Mean Square Convergence Research Articles

Convergence of a exponential tamed method for a general interest rate model

We prove mean-square convergence of a exponential tamed method, for a generalized Ait-Sahalia interest rate model. The method is based on a Lamperti transform, splitting and applying a tamed numerical method for the nonlinearity. The main difficulty in the analysis is caused by the non-globally Lipschitz drift coefficients of the model. We consider the existence, uniqueness of the solution and boundedness of moments for the transformed SDE before proving bounded moments and inverse moment bounds for the numerical approximation. The exponential tamed method is a hybrid method in the sense that a backstop method is invoked to prevent solutions from overshooting zero and becoming negative. We successfully recover the strong convergence rate of order one for the exponential tamed method. In addition we prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. In our numerical experiments we compare to other numerical methods in the literature for realistic parameter values.

No-regret learning for repeated non-cooperative games with lossy bandits

This paper considers no-regret learning for repeated continuous-kernel games with lossy bandit feedback. Since it is difficult to give an explicit model of the utility functions in dynamic environments, the players’ actions can only be learned with bandit feedback. Moreover, due to unreliable communication channels or privacy protection, the bandit feedback may be lost or dropped at random. Therefore, we study the asynchronous online learning strategy of the players to adaptively adjust the next actions for minimizing the long-term regret loss. The paper provides a novel no-regret learning algorithm, called Online Gradient Descent with lossy bandits (OGD-lb). We first give the regret analysis for concave games with differentiable and Lipschitz utilities. Then we show that the action profile converges to a Nash equilibrium with probability 1 when the game is also strictly monotone. We further provide the mean-squared convergence rate ONpi−2k−1/3 when the game is β-strongly monotone, where N denotes the number of players and pi is the update probability. In addition, we extend the algorithm to the case when the loss probability of the bandit feedback is unknown, and prove its almost sure convergence to Nash equilibrium for strictly monotone games. Finally, we take the resource management in fog computing as an application example, and carry out numerical experiments to empirically demonstrate the algorithm performance.

Variable substitution methods in the solution of stochastic ordinary differential equations and their applications

Abstract In this paper, we first propose numerical solution methods for stochastic ordinary differential equations by using the two-step Maruyama method and Euler-Maruyama method in variable substitution, and analyze the mean-square compatibility, mean-square convergence and mean-square linear stability of the corresponding numerical methods, respectively. Finally, 10,000 times value experiments are conducted to verify the convergence accuracy and stability of the variable substitution methods. The results show that the Euler method simulates this equation when taking steps h = 0.05, β = 5 and σ = 2 for numerical experiments, and the numerical results obtained are convergent but unstable. On the other hand, the Euler-Maruyama method with variable substitution is consistent with the real solution. It shows that variable substitution is an important method for solving stochastic ordinary differential equations with high convergence and stability, which is of great significance for the solution of stochastic ordinary differential equations.

Theoretical convergence analysis of the FXLMS-based feedforward hybrid active noise control system

Real noise typically includes both broadband and narrowband components, such as noise generated by rotating machines. The feedforward hybrid active noise control (FFHANC) system reduces both components separately by arranging broadband and narrowband controllers in parallel. Although considerable efforts have been made to improve the FFHANC system, a theoretical analysis of its performance is still lacking. Consequently, the application of the FFHANC system lacks a solid foundation. Furthermore, the stability bounds of step sizes that guarantee convergence are not available in the literature. To address these problems, a convergence analysis of the FFHANC system based on the filtered-x least-mean-square algorithm is proposed. The analysis of the mean and mean-square convergence behaviors of the weight errors theoretically illustrates the advantages of the FFHANC system in terms of fast convergence and high reduction. The mutual coupling of parallel controllers in the mean-square sense is derived and discussed. By predicting the steady-state mean-square error and stability bounds, the proposed model can also guide the choice of step sizes. Our analysis lays the theoretical foundation for the practical application of the FFHANC system. Extensive simulations are performed to demonstrate the validity of the proposed theoretical theory.

