Rate Of Convergence Results Research Articles

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

Error estimates of unilateral piezoelectric contact problem in a curved and smooth boundary domain

We study the linear finite element approximation of piezoelectric unilateral contact problem in a curved and smooth boundary domain. The unilateral contact conditions will be weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter and the mesh size. In fact, under regularity of the solution, we prove some convergence rate results.

Shuffling-type gradient method with bandwidth-based step sizes for finite-sum optimization

Shuffling-type gradient method is a popular machine learning algorithm that solves finite-sum optimization problems by randomly shuffling samples during iterations. In this paper, we explore the convergence properties of shuffling-type gradient method under mild assumptions. Specifically, we employ the bandwidth-based step size strategy that covers both monotonic and non-monotonic step sizes, thereby providing a unified convergence guarantee in terms of step size. Additionally, we replace the lower bound assumption of the objective function with that of the loss function, thereby eliminating the restrictions on the variance and the second-order moment of stochastic gradient that are difficult to verify in practice. For non-convex objectives, we recover the last iteration convergence of shuffling-type gradient algorithm with a less cumbersome proof. Meanwhile, we also establish the convergence rate for the minimum iteration of gradient norms. Under the Polyak-Łojasiewicz (PL) condition, we prove that the function value of last iteration converges to the lower bound of the objective function. By selecting appropriate boundary functions, we further improve the previous sublinear convergence rate results. Overall, this paper contributes to the understanding of shuffling-type gradient method and its convergence properties, providing insights for optimizing finite-sum problems in machine learning. Finally, numerical experiments demonstrate the efficiency of shuffling-type gradient method with bandwidth-based step size and validate our theoretical results.

R-linear and strong convergence of Tseng methods for solving certain split and non-split problems

In the recent past, authors have fondly assumed the co-coercive condition for solving certain class of split inverse problems. Recently, some attempts have been made to relax this very strict condition, however, only few results exists in literature in such case. In this paper, we present some new Mann-type Tseng methods for solving split monotone variational inclusion and common fixed point problems for a finite collection of quasi-pseudocontractive operators. We prove strong convergence and R − linear convergence rate results of our methods, while the co-coerciveness property is dispensed with. Our methods incorporate the relaxation, inertial extrapolation and self-adaptive step size techniques to achieve better performance and faster convergence. With the aid of relevant numerical experiments, we showcase the performance profile of our methods in comparison with recent methods in literature.

A New Robust Iterative Scheme Applied in Solving a Fractional Diffusion Model for Oxygen Delivery via a Capillary of Tissues

In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green’s functions with an example.

On Convergence Rates of Proximal Alternating Direction Method of Multipliers

In this paper we consider from two different aspects the proximal alternating direction method of multipliers (ADMM) in Hilbert spaces. We first consider the application of the proximal ADMM to solve well-posed linearly constrained two-block separable convex minimization problems in Hilbert spaces and obtain new and improved non-ergodic convergence rate results, including linear and sublinear rates under certain regularity conditions. We next consider the proximal ADMM as a regularization method for solving linear ill-posed inverse problems in Hilbert spaces. When the data is corrupted by additive noise, we establish, under a benchmark source condition, a convergence rate result in terms of the noise level when the number of iterations is properly chosen.

Pattern formation of the Cucker–Smale type kinetic models based on gradient flow

In this paper, we study the pattern formation of the Cucker–Smale type kinetic models. Two distributed Cucker–Smale type kinetic models for formation control are introduced based on gradient flow. We provide rigorous proof to prove that the above two kinetic models will achieve the desired position with the same velocity over a long time. In particular, the exponential convergence rate of the pattern formation on the corresponding particle models is obtained. Our analysis shows the gradient flow structure of the velocity field is important for establishing the convergence rate results of distributed control kinetic models. Finally, some numerical simulations are performed to illustrate our theoretical results.

