A generalized data windowing scheme for adaptive conjugate gradient algorithms

An adaptive array code acquisition for direct-sequence/ code-division multiple access (DS/CDMA) systems was recently proposed to enhance the performance of the conventional correlator-based method. The scheme consists of an adaptive spatial and an adaptive temporal filter, and can simultaneously perform beamforming and code-delay estimation. Unfortunately, the scheme uses a least-mean-square (LMS) adaptive algorithm, and its convergence is slow. Although the recursive-least-squares (RLS) algorithm can be applied, the computational complexity will greatly increase. In this paper, we solve the dilemma with a low-complexity conjugate gradient (LCG) algorithm, which can be considered as a special case of a modified conjugate gradient (MCG) algorithm. Unlike the original conjugate gradient (CG) algorithm developed for adaptive applications, the proposed method, exploiting the special structure inherent in the input correlation matrix, requires a low computational-complexity. It can be shown that the computational complexity of the proposed method is on the same order of the LMS algorithm. However, the convergence rate is improved significantly. Simulation results show that the performance of adaptive array code acquisition with the proposed CG algorithm is comparable to that with the original CG algorithm.

Self-scaled conjugate gradient training algorithms

The advantage of the electromagnetic tomography is introduced briefly. Based on conjugate gradient algorithm, modified conjugate gradient algorithm for electromagnetic tomography (EMT) is proposed, which improves quality of reconstructed image and convergence speed efficiently. In the light of the lab electromagnetic tomography system, modified conjugate gradient for reconstructing images is verified. By evaluation of image error and the relevance, regularization, Landweber, conjugate gradient and modified conjugate gradient algorithms are compared. It can draw the conclusion that for different flow models, image error and the correlation using modified conjugate gradient algorithm is superior to the others in lab EMT system.

In this work, we propose an adaptive linear interference suppression scheme based on single-carrier frequency domain equalization (SC-FDE) for multiuser direct-sequence ultra-wideband (DS-UWB) systems. In a block-by-block based transmission, by inserting the cyclic prefix (CP) between each data block, the inter-block interference (IBI) can be ignored and the Toeplitz channel matrix can be expressed as an equivalent circulant channel matrix. With this feature, we propose a simple signal model, in which only a vector formed adaptive filter needs to be trained in the frequency domain. Based on the new signal model, an adaptive block conjugate gradient (BCG) algorithm is developed. Complexity analysis shows that the proposed BCG algorithm is significantly simpler than the recursive least squares (RLS) algorithm. The bit-error rate (BER) performance of the proposed scheme is compared with the least-mean-square (LMS) and RLS algorithm in a number of scenarios and the simulation results show fast convergence speed and excellent performance of the proposed BCG algorithm.

The Multi-Stage Nested Wiener Filter (MSNWF) and the Conjugate Gradient (CG) method yield the solution of the Wiener-Hopf equation in the Krylov subspace of the covariance matrix of the observation and the crosscorrelation vector between the observation and the desired signal. Using the Lanczos algorithm instead of the Arnoldi algorithm for the MSNWF simplifies the computation of the Krylov subspace basis. In this paper, we show the relationship between the CG method and the Lanczos based MSNWF and finally derive that the MSNWF may be mathematically transformed into the CG algorithm. Consequently, we present a new implementation of the MSNWF where the weight vector and the Mean Square Error (MSE) is directly updated as each new stage is added. The new algorithm is applied to an Enhanced Data rates for GSM Evolution (EDGE) system where it linearily equalizes the received signal. Simulation results demonstrate the ability of the MSNWF to reduce the receiver complexity while maintaining the same level of system performance.

In this paper, several reported blind adaptive beam- forming algorithms, including the stochastic gradient (SG) methods and the recursive least squares (RLS) methods are reviewed. Then, the conjugate gradient (CG) algorithm is introduced as a tradeoff between SG and RLS algorithms. On the basis of this idea, a novel constrained constant modulus (CCM) design criterion is described and employed for deriving a modified conjugate gradient (CG) optimization algorithm, which does not require the information of the desired user beforehand. Simulations are performed to compare the performance of the proposed method with both SG and RLS methods. The results indicate that the proposed approach has a significant faster convergence rate and superior resolution compared with the SG algorithms and lower computation complexity compared with the RLS algorithms.