Frequency-domain Volterra kernel-based adaptation: Formulations and algorithms

For the correlated input, the Volterra kernel-based least mean-square (LMS) algorithm in the time-domain exhibits a slow learning rate caused by the large eigenvalue spread of the input covariance matrix. To tackle such an issue, this paper develops a novel frequency-domain Volterra kernel-based filter, resulting in the periodic update constrained frequency-domain second-order Volterra normalized LMS (named as P-CFDSOV-NLMS1) algorithm. Subsequently, by using one- and two-dimensional discrete Fourier transforms (DFTs) simultaneously, another frequency-domain implementation and corresponding P-CFDSOV-NLMS2 algorithm are constructed. In contrast, the P-CFDSOV-NLMS1 scheme only requires one-dimensional DFT operations and takes advantage of the joint information between the block input vectors. Then, the mean and mean-square convergence behaviors of the P-CFDSOV-NLMS1 algorithm are investigated. Furthermore, the designed frequency-domain method is extended to three different widely complex-valued Volterra kernel-based models. Finally, computer simulations reveal that the suggested algorithms outperform the previously reported frequency-domain techniques in terms of convergence speed and tracking ability.

On Traces of Linear Operators with Symmetrized Volterra-Type Kernels

A solution to the trace convergence problem, which arises in proving the mean-square convergence for the approximation of iterated Stratonovich stochastic integrals, is proposed. This approximation is based on the representation of factorized Volterra-type functions as the orthogonal series. Solving the trace convergence problem involves the theory of trace class operators for symmetrized Volterra-type kernels. The main results are primarily focused on the approximation of iterated Stratonovich stochastic integrals, which are used to implement numerical methods for solving stochastic differential equations based on the Taylor–Stratonovich expansion.

Averaging principles for two-time-scale neutral stochastic delay partial differential equations driven by fractional Brownian motions

We prove the validity of averaging principles for two-time-scale neutral stochastic delay partial differential equations (NSDPDEs) driven by fractional Brownian motions (fBms) under two-time-scale formulation. Firstly, in the sense of mean-square convergence, we obtain not only the averaging principles for NSDPDEs involving two-time-scale Markov switching with a single weakly recurrent class but also for the case of two-time-scale Markov switching with multiple weakly irreducible classes. Secondly, averaging principles for NSDPDEs driven by fBms with random delay modulated by two-time-scale Markovian switching are established. We proved that there is a limit process in which the fast changing noise is averaged out. The limit process is substantially simpler than that of the original full fast–slow system.

The regularised inertial Dean–Kawasaki equation: discontinuous Galerkin approximation and modelling for low-density regime

The Regularised Inertial Dean–Kawasaki model (RIDK) – introduced by the authors and J. Zimmer in earlier works – is a nonlinear stochastic PDE capturing fluctuations around the meanfield limit for large-scale particle systems in both particle density and momentum density. We focus on the following two aspects. Firstly, we set up a Discontinuous Galerkin (DG) discretisation scheme for the RIDK model: we provide suitable definitions of numerical fluxes at the interface of the mesh elements which are consistent with the wave-type nature of the RIDK model and grant stability of the simulations, and we quantify the rate of convergence in mean square to the continuous RIDK model. Secondly, we introduce modifications of the RIDK model in order to preserve positivity of the density (such a feature only holds in a “high-probability sense” for the original RIDK model). By means of numerical simulations, we show that the modifications lead to physically realistic and positive density profiles. In one case, subject to additional regularity constraints, we also prove positivity. Finally, we present an application of our methodology to a system of diffusing and reacting particles. Our Python code is available in open-source format.

Mean-square convergent continuous state estimation of randomly switched linear systems with unobservable subsystems and stochastic output noises

This paper studies mean-square (MS) convergent observers for estimating continuous states of randomly switched linear systems (RSLSs) with unobservable subsystems that are subject to stochastic output observation noises. When subsystems are unobservable and switching sequences are random, the classical Kalman–Bucy filters that are applied to observable sub-states are shown to be potentially divergent. It is also shown that unless the switching interval T can be selected to be sufficiently small from the outset, MS convergence may never be achieved, regardless of how the observers for the subsystems are designed. The critical threshold Tmax on T is derived for MS convergent observers to be achievable. Under the condition T<Tmax, this paper introduces design algorithms for subsystem observers to generate a globally MS convergent observer for the entire continuous state. A fundamental design tradeoff between convergence speeds and steady-state estimation errors is analyzed. This paper extends our recent new framework and algorithms for strong convergent observer design in RSLSs by including observation noises, considering multi-output systems, establishing new algorithms for MS convergence, and developing design tradeoff analysis. Examples and a practical case study are presented to illustrate the design procedures, main convergence properties, and error analysis.

