Rate Of Convergence Results Research Articles

On Convergence Rate of Weighted-Averaging Dynamics for Consensus Problems

This paper investigates the weighted-averaging dynamic for unconstrained and constrained consensus problems. Through the use of a suitably defined adjoint dynamic, quadratic Lyapunov comparison functions are constructed to analyze the behavior of weighted-averaging dynamic. As a result, new convergence rate results are obtained that capture the graph structure in a novel way. In particular, the exponential convergence rate is established for unconstrained consensus with the exponent of the order of $1-O(1/(m\log_{2}m))$ for special tree-like regular graphs. Also, the exponential convergence rate is established for constrained consensus over time-varying graphs, which nontrivially extends the existing result limited to the static graph case and the use of uniform weight matrices. Our main results are developed for directed graphs that are weakly connected, and we also provide statements regarding the rates in case of joint connectivity.

An optimal order a posteriori parameter choice strategy with modified Newton iterative scheme for solving nonlinear ill-posed operator equations

Study of inverse problems are interesting and mathematically challenging due to the fact that in most of the situation they are unstable with respect to perturbations of the data. In this paper to solve such operator equations, we propose a modified form of Gauss-Newton method combined with an a posteriori parameter choice strategy with the inexact data. Convergence and the convergence rate results are proven. We consider both a-priori and a-posteriori choice rule of parameter that guarantees the scheme converges to the exact solution. The theoretical results are illustrated through numerical examples and compared with the standard scheme to demonstrate that the scheme is stable and achieves good computational output. The salient features of our proposed scheme are: 1) convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in the standard scheme in literature; 2) consideration of an adaptive and numerically stable a posteriori parameter strategy that gives the same order of convergence as that of an a priori method; 3) computation of an optimal order regularisation parameter of the order O(δ2/3) using a discrepancy principle.

A Support Function Based Algorithm for Optimization with Eigenvalue Constraints

Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalue functions and is of practical interest because of a wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytic and Hermitian matrix-valued function. We propose a numerical approach based on quadratic support functions that overestimate the smallest eigenvalue function globally. The quadratic support functions are derived by employing variational properties of the smallest eigenvalue function over a set of Hermitian matrices. We establish the local convergence of the algorithm under mild assumptions and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. The convergence analysis reveals that the algorithm is immune to the nonsmooth nature of the smallest eigenvalue. We illustrate the ...

Convergence of the Uzawa method for the Stokes equations with damping

The method we study in this paper is to combine the finite element method with the Uzawa algorithm for the Stokes equations with damping. It aims to establish convergence rate results of the method. Meanwhile, an optimal bound of the convergence rate is obtained.

Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems

ABSTRACTThis work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under a Hölder-type sourcewise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt, Lobatto, and Radau methods.

A global minimization algorithm for Tikhonov functionals with convex () penalty terms in Banach spaces

We extend the globally convergent TIGRA method in Ramlau (2003 Inverse Prob. 19 433–65) for the computation of a minimizer of the Tikhonov-type functional with the convex () penalty terms Θ for nonlinear forward operators in Banach spaces. The Θ are allowed to be non-smooth to include or TV (total variation) functionals, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. The proposed TIGRA-Θ method uses a dual gradient descent method in the inner iteration and linearly decreases the regularization parameter in the outer iteration. We present the global convergence analysis for the algorithm under suitable parameter selections, and the convergence rate results are provided under both a priori and a posteriori stopping rules. Two numerical examples—an auto-convolution problem and a parameter identification problem—are presented to illustrate the theoretic analysis and verify the effectiveness of the method.

Convergence Rate of a Modified Extragradient Method for Pseudomonotone Variational Inequalities

We establish R-linear convergence rate and Q-linear convergence rate of the iterative sequences generated by the modified extragradient method in Khanh (Acta Math. Vietnam. (2015)). By constructing a suitable example, we analyze the regularity assumption of Tseng (J. Comput. Appl. Math. 60, 237–252, 1995) and the need of it in the proof of our convergence rate result.

On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations

We extend the taming techniques for explicit Euler approximations of stochastic differential equations driven by Levy noise with superlinearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global assumptions and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations.

A note on the inertial proximal point method

The proximal point method (PPM) for solving maximal monotone operator inclusion problem is a highly powerful tool for algorithm design, analysis and interpretation. To accelerate convergence of the PPM, inertial PPM (iPPM) was proposed in the literature. In this note, we point out that some of the attractive properties of the PPM, e.g., the generated sequence is contractive with the set of solutions, do not hold anymore for iPPM. To partially inherit the advantages of the PPM and meanwhile incorporate inertial extrapolation steps, we propose an iPPM with alternating inertial steps. Our analyses show that the even subsequence generated by the proposed iPPM is contractive with the set of solutions. Moreover, we establish global convergence result under much relaxed conditions on the inertial extrapolation stepsizes, e.g., monotonicity is no longer needed and the stepsizes are significantly enlarged compared to existing methods. Furthermore, we establish certain $o(1/k)$ convergence rate results, where $k$ denotes the iteration counter. These features are new to inertial type PPMs.