This study introduces an improved three-term conjugate gradient (CG) algorithm which fulfills the descent conditions and exhibits good global convergence properties. The development of this new algorithm is based on the findings of recently introduced generalized RMIL CG technique. The algorithm has been modified so that the search direction would always meet the sufficient descent condition no matter the line-search techniques employed. Under some given assumptions, the algorithm’s global convergence result for the general nonconvex functions is established. The numerical efficacy of the suggested three-term CG algorithm is assessed through comparisons with four other CG algorithms. The assessment is performed using an array of unconstrained optimization benchmark functions. The acquired findings indicate that all the considered algorithms exhibit similar performance, particularly the variants of the RMIL method. One algorithm is more robust and slightly faster than the rest. However, it is essential to highlight that the newly developed CG formula outperforms all the other algorithms, including a recently presented three-term PRP algorithm. The study further expanded the proposed method to address an image restoration problem. The computational experiments yielded promising results, proving that the proposed CG algorithm surpasses others. It produces higher-quality output images, requires less CPU time (sec), and achieves higher PSNR values.

We consider the bilinear optimal control of an advection-reaction-diffusion system, where the control arises as the velocity field in the advection term. Such a problem is generally challenging from both theoretical analysis and algorithmic design perspectives, mainly because the state variable depends nonlinearly on the control variable and, an additional divergence-free constraint on the control is coupled together with the state equation. Mathematically, the proof of the existence of optimal solutions is delicate, and, up to now, only some results have been known for a few special cases where additional restrictions are imposed on the space dimension and the regularity of the control. We prove the existence of optimal controls and derive the first-order optimality conditions in general settings without any extra assumptions. Computationally, the well-known conjugate gradient (CG) method can be applied conceptually. However, due to the additional divergence-free constraint on the control variable and the nonlinear relation between the state and control variables, it is challenging to compute the gradient and the optimal stepsize at each CG iteration, and thus nontrivial to implement the CG method. To address these issues, we advocate a fast inner preconditioned CG method to ensure the divergence-free constraint and an efficient inexactness strategy to determine an appropriate stepsize. An easily implementable nested CG method is thus proposed for solving such a complicated problem. For the numerical discretization, we combine finite difference methods for the time discretization and finite element methods for the space discretization. Efficiency of the proposed nested CG method is promisingly validated by the results of some preliminary numerical experiments.

In some types of magnetic resonance (MR) imaging, particularly functional brain scans, the conventional Fourier model for the measurements is inaccurate. Magnetic field inhomogeneities, which are caused by imperfect main fields and by magnetic susceptibility variations, induce distortions in images that are reconstructed by conventional Fourier methods. These artifacts hamper the use of functional MR imaging (fMRI) in brain regions near air/tissue interfaces. Recently, iterative methods that combine the conjugate gradient (CG) algorithm with nonuniform FFT (NUFFT) operations have been shown to provide considerably improved image quality relative to the conjugate-phase method. However, for non-Cartesian k-space trajectories, each CG-NUFFT iteration requires numerous k-space interpolations; these are operations that are computationally expensive and poorly suited to fast hardware implementations. This paper proposes a faster iterative approach to field-corrected MR image reconstruction based on the CG algorithm and certain Toeplitz matrices. This CG-Toeplitz approach requires k-space interpolations only for the initial iteration; thereafter, only fast Fourier transforms (FFTs) are required. Simulation results show that the proposed CG-Toeplitz approach produces equivalent image quality as the CG-NUFFT method with significantly reduced computation time.

