases. This adaptive approach maintains a high initial convergence speed while reducingss the steady-state error. Moreover, the computational complexity decreases during convergence. Additionally, we propose a combination of proportionate and non-proportionate adaptive algorithms, leveraging the variation of MSD during convergence. Simulation results in identification of sparse/dispersive channels confirm the improved convergence performance of the proposed algorithms compared to competing adaptive algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a Non-Lipschitz stepsize strategy, new parameters and inertial effects in projection and contraction approaches to solve split feasibility problems in real Hilbert spaces. Weak and strong convergence theorems for our methods are established with non-Lipschitz continuity and expansiveness of the monotone operator and related projection mappings, respectively. Further, we discuss a nonasymptotic O(1/n) convergence rate for the proposed algorithms. Finally, signal recovery and image deblurring problems that we provide for demonstration and comparison. [ABSTRACT FROM AUTHOR]

‎In this paper, we delve into frame theory to create an innovative iterative method for resolving the operator equation $ Lu=f $. In this case, $ L:H\rightarrow H $, a bounded, invertible, and self-adjoint linear operator, operates within a separable Hilbert space denoted by $H$. Our methodology, which is based on the GMRES projective method, introduces an alternate search space, which brings another dimension to the problem-solving process. Our investigation continues with the assessment of convergence, where we look at the corresponding convergence rate. This rate is intricately influenced by the frame bounds, shedding light on the effectiveness of our approach. Furthermore, we investigate the ideal scenario in which the equation finds an exact solution, providing useful insights into the practical implications of our work.

Abstract This article presents the HDG approximation as a solution to the unilateral contact problem, leveraging the regularization method and an iterative procedure for resolution. In our study, u represents the potential (displacement of the elastic body) and q represents the flux (the force exerted on the body). Our analysis establishes that the utilization of polynomials of degree k ( k ≥ 1 ) $k (k \ge 1)$ leads to achieving an optimal convergence rate of order k + 1 $k+1$ in L 2 $L^{2}$ -norm for both u and q. Importantly, this optimal convergence is maintained irrespective of whether the domain is discretized through a structured or unstructured grid. The numerical results consistently align with the theoretical findings, underscoring the effectiveness and reliability of the proposed HDG approximation method for unilateral contact problems.

This paper considers nonparametric estimations of a density function in a two-class mixture model. A linear wavelet estimator and an adaptive wavelet estimator are constructed. Upper bound estimations over $ L^{p}\; (1\leq p < +\infty) $ risk of those wavelet estimators are proved in Besov spaces. When $ \tilde{p}\geq p\geq1 $, the convergence rate of adaptive wavelet estimator is the same as the linear estimator up to a $ \ln n $ factor. The adaptive wavelet estimator can get better than the linear estimator in the case of $ 1\leq \tilde{p} < p $. Finally, some numerical experiments are presented to validate the theoretical results.

The integrals of diffusion processes are of significant importance in the field of finance, particularly in relation to stochastic volatility models, which are frequently employed to represent the temporal variability of stock prices. In this paper, we consider the strong consistency of the nonparametric kernel estimator of the transition density for second-order diffusion processes, using the moment inequalities of $ \rho $-mixing sequences to demonstrate the strong consistency under some regularity conditions. Furthermore, the asymptotic mean square error is provided based on the deviation and variance of the transition density kernel estimator. The optimal bandwidth is found and thus the convergence rate of the kernel estimator is obtained. At the same time, our results are compared with the conclusions of the univariate density function.

This paper is concerned with the inverse problem of determining the initial value of the distributed‐order time‐fractional diffusion equation from the final time observation data, which arises in some ultra‐slow diffusion phenomena in applied areas. Since the problem is ill‐posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the a priori and a posteriori regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial‐boundary value problem for the distributed‐order time‐fractional diffusion equation. Some numerical examples including one‐dimensional and two‐dimensional cases are presented to illustrate the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a variation of the forward–backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward–backward splitting scheme and an accelerated O 1 n 2 -convergent forward–backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization. [ABSTRACT FROM AUTHOR]

In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper. [ABSTRACT FROM AUTHOR]

