Asymptotic Convergence Rate Research Articles

Wavelet-based estimation of regression function with strong mixing errors under fixed design

ABSTRACTWe consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes Bsp, q. The theory is illustrated with some numerical examples.A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.

Tensor norm and maximal singular vectors of nonnegative tensors — A Perron–Frobenius theorem, a Collatz–Wielandt characterization and a generalized power method

We study the ℓp1,…,pm-singular value problem for nonnegative tensors. We prove a general Perron–Frobenius theorem for weakly irreducible and irreducible nonnegative tensors and provide a Collatz–Wielandt characterization of the maximal singular value. Additionally, we propose a higher order power method for the computation of the maximal singular vectors and show that it has an asymptotic linear convergence rate.

Efficiency of the estimate refinement method for polyhedral approximation of multidimensional balls

The estimate refinement method for the polyhedral approximation of convex compact bodies is analyzed. When applied to convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. In previous studies, for the approximation of a multidimensional ball, the convergence rates of the method were estimated in terms of the number of faces of all dimensions and the cardinality of the facial structure (the norm of the f-vector) of the constructed polytope was shown to have an optimal rate of growth. In this paper, the asymptotic convergence rate of the method with respect to faces of all dimensions is compared with the convergence rate of best approximation polytopes. Explicit expressions are obtained for the asymptotic efficiency, including the case of low dimensions. Theoretical estimates are compared with numerical results.

Discrete analysis of domain decomposition approaches for mesh generation via the equidistribution principle

Moving mesh methods based on the equidistribution principle are powerful techniques for the space-time adaptive solution of evolution problems. Solving the resulting coupled system of equations, namely the original PDE and the mesh PDE, however, is challenging in parallel. Recently several Schwarz domain decomposition algorithms were proposed for this task and analyzed at the continuous level. However, after discretization, the resulting problems may not even be well posed, so the discrete algorithms requires a different analysis, which is the subject of this paper. We prove that when the number of grid points is large enough, the classical parallel and alternating Schwarz methods converge to the unique monodomain solution. Thus, such methods can be used in place of Newton’s method, which can suffer from convergence difficulties for challenging problems. The analysis for the nonlinear domain decomposition algorithms is based on M–function theory and is valid for an arbitrary number of subdomains. An asymptotic convergence rate is provided and numerical experiments illustrate the results.

On the convergence of the minimally irreducible Markov chain method with applications to PageRank

PageRank is an important technique for determining the most important nodes in the Web. In this paper, we point out that the main result derived in Bao and Zhu [Acta Math Appl Sin (Engl Ser) 22:517---528, 2006] is incorrect, and establish new lower and upper bounds on the convergence of the minimally irreducible Markov chain method for PageRank. We show that the asymptotic convergence rates of the maximally and the minimally irreducible Markov chains are the same when the damping factor $$\frac{1}{2}\le \alpha <1$$12≤?<1, however, they are not mathematically equivalent any more as $$0<\alpha <\frac{1}{2}$$0<?<12. Numerical experiments validate our theoretical results.

Modelling Function-Valued Stochastic Processes, with Applications to Fertility Dynamics

Summary We introduce a simple and interpretable model for functional data analysis for situations where the observations at each location are functional rather than scalar. This new approach is based on a tensor product representation of the function-valued process and utilizes eigenfunctions of marginal kernels. The resulting marginal principal components and product principal components are shown to have nice properties. Given a sample of independent realizations of the underlying function-valued stochastic process, we propose straightforward fitting methods to obtain the components of this model and to establish asymptotic consistency and rates of convergence for the estimates proposed. The methods are illustrated by modelling the dynamics of annual fertility profile functions for 17 countries. This analysis demonstrates that the approach proposed leads to insightful interpretations of the model components and interesting conclusions.

High-Performance Consensus Control in Networked Systems With Limited Bandwidth Communication and Time-Varying Directed Topologies.

Communication data rates and energy constraints are two important factors that have to be considered in the coordination control of multiagent networks. Although some encoder-decoder-based consensus protocols are available, there still exists a fundamental theoretical problem: how can we further reduce the update rate of control input for each agent without the changing consensus performance? In this paper, we consider the problem of average consensus over directed and time-varying digital networks of discrete-time first-order multiagent systems with limited communication data transmission rates. Each agent has a real-valued state but can only exchange binary symbolic sequence with its neighbors due to bandwidth constraints. A class of novel event-triggered dynamic encoding and decoding algorithms is proposed, based on which a kind of consensus protocol is presented. Moreover, we develop a scheme to select the numbers of time-varying quantization levels for each connected communication channel in the time-varying directed topologies at each time step. The analytical relation among system and network parameters is characterized explicitly. It is shown that the asymptotic convergence rate is related to the scale of the network, the number of quantization levels, the system parameter, and the network structure. It is also found that under the designed event-triggered protocol, for a directed and time-varying digital network, which uniformly contains a spanning tree over a time interval, the average consensus can be achieved with an exponential convergence rate based on merely 1-b information exchange between each pair of adjacent agents at each time step.