The role of local steps in local SGD

We consider the distributed stochastic optimization problem where n agents want to minimize a global function given by the sum of agents' local functions and focus on the heterogeneous setting when agents' local functions are defined over non-i.i.d. datasets. We study the Local SGD method, where agents perform a number of local stochastic gradient steps and occasionally communicate with a central node to improve their local optimization tasks. We analyze the effect of local steps on the convergence rate and the communication complexity of Local SGD. In particular, instead of assuming a fixed number of local steps across all communication rounds, we allow the number of local steps during the jth communication round, , to be different and arbitrary numbers. Our main contribution is to characterize the convergence rate of Local SGD as a function of under various settings of strongly convex, convex, and nonconvex local functions, where R is the total number of communication rounds. Based on this characterization, we provide sufficient conditions on the sequence such that Local SGD can achieve linear speedup with respect to the number of workers. Furthermore, we propose a new communication strategy with increasing local steps that is superior to constant local steps for strongly convex local functions. On the other hand, for convex and nonconvex local functions, we argue that fixed local steps are the best communication strategy for Local SGD and recover state-of-the-art convergence rate results. Finally, we justify our theoretical results through extensive numerical experiments.

Modified bee colony optimization with opposition learning algorithm on use of medical data clustering

Clustering has gained popularity in the data mining field as one of the primary approaches for obtaining data distribution and data analysis. The medical data analysis for different diseases is a great challenge in current research. The benefits of opposition based learning such as faster convergence rate and better approximate result in finding global optimum can be helpful in this area. To achieve faster convergence and better clustering results for medical data, in this work, the authors have proposed an approach utilising modified bee colony optimization with opposition based learning and k-medoids technique. The initial centroid plays an important role in the bee colony optimization based clustering. The proposed approach uses k-medoids algorithm for this task. In order to facilitate faster convergence, it adds the opposite bees which are located at exactly the opposite location of the initial bees. The exploration task is performed by both of these kinds of bees to find potential solutions. This increases the algorithm’s capacity for exploration and, consequently, the rate of convergence. Five distinct medical datasets collected from the UCI library are investigated to demonstrate the algorithm’s efficacy. The implementation results demonstrate that the algorithm gives better convergence rate and clustering quality compared to some the existing algorithms.

Differentially Private Distributed Optimization With an Event-Triggered Mechanism

This study concentrates on the differential private distributed optimization problem with an event-triggered mechanism, whose goals include preserving the privacy of agents’ initial states and local cost functions and improving communication efficiency. A distributed event-triggered mechanism is integrated into the differentially private subgradient-push distributed optimization algorithm and then a new algorithm named as DP-ETSP is designed, where the real-time information propagation among agents is avoided. Additionally, under the proposed event-triggered mechanism, an analysis of mean-square consensus and optimality over time-varying directed networks is made when the added Laplace noises meet some specific decaying conditions. Convergence rate results are further established under a specific stepsize, which are equal to the rate of stochastic gradient-push algorithm without event-triggered communication. Moreover, the differential privacy preservation performance is analyzed and the rule for selecting privacy level is discussed. Finally, the feasibility and effectiveness of DP-ETSP are verified in two simulation cases.

Asymptotic properties of the maximum smoothed partial likelihood estimator in the change‐plane Cox model

AbstractThe change‐plane Cox model is a popular tool for the subgroup analysis of survival data. Despite the rich literature on this model, there has been limited investigation into the asymptotic properties of the estimators of the finite‐dimensional parameter. Particularly, the convergence rate, not to mention the asymptotic distribution, has not been fully characterized for the general model where classification is based on multiple covariates. To bridge this theoretical gap, this study proposes a maximum smoothed partial likelihood estimator and establishes the following asymptotic properties. First, it shows that the convergence rate for the classification parameter can be arbitrarily close to up to a logarithmic factor under a certain condition on covariates and the choice of tuning parameter. Given this convergence rate result, it also establishes the asymptotic normality for the regression parameter.