A new approach to decentralized constrained nonlinear estimation over noisy communication links

This paper investigates constrained nonlinear estimation over noisy communication links. The estimation is solved by a set of nodes in a fully distributed fashion. The unknown parameter of interest is assumed to be belonging to a closed convex set. The measurement of each node is nonlinearly related to the unknown parameter. The communication among adjacent nodes is corrupted by the additive communication noises. We propose a decentralized projection consensus+innovation algorithm with communication noises to solve the nonlinear estimation problem and develop a novel approach to analyze its convergence. For the case of fixed graph, by introducing an auxiliary matrix and combination of the graph Laplacian and the auxiliary matrix, we prove that the algorithm converges in mean square and almost surely if the combined persistence of excitation (CPE) condition holds and the measurement function satisfies the Lipschitz continuity and monotonicity conditions. Furthermore, for the case of the time-varying graphs, we establish the jointly combined persistence of excitation (JCPE) condition guaranteeing convergence in mean square. Both the CPE and JCPE conditions are proposed for the first time and do not require that the graph is balanced. The JCPE condition holds even when the graph is disconnected at infinitely many time instants. A simulation example is presented to demonstrate our theoretical results.

Correction: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Correction: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Mean-square convergence and stability of compensated stochastic theta methods for jump-diffusion SDEs with super-linearly growing coefficients

Mean-square convergence and stability of compensated stochastic theta methods for jump-diffusion SDEs with super-linearly growing coefficients

Rain process models and convergence to point processes

Abstract. A variety of stochastic models have been used to describe time series of precipitation or rainfall. Since many of these stochastic models are simplistic, it is desirable to develop connections between the stochastic models and the underlying physics of rain. Here, convergence results are presented for such a connection between two stochastic models: (i) a stochastic moisture process as a physics-based description of atmospheric moisture evolution and (ii) a point process for rainfall time series as spike trains. The moisture process has dynamics that switch after the moisture hits a threshold, which represents the onset of rainfall and thereby gives rise to an associated rainfall process. This rainfall process is characterized by its random holding times for dry and wet periods. On average, the holding times for the wet periods are much shorter than the dry ones, and, in the limit of short wet periods, the rainfall process converges to a point process that is a spike train. Also, in the limit, the underlying moisture process becomes a threshold model with a teleporting boundary condition. To establish these limits and connections, formal asymptotic convergence is shown using the Fokker–Planck equation, which provides some intuitive understanding. Also, rigorous convergence is proved in mean square with respect to continuous functions of the moisture process and convergence in mean square with respect to generalized functions of the rain process.

Proportionate Robust Diffusion Recursive Least Exponential Hyperbolic Cosine Algorithm: Optimum Parameter Selection and Convergence Analysis

In this brief, an improved version of the proportionate robust diffusion recursive least exponential hyperbolic cosine algorithm is suggested. This improved version is obtained by optimally selecting the step-size parameter of this algorithm using a minimization of the squared norm of the error vector. Moreover, a complete theoretical analysis of the presented algorithm is performed. This theoretical analysis consists of mean-square convergence analysis, mean-square steady-state analysis, and a discussion on the forgetting factor parameter selection. Moreover, simulation experiments show that the improved algorithm is superior than the original algorithm and other state-of-the-art algorithms in the literature in terms of speed of convergence.

Distributed Estimation With Adaptive Cluster Learning Over Asynchronous Data Fusion

In the electronic information era, wireless sensor network (WSN) has always been an essential foundation for information collection, processing and communication. In WSN with multi-task estimation, distributed cooperation estimation with cluster learning has always been an attractive topic. When the unknown estimation parameters become complex, some cluster learning algorithms may not work, and their estimation performance could degrade. In addition, the problems of time delay, caused by synchronous data fusion, and different sampling rates between different types of sensors are usually neglected in practical applications. To solve these problems, an unsupervised distributed multi-task estimation algorithm with adaptive cluster learning over asynchronous data is proposed to obtain a more accurate estimation. In the proposed algorithm, the time delay and different sampling rates are fully considered and investigated. The mean stability, mean-square convergence, and behavior of adaptive cluster learning are analyzed for the proposed algorithm with asynchronous data. Finally, simulations are provided to demonstrate the robustness and effectiveness of the proposed algorithm.