Accelerated PMHSS iteration methods for complex symmetric linear systems

In this paper, we introduce and analyze an accelerated preconditioning modification of the Hermitian and skew-Hermitian splitting (APMHSS) iteration method for solving a broad class of complex symmetric linear systems. This accelerated PMHSS algorithm involves two iteration parameters ź,β and two preconditioned matrices whose special choices can recover the known PMHSS (preconditioned modification of the Hermitian and skew-Hermitian splitting) iteration method which includes the MHSS method, as well as yield new ones. The convergence theory of this class of APMHSS iteration methods is established under suitable conditions. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. Theoretical analyses show that the upper bound ź1(ź,β) of the asymptotic convergence rate of the APMHSS method is smaller than that of the PMHSS iteration method. This implies that the APMHSS method may converge faster than the PMHSS method. Numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method.

A nonlinear finite element plasticity formulation without matrix inversions

In the present paper a robust computational procedure is proposed for the finite element solution of elasto-plastic boundary value problems in the structural analysis of ductile materials. An implicit numerical scheme is detailed based on a return mapping algorithm. In the present computational scheme the local constitutive equations lead to the solution of a nonlinear scalar equation which ultimately reduces to a single variable algebraic (polynomial) equation in the plastic rate parameter. The proposed integration scheme allows to find the analytical solution of the algebraic equation in closed form. Consequently, in the present algorithmic procedure no iterative method is required to solve the local constitutive problem. The algorithmic scheme presented herein is also characterized by an alternative formulation of the algorithmic counterpart of the plastic consistency condition. Furthermore, in the algorithmic strategy a suitable procedure is presented for the consistent linearization of elasto-plastic boundary value problems. In fact an alternative procedure is proposed herein for the consistent derivation of the tangent operator associated to the algorithmic scheme so that in the present approach the consistent tangent operator is determined without the necessity of computing matrix inversions. The presented computational approach combined with the proposed consistent tangent operator ensures a quadratic rate of asymptotic convergence when an iterative method is pursued for the global solution procedure of elasto-plastic boundary value problems. Numerical applications and illustrative examples are finally reported to show the robustness and effectiveness of the proposed computational procedure in the finite element analysis of elasto-plastic boundary value problems.

Optimal linear drift for the speed of convergence of an hypoelliptic diffusion

Among all generalized Ornstein-Uhlenbeck processes which sample the same invariant measure and for which the same amount of randomness (a $N$-dimensional Brownian motion) is injected in the system, we prove that the asymptotic rate of convergence is maximized by a non-reversible hypoelliptic one.

A Simulation Budget Allocation Procedure for Enhancing the Efficiency of Optimal Subset Selection

Selecting the optimal subset is highly beneficial to numerous developments in simulation optimization. This paper studies the problem of maximizing the probability of correctly selecting the top- $m$ designs out of $k$ designs under a computing budget constraint. We develop a new procedure which is more efficient and robust than currently existing procedures in the literature. We also provide an analysis on its asymptotic convergence rate. Based on this analysis, we show that our new procedure achieves a higher convergence rate than other procedures under certain conditions. Numerical testing supports our analytical analysis and shows that the new procedure is significantly more efficient and robust.

Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem

We aim to provide a finite element analysis for the elastoacoustic vibration problem. We use a dual-mixed variational formulation for the elasticity system and combine the lowest order Lagrange finite element in the fluid domain with the reduced symmetry element known as PEERS and introduced for linear elasticity in [1]. We show that the resulting global nonconforming scheme provides a correct spectral approximation and we prove quasi-optimal error estimates. Finally, we confirm the asymptotic rates of convergence by numerical experiments.

Steepest Descent Method with Random Step Lengths

The paper studies the steepest descent method applied to the minimization of a twice continuously differentiable function. Under certain conditions, the random choice of the step length parameter, independent of the actual iteration, generates a process that is almost surely R-convergent for quadratic functions. The convergence properties of this random procedure are characterized based on the mean value function related to the distribution of the step length parameter. The distribution of the random step length, which guarantees the maximum asymptotic convergence rate independent of the detailed properties of the Hessian matrix of the minimized function, is found, and its uniqueness is proved. The asymptotic convergence rate of this optimally created random procedure is equal to the convergence rate of the Chebyshev polynomials method. Under practical conditions, the efficiency of the suggested random steepest descent method is degraded by numeric noise, particularly for ill-conditioned problems; furthermore, the asymptotic convergence rate is not achieved due to the finiteness of the realized calculations. The suggested random procedure is also applied to the minimization of a general non-quadratic function. An algorithm needed to estimate relevant bounds for the Hessian matrix spectrum is created. In certain cases, the random procedure may surpass the conjugate gradient method. Interesting results are achieved when minimizing functions having a large number of local minima. Preliminary results of numerical experiments show that some modifications of the presented basic method may significantly improve its properties.