Convergence Rate Analysis and Improved Iterations for Numerical Radius Computation

The current two main methods for computing the numerical are the level-set approach of Mengi and Overton and the cutting-plane method of Uhlig, but until now, no formal convergence rate results have been established for either. In practice, which is faster is also unclear, with Uhlig's method sometimes being much faster than Mengi and Overton's approach, and on other problems, much slower. In this paper, we clarify this issue and also propose three improved methods. We show that Mengi and Overton's method converges quadratically, as has been suspected, while we completely characterize the total cost of Uhlig's method for so-called disk matrices. Then, for arbitrary fields of values, we derive the exact Q-linear local convergence rate of Uhlig's cutting procedure. Together, this establishes that Uhlig's method is extremely expensive when the field of values is a disk centered at the origin, but that the local rate of convergence of his cutting procedure actually varies from linear to superlinear depending on the shape and location of the field of values, which we show can be encapsulated by a single parameter via introducing the notion of normalized curvature. These results fully explain why Uhlig's method can both be exceptionally fast and exceptionally slow. With this insight, we propose an improved level-set method and an improved cutting-plane method, both of which can be significantly faster than their earlier counterparts, while also establishing analogous convergence rate results for both. Moreover, in order to remain efficient for any field of values configuration, we introduce a third algorithm that leverages the concept of normalized curvature and combines both of our improved iterations.

A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems

This paper examines inverse Cauchy problems that are governed by a kind of elliptic partial differential equation. The inverse problems involve recovering the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By using the coupled complex boundary method (CCBM), which integrates both Dirichlet and Neumann data into a single Robin boundary condition, we reformulate the underlying problem into an operator equation. Based on this new formulation, we study the solution existence issue of the reduced problem with noisy data. A Golub–Kahan bidiagonalization (GKB) process together with Givens rotation is employed for iteratively solving the proposed operator equation. The regularizing property of the developed method, called CCBM-GKB, and its convergence rate results are proved under a posteriori stopping rule. Finally, a linear finite element method is used for the numerical realization of CCBM-GKB. Various numerical experiments demonstrate that CCBM-GKB is a kind of accelerated iterative regularization method, as it is much faster than the classic Landweber method.

Differential Evolution Optimization of Microwave Focused Hyperthermia Phased Array Excitation for Targeted Breast Cancer Heating.

Microwave hyperthermia using the phased array applicator is a non-invasive treatment modality for breast cancer. Hyperthermia treatment planning (HTP) is critical to accurately treating breast cancer and avoiding damage to the patient's healthy tissue. A global optimization algorithm, differential evolution (DE) algorithm, has been applied to optimize HTP for breast cancer and its ability to improve the treatment effect was proved by electromagnetic (EM) and thermal simulation results. DE algorithm is compared to time reversal (TR) technology, particle swarm optimization (PSO) algorithm, and genetic algorithm (GA) in HTP for breast cancer in terms of convergence rate and treatment results, such as treatment indicators and temperature parameters. The current approaches in breast cancer microwave hyperthermia still have the problem of hotspots in healthy tissue. DE enhances focused microwave energy absorption into the tumor and reduces the relative energy of healthy tissue during hyperthermia treatment. By comparing the treatment results of each objective function used in DE, the DE algorithm with hotspot to target quotient (HTQ) as the objective function has outstanding performance in HTP for breast cancer, which can increase the focused microwave energy of the tumor and decrease the damage to healthy tissue.

Identification of the time-dependent source term in a Kuramoto–Sivashinsky equation

Abstract The determination of an unknown time-dependent source term is investigated in a Kuramoto–Sivashinsky equation from given additional integral-type measurement. Based on Schauder’s fixed point theorem, the existence and uniqueness of such inverse problem are obtained under certain assumptions on the input data. In order to calculate the unknown source term, a time-discrete system is established, and its solution shall be applied to approximate the unknown quantity. The existence, uniqueness and some estimates to the time-discrete system are derived, and the convergence rates are deduced rigorously for both exact and noisy observation, respectively. Finally, the theoretical convergence rate results are verified, and accurate and stable solutions to the inverse problem are computed numerically by two numerical experiments.

