Individual Step Sizes Research Articles

Robust augmented complex-valued normalized M-estimate subband adaptive filtering algorithm against colored non-circular inputs and impulsive noise

Recently, the augmented complex-valued normalized subband adaptive filtering (ACNSAF) algorithm has been proposed to process colored non-circular signals. However, its performance will deteriorate severely under impulsive noise interference. To overcome this issue, a robust augmented complex-valued normalized M-estimate subband adaptive filtering (ACNMSAF) algorithm is proposed, which is obtained by modifying the subband constraints of the ACNSAF algorithm using the complex-valued modified Huber (MH) function and is derived based on CR calculus and Lagrange multipliers. In order to improve both the convergence speed and steady-state accuracy of the fixed step size ACNMSAF algorithm, a variable step size (VSS) strategy based on the minimum mean squared deviation (MSD) criterion is devised, which allocates individual adaptive step size to each subband, fully exploiting the structural advantages of SAF and significantly improving the convergence performance of the ACNMSAF algorithm as well as its tracking capability in non-stationary environment. Then, the stability, transient and steady-state MSD performance of the ACNMSAF algorithm in the presence of colored non-circular inputs and impulsive noise are analyzed, and the stability conditions, transient and steady-state MSD formulas are also derived. Computer simulations in impulsive noise environments verify the accuracy of theoretical analysis results and the effectiveness of the proposed algorithms compared to other existing complex-valued adaptive algorithms.

A Novel Individual Variable Step-Size Subband Adaptive Filter Algorithm Robust to Impulsive Noises

This paper proposes a novel individual variable step-size subband adaptive filter algorithm robust to impulsive noises. A fixed step-size subband adaptive filter algorithm that is robust against impulsive noises is newly derived by obtaining the optimal solution from a constrained optimization problem through the Lagrange multiplier. In addition, in order to further improve the convergence performance of the proposed algorithm, the weight update formula with a single fixed step size is modified to have multiple individual step sizes. By analyzing its mean-square-deviation (MSD), the optimal individual step size is designed. Simulation results show that the proposed algorithm outperforms the algorithms robust to impulsive noises in the literature.

A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay

Inspired by the collective behavior of fish schools, the fish school search (FSS) algorithm is a technique for finding globally optimal solutions. The algorithm is characterized by its simplicity and high performance; FSS is computationally inexpensive, compared to other evolution-inspired algorithms. However, the premature convergence problem is inherent to FSS, especially in the optimization of functions that are in very-high-dimensional spaces and have plenty of local minima or maxima. The accuracy of the obtained solution highly depends on the initial distribution of agents in the search space and on the predefined initial individual and collective-volitive movement step sizes. In this paper, we provide a study of different chaotic maps with symmetric distributions, used as pseudorandom number generators (PRNGs) in FSS. In addition, we incorporate exponential step decay in order to improve the accuracy of the solutions produced by the algorithm. The obtained results of the conducted numerical experiments show that the use of chaotic maps instead of other commonly used high-quality PRNGs can speed up the algorithm, and the incorporated exponential step decay can improve the accuracy of the obtained solution. Different pseudorandom number distributions produced by the considered chaotic maps can positively affect the accuracy of the algorithm in different optimization problems. Overall, the use of the uniform pseudorandom number distribution generated by the tent map produced the most accurate results. Moreover, the tent-map-based PRNG achieved the best performance when compared to other chaotic maps and nonchaotic PRNGs. To demonstrate the effectiveness of the proposed optimization technique, we provide a comparison of the tent-map-based FSS algorithm with exponential step decay (ETFSS) with particle swarm optimization (PSO) and with the genetic algorithm with tournament selection (GA) on test functions for optimization.

Multi-Swarm Cuckoo Search Algorithm with Q-Learning Model

Abstract As a novel swarm intelligence optimization algorithm, cuckoo search (CS) has been successfully applied to solve diverse problems in the real world. Despite its efficiency and wide use, CS has some disadvantages, such as premature convergence, easy to fall into local optimum and poor balance between exploitation and exploration. In order to improve the optimization performance of the CS algorithm, a new CS extension with multi-swarms and Q-Learning namely MP-QL-CS is proposed. The step size strategy of the CS algorithm is that an individual fitness value is examined based on a one-step evolution effect of an individual instead of evaluating the step size from the multi-step evolution effect. In the MP-QL-CS algorithm, a step size control strategy is considered as action, which is used to examine the individual multi-stepping evolution effect and learn the individual optimal step size by calculating the Q function value. In this way, the MP-QL-CS algorithm can increase the adaptability of individual evolution, and a good balance between diversity and intensification can be achieved. Comparing the MP-QL-CS algorithm with various CS algorithms, variants of differential evolution (DE) and improved particle swarm optimization (PSO) algorithms, the results demonstrate that the MP-QL-CS algorithm is a competitive swarm algorithm.

