Convergence Rate Estimates Research Articles

Linear-quadratic Tobit regression model with a change point due to a covariate threshold

This paper considers a linear-quadratic Tobit regression model, which is developed for modelling the mixture structure with a line segment and a quadratic segment intersecting at an unknown change point. Due to the smoothness of such model structure, the regression coefficients and the change point can be obtained by directly maximising the likelihood function. The proposed estimator has rapid convergence rate and high estimation accuracy, without other computation burden. The asymptotic properties for the proposed estimator are derived by using empirical process theory. A sup-likelihood ratio test procedure is developed for testing the existence of a change point, and its limiting distributions are derived. The good finite sample performances of the proposed estimator are illustrated by numerical studies and empirical applications to the MGUS and GDP data.

Pickl’s proof of the quantum mean-field limit and quantum Klimontovich solutions

This paper discusses the mean-field limit for the quantum dynamics of N identical bosons in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{R}}^3$$\\end{document} interacting via a binary potential with Coulomb-type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in Golse and Paul (Commun Math Phys 369:1021–1053, 2019) . Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in Kato (Trans Am Math Soc 70:195–211, 1951). Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb-type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in Pickl (Lett Math Phys 97:151-164, 2011), resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.

An Adaptive V2G Capacity-Based Frequency Regulation Scheme With Integral Reinforcement Learning Against DoS Attacks

Aggregated electrical vehicles (EVs) can flexibly serve grid frequency regulation (FR) with adjustable FR capacity (FRC) via vehicle-to-grid (V2G) technology. Therefore, current research applies integral reinforcement learning (IRL) to FR as it can easily solve difficult modeling and optimization problems. However, communication between EVs is vulnerable to denial of service (DoS) attacks, which can significantly degrade FR performance and even destabilize V2G FR systems. This paper proposes an adaptive V2G FRC-based FR scheme with IRL to improve FR resilience and mitigate FR performance degradation against DoS attacks. The proposed scheme optimizes V2G control by IRL without analytical models, integrating an event-triggered mechanism to reduce communication burden. It adapts V2G FRC to attack intensity to minimize frequency deviation and mitigate the impact of DoS attacks. The analytical estimation of convergence rate establishes the quantitative relationships among control performance, V2G FRC, and attack intensity, thereby deriving an adaptive mechanism. The theoretical analysis of the proposed scheme yields stability conditions that guarantee the FR performance. The simulations were performed on the IEEE 39-bus test system to verify the effectiveness and advantages of the proposed scheme. The results indicate that V2G FRC should be scaled down to mitigate performance degradation when intensive attacks occur.

Upper Plate and Subduction Interface Deformation Models in the 2022 Revision of the Aotearoa New Zealand National Seismic Hazard Model

ABSTRACT As part of the 2022 revision of the Aotearoa New Zealand National Seismic Hazard Model (NZ NSHM 2022), deformation models were constructed for the upper plate faults and subduction interfaces that impact ground-shaking hazard in New Zealand. These models provide the locations, geometries, and slip rates of the earthquake-producing faults in the NZ NSHM 2022. For upper plate faults, two deformation models were developed: a geologic model derived directly from the fault geometries and geologic slip rates in the NZ Community Fault Model version 1.0 (NZ CFM v.1.0); and a geodetic model that uses the same faults and fault geometries and derives fault slip-deficit rates by inverting geodetic strain rates for back slip on those specified faults. The two upper plate deformation models have similar total moment rates, but the geodetic model has higher slip rates on low-slip-rate faults, and the geologic model has higher slip rates on higher-slip-rate faults. Two deformation models are developed for the Hikurangi–Kermadec subduction interface. The Hikurangi–Kermadec geometry is a linear blend of the previously published interface models. Slip-deficit rates on the Hikurangi portion of the deformation model are updated from the previously published block models, and two end member models are developed to represent the alternate hypotheses that the interface is either frictionally locked or creeping at the trench. The locking state in the Kermadec portion is less well constrained, and a single slip-deficit rate model is developed based on plate convergence rate and coupling considerations. This single Kermadec realization is blended with each of the two Hikurangi slip-deficit rate models to yield two overall Hikurangi–Kermadec deformation models. The Puysegur subduction interface deformation model is based on geometry taken directly from the NZ CFM v.1.0, and a slip-deficit rate derived from published geodetic plate convergence rate and interface coupling estimates.