CGIHT: conjugate gradient iterative hard thresholding for compressed sensing and matrix completion

We introduce the conjugate gradient iterative hard thresholding (CGIHT) family of algorithms for the efficient solution of constrained underdetermined linear systems of equations arising in compressed sensing, row-sparse approximation and matrix completion. CGIHT is designed to balance the low per iteration complexity of simple hard thresholding algorithms with the fast asymptotic convergence rate of employing the conjugate gradient method. We establish provable recovery guarantees and stability to noise for variants of CGIHT with sufficient conditions in terms of the restricted isometry constants of the sensing operators. Extensive empirical performance comparisons establish significant computational advantages for CGIHT both in terms of the size of problems, which can be accurately approximated and in terms of overall computation time.

An efficient return mapping algorithm for elastoplasticity with exact closed form solution of the local constitutive problem

Purpose – The purpose of this paper is to present an efficient return mapping algorithm for elastoplastic constitutive problems of ductile metals with an exact closed form solution of the local constitutive problem in the small strain regime. A Newton Raphson iterative method is adopted for the solution of the boundary value problem. Design/methodology/approach – An efficient return mapping algorithm is illustrated which is based on an elastic predictor and a plastic corrector scheme resulting in an implicit and accurate numerical integration method. Nonlinear kinematic hardening rules and linear isotropic hardening rules are used to describe the components of the hardening variables. In the adopted algorithmic approach the solution of the local constitutive equations reduces to only one straightforward nonlinear scalar equation. Findings – The presented algorithmic scheme naturally leads to a particularly simple form of the nonlinear scalar equation which ultimately scales down to an algebraic (polynomial) equation with a single variable. The straightforwardness of the present approach allows to find the analytical solution of the algebraic equation in a closed form. Further, the consistent tangent operator is derived as associated with the proposed algorithmic scheme and it is shown that the proposed computational procedure ensures a quadratic rate of asymptotic convergence when used with a Newton Raphson iterative method for the global solution procedure. Originality/value – In the present approach the solution of the algebraic nonlinear equation is found in a closed form and accordingly no iterative method is required to solve the problem of the local constitutive equations. The computational procedure ensures a quadratic rate of asymptotic convergence for the global solution procedure typical of computationally efficient solution schemes. In the paper it is shown that the proposed algorithmic scheme provides an efficient and robust computational solution procedure for elastoplasticity boundary value problems. Numerical examples and computational results are reported which illustrate the effectiveness and robustness of the adopted integration algorithm for the finite element analysis of elastoplastic structures also under elaborate loading conditions.

Formula omitted] regularization: Convergence of iterative thresholding algorithm

The L2/3 regularization is a nonconvex and nonsmooth optimization problem. Cao et al. (2013) investigated that the L2/3 regularization is more effective in imaging deconvolution. The convergence issue of the iterative thresholding algorithm of L2/3 regularization problem (the L2/3 algorithm) hasn’t been addressed in Cao et al. (2013). In this paper, we study the convergence of the L2/3 algorithm. As the main result, we show that under certain conditions, the sequence {x(n)} generated by the L2/3 algorithm converges to a local minimizer of L2/3 regularization, and its asymptotical convergence rate is linear. We provide a set of experiments to verify our theoretical assertions and show the performance of the algorithm on sparse signal recovery. The established results provide a theoretical guarantee for a wide range of applications of the algorithm.

Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis

The asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper. By a Cole–Hopf type transformation, the singular chemotaxis is converted into a non-singular hyperbolic system. Then we study the transformed system and establish the global existence, asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates. The main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established.

Extrinsic Local Regression on Manifold-Valued Data

We propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling i.i.d manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples are considered indicating the wide applicability of our approach.

Average Convergence Rate of Evolutionary Algorithms

In evolutionary optimization, it is important to understand how fast evolutionary algorithms converge to the optimum per generation, or their convergence rate. This paper proposes a new measure of the convergence rate, called average convergence rate. It is a normalised geometric mean of the reduction ratio of the fitness difference per generation. The calculation of the average convergence rate is very simple and it is applicable for most evolutionary algorithms on both continuous and discrete optimization. A theoretical study of the average convergence rate is conducted for discrete optimization. Lower bounds on the average convergence rate are derived. The limit of the average convergence rate is analysed and then the asymptotic average convergence rate is proposed.

Estimating Effective Connectivity from fMRI Data Using Factor-based Subspace Autoregressive Models

We consider the problem of identifying large-scale effective connectivity of brain networks from fMRI data. Standard vector autoregressive (VAR) models fail to estimate reliably networks with large number of nodes. We propose a new method based on factor modeling for reliable and efficient high-dimensional VAR analysis of large networks. We develop a subspace VAR (SVAR) model from a factor model (FM), where observations are driven by a lower-dimensional subspace of common latent factors with an AR dynamics. We consider two variants of principal components (PC) methods that provide consistent estimates for the FM hence the implied SVAR model, even of large dimensions. Information criterion is used to select the optimal subspace dimension. We established asymptotic normality and convergence rates for the estimated SVAR coefficients matrix. Evaluation on simulated resting-state fMRI shows that the SVAR models are more robust and produce better connectivity estimates than the classical model for a moderately-large network analysis. Results on real data by varying the subspace dimensions identify strong connections in the default mode network and reveal hierarchical connectivity of resting-state networks with distinct functional relevance.