Using Cuckoo Search Algorithm with Q-Learning and Genetic Operation to Solve the Problem of Logistics Distribution Center Location

Cuckoo search (CS) algorithm is a novel swarm intelligence optimization algorithm, which is successfully applied to solve some optimization problems. However, it has some disadvantages, as it is easily trapped in local optimal solutions. Therefore, in this work, a new CS extension with Q-Learning step size and genetic operator, namely dynamic step size cuckoo search algorithm (DMQL-CS), is proposed. Step size control strategy is considered as action in DMQL-CS algorithm, which is used to examine the individual multi-step evolution effect and learn the individual optimal step size by calculating the Q function value. Furthermore, genetic operators are added to DMQL-CS algorithm. Crossover and mutation operations expand search area of the population and improve the diversity of the population. Comparing with various CS algorithms and variants of differential evolution (DE), the results demonstrate that the DMQL-CS algorithm is a competitive swarm algorithm. In addition, the DMQL-CS algorithm was applied to solve the problem of logistics distribution center location. The effectiveness of the proposed method was verified by comparing with cuckoo search (CS), improved cuckoo search algorithm (ICS), modified chaos-enhanced cuckoo search algorithm (CCS), and immune genetic algorithm (IGA) for both 6 and 10 distribution centers.

Towards time symmetric N-body integration

Computational efficiency demands discretised, hierarchically organised, and individually adaptive time-step sizes (known as the block-step scheme) for the time integration of N-body models. However, most existing N-body codes adapt individual step sizes in a way that violates time symmetry (and symplecticity), resulting in artificial secular dissipation (and often secular growth of energy errors). Using single-orbit integrations, I investigate various possibilities to reduce or eliminate irreversibility from the time stepping scheme. Significant improvements over the standard approach are possible at little extra effort. However, in order to reduce irreversible step-size changes to negligible amounts, such as suitable for long-term integrations of planetary systems, more computational effort is needed, while exact time reversibility appears elusive for discretised individual step sizes.

A New Normalized Subband Adaptive Filter Algorithm with Individual Variable Step Sizes

Proposed is a novel variable step size normalized subband adaptive filter algorithm, which assigns an individual step size for each subband by minimizing the mean square of the noise-free a posterior subband error. Furthermore, a noniterative shrinkage method is used to recover the noise-free priori subband error from the noisy subband error signal. Simulation results using the colored input signals have demonstrated that the proposed algorithm not only has better tracking capability than the existing subband adaptive filter algorithms, but also exhibits lower steady-state error.

Variable individual step‐size subband adaptive filtering algorithm

A subband adaptive filtering algorithm is proposed which improves its performance by adjusting step sizes. The proposed algorithm derives the individual step sizes for each subband instead of using a common step size for multiple subbands. The derivation of the step sizes is based on the mean-square deviation minimisation in order to achieve the fastest convergence at the instant. Furthermore, the individual step sizes contain the squared norm of the input vector, hence it leads to the regularisation effect that helps the algorithm work well in the case of badly excited input signals. The simulation results show that the proposed algorithm achieves a faster convergence rate and a smaller steady-state estimation error than the existing algorithms.

Variable matrix-type step-size affine projection algorithm with orthogonalized input vectors

In this paper, we propose a variable matrix-type step-size affine projection algorithm (APA) with orthogonalized input vectors. We generate orthogonalized input vectors using the Gram–Schmidt process to implement the weight update equation of the APA using the sum of normalized least mean squares (NLMS)-like updating equations. This method allows us to use individual step sizes corresponding to each NLMS-like equation, which is equivalent to adopting the step size in the form of a diagonal matrix in the APA. We adopt a variable step-size scheme, in which the individual step sizes are determined to minimize the mean square deviation of the APA in order to achieve the fastest convergence on every iteration. Furthermore, because of the weight vector updated successively only along each innovative one among the reused inputs and effect of the regularization absorbed into the derived step size, the algorithm works well even for badly excited input signals. Experimental results show that our proposed algorithm has almost optimal performance in terms of convergence rate and steady-state estimation error, and these results are remarkable especially for badly excited input signals.

Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

This article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints. Asynchronous discretizations allow individual time step sizes for each spatial region, improving the efficiency of explicit time stepping for finite element meshes with heterogeneous element sizes. The article first introduces asynchronous variational integration being expressed by drift and kick operators. Linear nodal restraint conditions are solved by a simple projection of the forces that is shown to be equivalent to RATTLE. Unilateral contact is solved by an asynchronous variant of decomposition contact response. Therein, velocities are modified avoiding penetrations. Although decomposition contact response is solving a large system of linear equations (being critical for the numerical efficiency of explicit time stepping schemes) and is needing special treatment regarding overconstraint and linear dependency of the contact constraints (for example from double-sided node-to-surface contact or self-contact), the asynchronous strategy handles these situations efficiently and robust. Only a single constraint involving a very small number of degrees of freedom is considered at once leading to a very efficient solution. The treatment of friction is exemplified for the Coulomb model. Special care needs the contact of nodes that are subject to restraints. Together with the aforementioned projection for restraints, a novel efficient solution scheme can be presented. The collision integrator does not influence the critical time step. Hence, the time step can be chosen independently from the underlying time-stepping scheme. The time step may be fixed or time-adaptive. New demands on global collision detection are discussed exemplified by position codes and node-to-segment integration. Numerical examples illustrate convergence and efficiency of the new contact algorithm. Copyright © 2013 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.