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions. [ABSTRACT FROM AUTHOR]

We present three novel algorithms based on the forward-backward splitting technique for the solution of monotone inclusion problems in real Hilbert spaces. The proposed algorithms work adaptively in the absence of the Lipschitz constant of the single-valued operator involved thanks to the fact that there is a non-monotonic step size criterion used. The weak and strong convergence and the R-linear convergence of the developed algorithms are investigated under some appropriate assumptions. Finally, our algorithms are put into practice to address the restoration problem in the signal and image fields, and they are compared to some pertinent algorithms in the literature. [ABSTRACT FROM AUTHOR]

The study of signal processing has recently devoted significantly more attention to adaptive filtering techniques. By addressing the shortcoming of the conventional geometric algebra-based fixed step-size least mean square algorithm that is unable to satisfy in terms of both reducing steady-state error and a faster convergence speed, simultaneously, this study presents an improved logarithmic function-based variable step-size least mean square geometric algebra adaptive filtering algorithm by establishing the step-size factor μ and error signal e(n) nonlinear function relationship. The instantaneous values of a current error estimate e(n) and the previous error estimate e (n - 1) are used to determine the step size of the defined algorithm. Besides, an extensive discussion is given on the performance of algorithm under influence of parameters γ and T as well as comparative analysis with other existing geometric algebra-based adaptive filters. Computer simulation reveals that the proposed approach not only has a low steady-state error, robustness against impulsive noise, and fast convergence speed, but it also overcomes some existing algorithm's instability under steady-state phase. [ABSTRACT FROM AUTHOR]

The activation functions are extremely important to neural networks since they are responsible for learning the abstract characteristics of the data through nonlinear modification. The paper presents a new activation function, which is referred to as modSwish. It is continuously differentiable, unbounded above, bounded below, and non-monotonic. Our results demonstrate that modSwish outperforms ReLU on a number of challenging datasets and neural network models. In the beginning of the experiment, Neural Networks are trained and classified using benchmark data like Diagnostic Wisconsin Breast Cancer and Iris and modSwish achieved 93.57% and 95.56% accuracy, respectively. Secondly, experiments were conducted on five distinct neural network depths that ranged from five to eight layers over MNIST datasets. The modSwish activation function obtained 95.57%, 95.29%, 94.93%, 94.69%, and 95.03% accuracy on 8 layers, 7 layers, 6 layers, 5 thinner layers, and wider 5 layers neural networks respectively. Finally, experiments were conducted on two distinct Convolution Neural Network with two convolution layer and four convolution layers over CIFAR-10 datasets. The modSwish activation function obtained 60.04% and 69.22% accuracy on two convolution layer and four convolution layer model respectively. Statistical feature measurements demonstrate that modSwish has the best mean accuracy, lowest Root Mean squared Error, lowest standard deviation, lowest variance, and lowest Mean squared Error. The study indicated that modSwish has faster convergence compared to ReLU, making it a valuable factor in deep learning. The results of the experiments suggest that modSwish can be a promising substitute for ReLU, leading to better performance in neural network models. [ABSTRACT FROM AUTHOR]

This article constructs a new two‐level explicit/implicit numerical scheme in an approximate solution for the two‐dimensional time fractional mobile/immobile advection‐dispersion problem. The stability and error estimates of the proposed technique are deeply analyzed in the L∞(0,T;L2)$$ {L}^{\infty}\left(0,T;{L}^2\right) $$‐norm. The developed approach is less time consuming, fourth‐order in space and temporal accurate of order O(k2−λ2)$$ O\left({k}^{2-\frac{\lambda }{2}}\right) $$, where k$$ k $$ is the time step and λ$$ \lambda $$ denotes a positive parameter less than 1. This result shows that the two‐level explicit/implicit formulation is faster and more efficient than a large class of numerical schemes widely discussed in the literature for the considered problem. Numerical experiments are performed to verify the theoretical studies and to demonstrate the efficiency of the new numerical method. [ABSTRACT FROM AUTHOR]