Convergence Rate Estimate of Distributed Localization Algorithms in Wireless Sensor Networks

Localization is one of the most important problems in wireless sensor networks. In this paper, we investigate the convergence rate estimate problem of a distributed localization algorithm which approximately formulates the localization problem as the convex feasibility problem including the consistent case and the inconsistent case. Although existing works established optimal consensus convergence analysis for this algorithm, they did not provide the convergence rate estimate. In this paper, we mainly show that for the consistent case the convergence rate of the optimal consensus will be exponential under some basic conditions, while for the inconsistent case we provide a necessary condition for the optimal consensus and a convergence rate estimate inequality. Furthermore, numerical examples are also provided to validate the established convergence and convergence rate results.

A General Inertial Proximal Point Algorithm for Mixed Variational Inequality Problem

In this paper, we first propose a general inertial proximal point algorithm (PPA) for the mixed variational inequality (VI) problem. Based on our knowledge, without stronger assumptions, a convergence rate result is not known in the literature for inertial type PPAs. Under certain conditions, we are able to establish the global convergence and nonasymptotic $O(1/k)$ convergence rate result (under a certain measure) of the proposed general inertial PPA. We then show that both the linearized augmented Lagrangian method (ALM) and the linearized alternating direction method of multipliers (ADMM) for structured convex optimization are applications of a general PPA, provided that the algorithmic parameters are properly chosen. Consequently, global convergence and convergence rate results of the linearized ALM and ADMM follow directly from results existing in the literature. In particular, by applying the proposed inertial PPA for mixed VI to structured convex optimization, we obtain inertial versions of the linearized ALM and ADMM whose global convergence is guaranteed. We also demonstrate the effect of the inertial extrapolation step via experimental results on the compressive principal component pursuit problem.

On the Uniqueness of Tikhonov Regularized Local Volatility Surfaces

We state sufficient conditions for the uniqueness of minimizers of the Tikhonov functional corresponding to the local volatility surface calibration problem. We use two different approaches. In the first one, using smoothness properties of the forward operator, we prove that, if the regularization parameter is sufficiently large, then, uniqueness hold. Secondly, we find sufficient conditions for the local uniqueness of Tikhonov minimizers by analyzing the geometrical properties of the image through the forward operator of line segments. We also study some theoretical consequences, such as, convergence-rate results with respect to the noise level, as well as, the well-posedness of an a posteriori rule for choosing the regularization parameter. In addition, we provide some numerical examples.

On the convergence of a class of the regularization methods for ill-posed quadratic minimization problems with constraint

We study a class of regularization methods for solving least-squares ill-posed problem with a convex constraint. Convergence and convergence rate results are proven for the problems which satisfy so called power source condition. All the results are obtained under the assumptions that, instead of exact initial data, only their approximations are known.

Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ satisfying c=τ−1/2 when the relaxation time τ tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞. In addition, the related convergence rate results are also obtained.

Some composite-step constrained optimization methods interpreted via the perturbed sequential quadratic programming framework

We consider the inexact restoration and the composite-step sequential quadratic programming (SQP) methods, and relate them to the so-called perturbed SQP framework. In particular, iterations of the methods in question are interpreted as certain structured perturbations of the basic SQP iterations. This gives a different insight into local behaviour of those algorithms, as well as improved or different local convergence and rate of convergence results.

Distributed solutions for loosely coupled feasibility problems using proximal splitting methods

In this paper, we consider convex feasibility problems (CFPs) where the underlying sets are loosely coupled, and we propose several algorithms to solve such problems in a distributed manner. These algorithms are obtained by applying proximal splitting methods to convex minimization reformulations of CFPs. We also put forth distributed convergence tests which enable us to establish feasibility or infeasibility of the problem distributedly, and we provide convergence rate results. Under the assumption that the problem is feasible and boundedly linearly regular, these convergence results are given in terms of the distance of the iterates to the feasible set, which are similar to those of classical projection methods. In case the feasibility problem is infeasible, we provide convergence rate results that concern the convergence of certain error bounds.

Rates of Convergence in the Blume–Emery–Griffiths Model

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature \(\beta \) and the interaction strength \(K\). The rates of convergence results are obtained as \((\beta ,K)\) converges along appropriate sequences \((\beta _n,K_n)\) to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.

Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression

This paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression. We propose a regularized Newton-type iteration and establish convergence and convergence rate results. A particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption. We demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model. This is illustrated in computed examples with simulated data.

Convergence Rates of the POD–Greedy Method

Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD–Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD–Greedy algorithm.

Iteration-Complexity of Block-Decomposition Algorithms and the Alternating Direction Method of Multipliers

In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point-to-set maximal monotone operator with a separable two-block structure and introduce a framework of block-decomposition prox-type algorithms for solving it which allows for each one of the single-block proximal subproblems to be solved in an approximate sense. Moreover, by showing that any method in this framework is also a special instance of the hybrid proximal extragradient (HPE) method introduced by Solodov and Svaiter, we derive corresponding convergence rate results. We also describe some instances of the framework based on specific and inexpensive schemes for solving the single-block proximal subproblems. Finally, we consider some applications of our methodology to establish for the first time (i) the iteration-complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and, as a consequence, the ergodic iteration-complexity of the Douglas-...