Multilevel Monte Carlo FEM for elliptic PDEs with Besov random tree priors

We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain D⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}\\subset {\\mathbb {R}}^d$$\\end{document}, with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for (a) nonuniform pathwise upper and lower coefficient bounds, and for (b) low path-regularity of the Besov-tree coefficients. Admissible non-parametric random coefficients correspond to random functions exhibiting singularities on random fractals with tunable fractal dimension, but involve no a-priori specification of the fractal geometry of singular supports of sample paths. Optimal complexity and convergence rate estimates for quantities of interest and for their second moments are proved. A convergence analysis for MLMC-FEM is performed which yields choices of the algorithmic steering parameters for efficient implementation. A complexity (“error vs work”) analysis of the MLMC-FEM approximations is provided.

Convergence rate for degenerate partial and stochastic differential equations via weak Poincaré inequalities

We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the L2-convergence rate by using weak Poincaré inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and of weak solutions to the corresponding degenerate stochastic differential equation with multiplicative noise.

Convergence rate estimates for penalty methods revisited

Convergence rate estimates for penalty methods revisited

Decoupled adaptive terminal sliding mode control strategy for a 6-DOF electro-hydraulic suspension test rig with RBF coupling force compensator

Aiming at the problems of poor tracking accuracy, low convergence speed, weak robustness of a 6-DOF electro-hydraulic suspension test system under system uncertainties and external disturbance, this article proposes double closed-loop control scheme. In the outer loop controller, this article designs a new decoupled adaptive fast nonsingular terminal sliding mode (DAFNTSM) control to achieve fast convergence rate, high accurate and adaptive disturbances estimation during trajectory tracking. Additionally, a coupling force compensator based on RBF neural network is employed, which can reduce online calculation quantity and guarantee accuracy. For six inner loop controllers, each hydraulic actuator can precisely follow the required force solved by outer loop controller. The hard-in-loop experiment results reveal that the control approach we proposed in this paper realize superior trajectory tracking characteristics in comparison with the conventional nonsingular terminal sliding mode (CNTSM) method.

Convergence of trigonometric and finite-difference discretization schemes for FFT-based computational micromechanics

This paper studies the convergences of several FFT-based discretization schemes that are widely applied in computational micromechanics for deriving effective coefficients, and “convergence” here means the limiting behaviors as spatial resolutions tending to infinity. Those schemes include Moulinec–Suquet’s scheme, Willot’s scheme and the FEM scheme. Under some reasonable assumptions, we prove that the effective coefficients obtained by those schemes all converge to the theoretical ones. Moreover, for the FEM scheme, we can present several convergence rate estimates under additional regularity assumptions.

Некоторые вопросы вероятности на конечных квазигруппах

We consider transformations of independent random variables by quasigroup operations. We explore the role of non-uniformity measures in the proof of convergence to uniformity and the estimation of convergence rate. We suggest a non-uniformity measure that allows an elementary proof of exponential convergence rate. The results on convergence rate are juxtaposed with known results on transformations of random variables in finited groups.

Estimate of convergence rate for remote projections on three subspaces

Получена оценка скорости сходимости к нулю норм дальних проекций на три подпространства гильбертова пространства с нулевым пересечением для начальных векторов из суммы ортогональных дополнений к этим подпространствам.