This article presents the HDG approximation as a solution to the unilateral contact problem, leveraging the regularization method and an iterative procedure for resolution. In our study, u represents the potential (displacement of the elastic body) and q represents the flux (the force exerted on the body). Our analysis establishes that the utilization of polynomials of degree k (k ≥ 1) leads to achieving an optimal convergence rate of order k + 1 in L 2 -norm for both u and q. Importantly, this optimal convergence is maintained irrespective of whether the domain is discretized through a structured or unstructured grid. The numerical results consistently align with the theoretical findings, underscoring the effectiveness and reliability of the proposed HDG approximation method for unilateral contact problems. [ABSTRACT FROM AUTHOR]

We study the efficiency of the vector greedy algorithms for simultaneous sparse approximation in a Banach space. We employ a unified way of analyzing the approximation properties of the Vector Weak Chebyshev Greedy Algorithm, the Vector Weak Greedy Algorithm with Free Relaxation and the Vector Rescaled Weak Relaxed Greedy Algorithm. We obtain the necessary and sufficient conditions for the convergence and the optimal convergence rate on the basic sparse class for these algorithms. It shows that these vector greedy algorithms are simple but highly efficient in dealing with multi-target element. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to study large-time behavior of solutions for an outflow problem to the reduced gravity two and a half layer model in a half space. We establish the existence of stationary solutions and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small. Moreover, we obtain the convergence rate of the solution toward the stationary solution in some weighted Sobolev spaces only for the supersonic case. The proof is based on the basic energy method. A key point for the proof of convergence rates is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity satisfy some additional conditions. [ABSTRACT FROM AUTHOR]

We study the problem of the nonparametric estimation for the density π$$ \pi $$ of the stationary distribution of a d$$ d $$‐dimensional stochastic differential equation (Xt)t∈[0,T]$$ {\left({X}_t\right)}_{t\in \left[0,T\right]} $$. From the continuous observation of the sampling path on [0,T]$$ \left[0,T\right] $$, we study the estimation of π(x)$$ \pi (x) $$ as T$$ T $$ goes to infinity. For d≥2$$ d\ge 2 $$, we characterize the minimax rate for the L2$$ {\mathbf{L}}^2 $$‐risk in pointwise estimation over a class of anisotropic Hölder functions π$$ \pi $$ with regularity β=(β1,…,βd)$$ \beta =\left({\beta}_1,\dots, {\beta}_d\right) $$. For d≥3$$ d\ge 3 $$, our finding is that, having ordered the smoothness such that β1≤⋯≤βd$$ {\beta}_1\le \cdots \le {\beta}_d $$, the minimax rate depends on whether β2<β3$$ {\beta}_2<{\beta}_3 $$ or β2=β3$$ {\beta}_2={\beta}_3 $$. In the first case, this rate is logTTγ$$ {\left(\frac{\log T}{T}\right)}^{\gamma } $$, and in the second case, it is (1T)γ$$ {\left(\frac{1}{T}\right)}^{\gamma } $$, where γ$$ \gamma $$ is an explicit exponent dependent on the dimension and β‾3$$ {\overline{\beta}}_3 $$, the harmonic mean of smoothness over the d$$ d $$ directions after excluding β1$$ {\beta}_1 $$ and β2$$ {\beta}_2 $$, the smallest ones. We also demonstrate that kernel‐based estimators achieve the optimal minimax rate. Furthermore, we propose an adaptive procedure for both L2$$ {L}^2 $$ integrated and pointwise risk. In the two‐dimensional case, we show that kernel density estimators achieve the rate logTT$$ \frac{\log T}{T} $$, which is optimal in the minimax sense. Finally we illustrate the validity of our theoretical findings by proposing numerical results. [ABSTRACT FROM AUTHOR]