A faster iterative scheme for solving nonlinear fractional differential equations of the Caputo type

<abstract><p>In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type.</p></abstract>

Convergence analysis of an ALF-based nonconvex splitting algorithm with SQP structure

<p style='text-indent:20px;'>In this paper, by combining the splitting method of augmented Lagrange function (ALF) with the sequential quadratic programming (SQP) approximation, a novel ALF-based splitting algorithm with SQP structure is proposed for multi-block linear constrained nonconvex separable optimization. The new algorithm uses ALF-based splitting idea to decompose the original problem into several small-scale subproblems. Meanwhile, the SQP approximation and Armijo-type line search are used to solve some subproblems with smoothness concurrently. Under the conventional weak hypothesis, the decreasing property of ALF as merit function is obtained. Furthermore, the global convergence, strong convergence and convergence rate results of the new algorithm in general sense are given.</p>

Convergence rate analysis of the coefficient identification problems in a Kuramoto–Sivashinsky equation

The convergence rate results of the mixed total variation (TV) and L p regularization to the identification of the unknown anti-diffusion coefficient, and the mixed H 2-L p regularization for the identification of the unknown diffusion coefficient in a Kuramoto–Sivashinsky (KS) equation are investigated with some additional observation. Under suitable source conditions, the convergence rates to regularized solutions for the TV-L p regularization of the anti-diffusion coefficient are obtained. Analogously, for the identification of diffusion coefficient with H 2-L p regularization, the convergence rates are derived with proper source conditions. In addition, simple and easily interpretable source conditions are also established to deduce the same convergence rates under some assumptions to the sought coefficients. For the source conditions in both two coefficient identification problems, no restrictive smallness condition to the source functions is required.

Generic Modal Design Variables for Efficient Aerodynamic Optimization

Orthogonal modes are an effective method for aerodynamic shape optimization due to their excellent design space compactness; however, all existing methods are generated from a database of representative geometry. Moreover, application to high-fidelity design spaces is not possible because high-frequency shape components are insufficiently bounded, leading to nonsmooth and oscillatory geometries. In this work, a new generic methodology for generating orthogonal shape modes is presented based on a purely geometric derivation, eliminating the need for geometric training data. The new method is a further development of the gradient-limiting method developed previously for constraining the design space in a geometrically meaningful way to reduce the effective degrees of freedom and improve optimization convergence rate and final result. Here, the gradient-limiting methodology is reformulated by transforming the constraints directly onto design variables to produce orthogonal shape modes with equivalent constraints for ensuring smooth and valid iterates. The new generic methodology requires no training data, can be applied to arbitrary topologies using different boundary conditions, and naturally includes translational modes as part of the orthogonal basis. When applied to two standard aerodynamic test cases, the new method has superior performance compared to library-derived modes. Importantly, the optimization convergence rate is independent of the number of design variables, and the optimized objective at high design fidelities is greatly improved by avoiding local minima corresponding to spurious geometries. A nonstandard test case is demonstrated, for which traditional library modes are not useable due to nontrivial topology, and it is shown to benefit from the high-fidelity design space and translational mode made possible with the novel methodology.

Cultural Value and Strategy of Ethnic Music Based on Distribution Estimation Algorithm

The current social construction process is gradually accelerating, and people’s growing spiritual and cultural needs are eager to be satisfied. Spiritual culture is conceived from the material production process of human beings. It is a unique ideology of human beings that is different from other animals. National music culture is a branch of spiritual culture. It occupies an indispensable position in China, and its development is closely related to the inheritance of national culture. At present, the cultural value of ethnic music has not been fully excavated. The reason is that the research methods used are relatively backward, resulting in insufficient optimization of the analysis method. Therefore, this paper applied the distribution estimation algorithm and the information index method to the analysis of the cultural value and strategy of ethnic music. After simulating 120 samples, this paper obtained four results: convergence rate results, average optimal value results, sorting effect results, and final optimization degree results. The experimental results showed that the research method after applying the distribution estimation algorithm can improve the optimization degree of the value and strategy analysis of ethnic music culture by 5.81%, which is conducive to the better value of ethnic music culture in social spiritual and cultural construction.