Multi-frequency periodic vibration suppressing in active magnetic bearing-rotor systems via response matching in frequency domain

A method for multi-frequency periodic vibration suppressing in active magnetic bearing (AMB)-rotor systems is proposed, which is based on an adaptive finite-duration impulse response (FIR) filter in time domain. Firstly, the theoretic feasibility of the method is proved. However, two problems would be unavoidable, if the conventional adaptive FIR filter is adopted in practical application. One is that the convergence rate of the different frequency components may be highly disparate in multi-frequency vibration control. The other is that the computational complexity is significantly increased because the long memory FIR filter is required to match the transient response time of the AMB-rotor system. To overcome the problems above, the Fast Block Least Mean Square (FBLMS) algorithm is adopted to efficiently implement the computation in frequency domain at a computational cost far less than that of the conventional FIR filter. By the FBLMS algorithm, regardless of the number of the considered frequency components in vibration disturbance, the computational complexity would be invariable. Moreover, filter’s weights in the FBLMS algorithm have the intuitional relation with signal’s frequency. As a result, the convergence rate of each frequency component can be adjusted by assigning the individual step size parameter for each weight.Experiments with the reciprocating simulating disturbance test and the rotating harmonic vibration test were carried out on an AMB-rigid rotor test rig with a vertical shaft. The experiment results indicate that the proposed method with the FBLMS algorithm can achieve the good effectiveness for suppressing the multi-frequency vibration. The convergence property of each frequency component can be adjusted conveniently. Each harmonic component of the vibration can be addressed, respectively, by reconfiguring the frequency components of the reference input signal.

A multivariate spectral projected gradient method for bound constrained optimization

In this paper, we consider a multivariate spectral projected gradient (MSPG) method for bound constrained optimization. Combined with a quasi-Newton property, the multivariate spectral projected gradient method allows an individual adaptive step size along each coordinate direction. On the basis of nonmonotone line search, global convergence is established. A numerical comparison with the traditional SPG method shows that the method is promising.

Evolutionary Shape Optimization With Self-adapting Mutation Distribution Based On The Cholesky Decomposition

The investigations in the paper have been carried out against the background of an integrated approach to the structural design problem. Due to the complex nature of the search space and the continous-discrete-binary system vector in the general case of structural optimization, conventional methods can not be used. Evolution strategies have shown their general capability to solve the sizing and the topology problem. On the other hand when employing evolution strategies to the subproblem of shape optimization they tend to get stuck due to an unsufficient step size adaptation. Therefore, a covariance matrix approach has been used for the self-adaptation of the mutation distribution during the evolutionary shape optimization process. The necessary decomposition of the covariance matrix has been carried out with the Cholesky method. The results of this approach are compared to the usual general and individual step size adaptation procedures of evolution strategies.

Evolution strategies for synthesis of mixed H 2 /H ∞ fixed-structure controllers

Abstract A new method for mixed H 2 / H ∞ controllers synthesis is presented in this paper. The problem is formulated as the minimization of a nominal H 2 performance index subject to a H ∞ robust stability constraint. The novel method, based on two derandomized Evolution Strategies (ESs) with individual step-sizes and correlated mutations using appropriate operators and a penalty function, is proposed for the solution of the constrained optimization problem. Simulation results are to illustrate the new approach, and to show the validity of the proposed synthesis method.

Self-avoiding Lévy flights at upper marginal dimensions.

We study the node-avoiding (NALF) and path-avoiding extensions of the L\'evy flights in terms of the critical exponents \ensuremath{\nu} and \ensuremath{\gamma} and the leading corrections to scaling, using Monte Carlo simulations with enrichment as the main technique. We focus on the upper marginal dimensions of NALF as predicted by the magnetic analogy where the renormalization-group results should be quantitatively exact for NALF if the method is valid at all. Similarly, we also focus on the boundary between the long-range and short-range behavior of NALF above four dimensions where the renormalization results should again be exact. Thus we investigate the self-avoiding L\'evy flights on hypercubic lattices from d=2 to 6 dimensions and obtain behavior consistent with logarithmic corrections to scaling in the moments of their end-to-end distance distributions. In addition, the effective L\'evy index ${\ensuremath{\mu}}_{\mathrm{eff}}$ is determined from the logarithmic averages of individual step sizes and compared for the two self-avoiding extensions.