Evolution Strategies under the 1/5 Success Rule

For large space dimensions, the log-linear convergence of the elitist evolution strategy with a 1/5 success rule on the sphere fitness function has been observed, experimentally, from the very beginning. Finding a mathematical proof took considerably more time. This paper presents a review and comparison of the most consistent theories developed so far, in the critical interpretation of the author, concerning both global convergence and the estimation of convergence rates. I discuss the local theory of the one-step expected progress and success probability for the (1+1) ES with a normal/uniform distribution inside the sphere mutation, thereby minimizing the SPHERE function, but also the adjacent global convergence and convergence rate theory, essentially based on the 1/5 rule. Small digressions into complementary theories (martingale, irreducible Markov chain, drift analysis) and different types of algorithms (population based, recombination, covariance matrix adaptation and self-adaptive ES) complete the review.

Sharp Estimates for Proximity of Geometric and Related Sums Distributions to Limit Laws

The convergence rate in the famous Rényi theorem is studied by means of the Stein method refinement. Namely, it is demonstrated that the new estimate of the convergence rate of the normalized geometric sums to exponential law involving the ideal probability metric of the second order is sharp. Some recent results concerning the convergence rates in Kolmogorov and Kantorovich metrics are extended as well. In contrast to many previous works, there are no assumptions that the summands of geometric sums are positive and have the same distribution. For the first time, an analogue of the Rényi theorem is established for the model of exchangeable random variables. Also within this model, a sharp estimate of convergence rate to a specified mixture of distributions is provided. The convergence rate of the appropriately normalized random sums of random summands to the generalized gamma distribution is estimated. Here, the number of summands follows the generalized negative binomial law. The sharp estimates of the proximity of random sums of random summands distributions to the limit law are established for independent summands and for the model of exchangeable ones. The inverse to the equilibrium transformation of the probability measures is introduced, and in this way a new approximation of the Pareto distributions by exponential laws is proposed. The integral probability metrics and the techniques of integration with respect to sign measures are essentially employed.

Bounds for the Rate of Convergence in the Generalized Rényi Theorem

In the paper, an overview is presented of the results on the convergence rate bounds in limit theorems concerning geometric random sums and their generalizations to mixed Poisson random sums, including the case where the mixing law is itself a mixed exponential distribution. The main focus is on the upper bounds for the Zolotarev ζ-metric as the distance between the pre-limit and limit laws. New results are presented that extend existing estimates of the rate of convergence of geometric random sums (in the well-known Rényi theorem) to a considerably more general class of random indices whose distributions are mixed Poisson, including generalized negative binomial (e.g., Weibull-mixed Poisson), Pareto-type (Lomax)-mixed Poisson, exponential power-mixed Poisson, Mittag-Leffler-mixed Poisson, and one-sided Linnik-mixed Poisson distributions. A transfer theorem is proven that makes it possible to obtain upper bounds for the rate of convergence in the law of large numbers for mixed Poisson random sums with mixed exponential mixing distribution from those for geometric random sums (that is, from the convergence rate estimates in the Rényi theorem). Simple explicit bounds are obtained for ζ-metrics of the first and second orders. An estimate is obtained for the stability of representation of the Mittag-Leffler distribution as a geometric convolution (that is, as the distribution of a geometric random sum).

Robust consensus control of nonlinear multi‐agent systems based on convergence rate estimation

AbstractConvergence rate is an important performance criterion for consensus control of multi‐agent systems (MASs). It has been an important and difficult problem to accelerate the convergence rate of MASs. In this article, a robust consensus control scheme based on the convergence rate estimation for a nonlinear high‐order MAS is proposed to accelerate the convergence rate as well as improve the control performance of the MAS. In the proposed control scheme, a novel convergence rate indicator (CRI) is defined to measure the influence of the model nonlinearity, the state couplings within a single agent and among multiple agents in the MAS on convergence rate. Then, a distributed consensus controller with a CRI‐based additive term is designed by the backstepping technique with disturbance observers to enhance the robustness of the closed‐loop system. Furthermore, from a theoretical perspective, the robustness and the stability of the MAS are proved. Finally, the consensus control problem of the multi‐agent control system composed of five single‐link robot arms is simulated to validate the effectiveness of the proposed controller. Simulation results illustrate that the proposed consensus control approach with CRI has better performance than conventional backstepping consensus control techniques of the MAS in the same conditions.