In the paper, an inverse source problem in bioluminescence tomography (BLT) is investigated. BLT is a method of light imaging and offers many advantages such as sensitivity, cost-effectiveness, high signal-to-noise ratio and non-destructivity. It thus has promising prospects for many applications such as cancer diagnosis, drug discovery and development as well as gene therapies. In the literature, BLT is extensively studied based on the (stationary) diffusion approximation (DA) equation, where the distribution of peak sources is reconstructed and no solution uniqueness is guaranteed without proper a priori information. In this work, motivated by solution uniqueness, a novel dynamic coupled DA model is proposed. Theoretical analysis including the well-posedness of the forward problem and the solution uniqueness of the inverse problem are given. Based on the new model, iterative inversion algorithms under the framework of regularizing schemes are introduced and applied to reconstruct the smooth and non-smooth sources. We discretize the regularization functional with the finite element method and give the convergence rate of numerical solutions. Several numerical examples are implemented to validate the effectiveness of the new model and the proposed algorithms. [ABSTRACT FROM AUTHOR]

The Internet of Things (IoT) is made up of billions of interconnected devices that can transmit and receive data over the Internet. IoT devices have many vulnerabilities that attackers could use to compromise their security because of the heterogeneity of device connectivity. Distributed denial-of-service (DDoS) attacks against those applications become more common as IoT applications continue to expand and devolve. Identifying DDoS attacks is a difficult process due to the variety of IoT devices connected. The present article proposed a new method to detect DDoS attacks using an optimized Elman recurrent neural network (ERNN) based on chaotic bacterial colony optimization (CBCO) called CBCO-ERNN. The proposed method uses CBCO for obtaining optimal parameters (weights and biases) and structure (number of hidden neurons) of ERNN architecture. The chaos theory is applied to improve BCO's exploration and exploitation capabilities by initializing the bacterial population and selecting the appropriate chemotaxis step size value. The CBCO approach is used to train the ERNN model to avoid local optima and enhance the convergence rate. The performance of the CBCO-ERNN is tested and evaluated using four benchmark attack datasets such as the BoT-IoT, CIC-IDS2017, CIC-DDoS2019, and IoTID20 datasets, and five performance metrics are considered: accuracy, sensitivity, specificity, precision, and F-Score. According to the experimental results, the CBCO-ERNN method provides a high detection and a faster convergence rate when compared to earlier algorithms. [ABSTRACT FROM AUTHOR]

Recently, structured nonlinear least-squares (NLS) based algorithms gained considerable emphasis from researchers; this attention may result from increasingly applicable areas of these algorithms in different science and engineering domains. In this article, we coined a new efficient structured-based NLS algorithm. We developed a diagonal Hessian-based formulation for solving NLS problems. We derived the quasi-Newton update based on a diagonal matrix scheme subject to a modified structured secant condition. Also, we show that the algorithm's search direction satisfies a sufficient descent condition under some standard assumptions. Subsequently, we also prove the global convergence of the algorithm and then eventually show its linear convergence rate for strongly convex functions. Furthermore, to show case the proposed algorithm's performance, we experimented numerically by comparing it with other approaches on some benchmark test functions available in the literature. Finally, the introduced scheme is applied to solve some data-fitting problems [ABSTRACT FROM AUTHOR]

The integrals of diffusion processes are of significant importance in the field of finance, particularly in relation to stochastic volatility models, which are frequently employed to represent the temporal variability of stock prices. In this paper, we consider the strong consistency of the nonparametric kernel estimator of the transition density for second-order diffusion processes, using the moment inequalities of ρ-mixing sequences to demonstrate the strong consistency under some regularity conditions. Furthermore, the asymptotic mean square error is provided based on the deviation and variance of the transition density kernel estimator. The optimal bandwidth is found and thus the convergence rate of the kernel estimator is obtained. At the same time, our results are compared with the conclusions of the univariate density function. [ABSTRACT FROM AUTHOR]

In this paper, we delve into frame theory to create an in- novative iterative method for resolving the operator equation Lu = f. In this case, L : H ! H, a bounded, invertible, and self-adjoint lin- ear operator, operates within a separable Hilbert space denoted by H. Our methodology, which is based on the GMRES projective method, in- troduces an alternate search space, which brings another dimension to the problem-solving process. Our investigation continues with the assess- ment of convergence, where we look at the corresponding convergence rate. This rate is intricately in uenced by the frame bounds, shedding light on the eflectiveness of our approach. Furthermore, we investigate the ideal scenario in which the equation finds an exact solution, providing useful insights into the practical implications of our work. [ABSTRACT FROM AUTHOR]