Projection pursuit adaptation on polynomial chaos expansions

The present work addresses the issue of accurate stochastic approximations in high-dimensional parametric space using tools from uncertainty quantification (UQ). The basis adaptation method and its accelerated algorithm in polynomial chaos expansions (PCE) were recently proposed to construct low-dimensional approximations adapted to specific quantities of interest (QoI). The present paper addresses one difficulty with these adaptations, namely their reliance on quadrature point sampling, which limits the reusability of potentially expensive samples. Projection pursuit (PP) is a statistical tool to find the “interesting” projections in high-dimensional data and thus bypass the curse-of-dimensionality. In the present work, we combine the fundamental ideas of basis adaptation and projection pursuit regression (PPR) to propose a novel method to simultaneously learn the optimal low-dimensional spaces and PCE representation from given data. While this projection pursuit adaptation (PPA) can be entirely data-driven, the constructed approximation exhibits mean-square convergence to the solution of an underlying governing equation and thus captures the supports and probability distributions associated with the physics constraints. The proposed approach is demonstrated on a borehole problem and a structural dynamics problem, demonstrating the versatility of the method and its ability to discover low-dimensional manifolds with high accuracy with limited data. In addition, the method can learn surrogate models for different quantities of interest while reusing the same data set.

Stochastic asymptotical regularization for linear inverse problems

We introduce stochastic asymptotical regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. We demonstrate that SAR can quantify the uncertainty in error estimates for inverse problems. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an order-optimal regularization method for linear ill-posed problems provided that the terminating time of SAR is chosen according to the smoothness of the solution. This result is proven for both a priori and a posteriori stopping rules under general range-type source conditions. Furthermore, some converse results of SAR are verified. Two iterative schemes are developed for the numerical realization of SAR, and the convergence analyses of these two numerical schemes are also provided. A toy example and a real-world problem of biosensor tomography are studied to show the accuracy and the advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can provide the uncertainty quantification of the quantity of interest, which can in turn be used to reveal and explicate the hidden information about real-world problems, usually obscured by the incomplete mathematical modeling and the ascendence of complex-structured noise.

Strong 1.5 order scheme for second-order stochastic differential equations without Levy area

In this work, a finite difference scheme combined with a truncated spectral expansion of white noise is presented for solving second-order stochastic differential equations (SDEs) with additive white noise. Consistency of the approximate equation, which is obtained by replacing the white noise with its spectral truncation, is considered in mean-square sense. Based on it, a full discrete scheme is constructed and the mean-square convergence order of numerical solution is investigated. Numerical examples show that the presented scheme is of 1.5-order convergence in mean-square sense, which verifies the theoretical findings and is a half order higher than converting the equation to a two-dimensional system with additive white noise and using the Euler-Maruyama scheme to solve the resulting problem. It is also illustrated that the piecewise version of the spectral approximation of white noise plus proper time discretization works well for long-time simulation.

Secure distributed estimation under Byzantine attack and manipulation attack

Wireless sensor networks (WSN) with distributed cooperation has been widely used in various fields due to their strong adaptive learning ability. However, WSN is vulnerable to malicious attacks, and the damaging behaviors of these attacks would make sensor nodes work unsatisfactorily and then contaminate the entire network. Although some security algorithms have been proposed to detect these malicious attacks, such as manipulation attack and Byzantine attack, they are not robust enough. To ameliorate this situation, a secure distributed diffusion least-mean-square (LMS) algorithm is designed, which adopts the dual detection mechanisms over the designed two subsystems. One subsystem is based on the LMS with cooperative strategy (L-CS), which uses an angle detector to filter manipulation attack, while the other subsystem is based on the LMS with non-cooperative strategy (L-NCS), where the sensors utilize non-cooperation to improve the detection effect on Byzantine attack. Moreover, the L-CS subsystem could further provide the secure estimation for the proposed algorithm by isolating malicious nodes. The performances are analyzed from the mean and mean-square convergence. Finally, some simulations are implemented to prove the effectiveness of the proposed algorithm.