Newton–Krylov-BDDC deluxe solvers for non-symmetric fully implicit time discretizations of the bidomain model

A novel theoretical convergence rate estimate for a Balancing Domain Decomposition by Constraints algorithm is proven for the solution of the cardiac bidomain model, describing the propagation of the electric impulse in the cardiac tissue. The non-linear system arises from a fully implicit time discretization and a monolithic solution approach. The preconditioned non-symmetric operator is constructed from the linearized system arising within the Newton–Krylov approach for the solution of the non-linear problem; we theoretically analyze and prove a convergence rate bound for the Generalised Minimal Residual iterations’ residual. The theory is confirmed by extensive parallel numerical tests, widening the class of robust and efficient solvers for implicit time discretizations of the bidomain model.

Variable Matrix-Type Step-Size Affine Projection Sign Algorithm for System Identification in the Presence of Impulsive Noise

This paper presents a novel variable matrix-type step-size affine projection sign algorithm (VMSS-APSA) characterized by robustness against impulsive noise. To mathematically derive a matrix-type step size, VMSS-APSA utilizes mean-square deviation (MSD) for the modified version of the original APSA. Accurately establishing the MSD of APSA is impossible. Therefore, the proposed VMSS-APSA derives the upper bound of the MSD using the upper bound of the L1-norm of the measurement noise. The optimal matrix-type step size is calculated at each iteration by minimizing the upper bound of the MSD, thereby improving the filter performance in terms of convergence rate and steady-state estimation error. Because a novel cost function of the proposed VMSS-APSA was designed to maintain a form similar to the original APSA, they have symmetric characteristics. Simulation results demonstrate that the proposed VMSS-APSA improves filter performance in a system-identification scenario in the presence of impulsive noise.

A Lyapunov Approach to Consensus of Linear Multi-Agent Systems with Infinite Distributed Communication Delays

This paper considers the leaderless consensus problem of linear time-invariant multi-agent systems with infinite distributed communication delays. A novel distributed low gain controller is proposed based on the solution to a parametric algebraic Riccati equation. It is shown via the newly developed Lyapunov-like method that not only the consensus of linear time-invariant multi-agent systems can be achieved exponentially under some mild assumptions but also an estimate of the exponential convergence rate of consensus is given in this work. The Lyapunov-like method is also extended to handle a special case of linear time-varying multi-agent systems. In addition, the obtained results include the results on the leaderless consensus of linear multi-agent systems with bounded distributed communication delays as special cases. To the best of our knowledge, this is the first work that develops the Lyapunov-like method for the leaderless consensus problems of both time-invariant and time-varying linear multi-agent systems with infinite distributed communication delays. Finally, a numerical example is presented to illustrate the effectiveness of the proposed controller.

Convergence rate of solutions to the generalized telegraph equation with an inhomogeneous force

The aim of this note is to study the convergence rate of the solution to the generalized telegraph equationεutt−div(|∇u|p(x)−2∇u)+ut=f. Roughly speaking, we borrow some ideas originated from the study of second order dynamical system to establish the strong convergence relation between u(x,t) and u⁎(x), in which u⁎ is the solution to the steady-state equation. Further, we also construct a new error functional to build up the first order differential inequality associated with the difference u(x,t)−u⁎(x), and then give explicit convergence rate estimates for the difference u(x,t)−u⁎(x). Namely, there exists a positive constant M such that∫Ω|∇u(x,t)−∇u⁎(x)|p(x)dx⩽Mt11−p+ with p+=max⁡p(x).

Convergence rate estimates of a higher-dimension reaction–diffusion system with density-dependent motility

Convergence rate estimates of a higher-dimension reaction–diffusion system with density-dependent motility