Kernel‐risk‐sensitive loss (KRSL) achieves an efficient performance surface, which has been applied in the kernel adaptive filters (KAFs) successfully. However, the KRSL based KAFs use the stochastic gradient descent (SGD) method in the optimization, which usually suffer from inadequate accuracy with the slow convergence speed. In this letter, the conjugate gradient method is adopted in the optimization of KRSL function, and the problem of non‐convexity in KRSL is addressed by twice half‐quadratic (HQ) methods. For sparsification, a novel Student's‐t distribution based random Fourier feature (St‐RFF) method for performance improvement of the conventional RFF method. Thus, a novel Student's‐t distribution based random Fourier features kernel‐risk‐sensitive conjugate gradient (St‐RFFKRSCG) algorithm is proposed. Simulations on Mackey‐Glass time series prediction under non‐Gaussian noises confirm the superiorities in terms of accuracy performance, robustness, and computational cost.

At present, the conjugate gradient (CG) method of Hager and Zhang (Hager and Zhang, SIAM Journal on Optimization, 16(2005)) is regarded as one of the most effective CG methods for optimization problems. In order to further study the CG method, we develop the Hager and Zhang's CG method and present two modified CG formulas, where the given formulas possess the value information of not only the gradient but also the function. Moreover, the sufficient descent condition will be holden without any line search. The global convergence is established for nonconvex function under suitable conditions. Numerical results show that the proposed methods are competitive to the normal conjugate gradient method.

The advances in kernel-based learning necessitate the study on solving a large-scale non-sparse positive definite linear system. To provide a deterministic approach, recent researches focus on designing fast matrix-vector multiplication techniques coupled with a conjugate gradient method. Instead of using the conjugate gradient method, our paper proposes to use a domain decomposition approach in solving such a linear system. Its convergence property and speed can be understood within von Neumann's alternating projection framework. We will report signi ficant and consistent improvements in convergence speed over the conjugate gradient method when the approach is applied to recent machine learning problems.

The performance of linear filters degrade drastically when applied to mitigate intersymbol interference caused by channels with frequent nulls in their spectral characteristics like, e.g., time-dispersive radio channels. In such cases, the well-known decision feedback equalizer (DFE) is one possible nonlinear approach to improve the quality of the receiver. However, adapting the DFE filter coefficients to equalize time-varying channels is computationally intense, especially if the dimension of the observation vector is very high. In this paper, we apply the conjugate gradient (CG) algorithm to a conventional minimum mean square error (MMSE) DFE in order to reduce its computational complexity. Moreover, we compare its performance to the one of MMSE DFE versions which are based on the computationally efficient least-mean-square (LMS) or recursive least-squares (RLS) algorithm, respectively. The analysis additionally includes a detailed investigation of computational complexity with respect to the required number of floating point operations (FLOPs). Simulation results when applied to a digital communications system show the ability of the CG based MMSE DFE to outperform either the LMS and the RLS based MMSE DFE although its computational complexity is even smaller in most of the cases.

Iterative solution algorithms are described for complex systems with a large number of degrees of freedom. Areas of application include partial differential equations, integral equations, and hybrid methods in electromagnetics. Classical (or stationary) methods and conjugate gradient (nonstationary) methods are presented. Classical methods are generalized in terms of matrix splittings. The conjugate gradient method is derived in detail from the point of view of Galerkin testing and Krylov spaces. Preconditioning is discussed, and some common preconditioners from classical methods are presented.

Presently, a preferred minimization for strong‐constraint four‐dimensional variational (4D‐Var) assimilation uses a Lanczos‐based conjugate gradient (CG) algorithm. This requires the availability of a square‐root of the background‐error covariance matrix (B). In the context of weak‐constraint 4D‐Var, this requirement might be too restrictive for the formulations of the model error term. It might therefore be desirable to avoid a square‐root decomposition of the augmented background term. An appealing minimization scheme is the double CG minimization employed, for example, in the grid‐point statistical interpolation (GSI) analysis. Realizing the double CG algorithm is a special case of the more general bi‐conjugate gradient (BiCG) method for solving non‐symmetric problems, the present work introduces a Lanczos‐based preconditioning strategy when B, instead of its square‐root, is used initially. Implementation of the scheme is done in the context of the GSI analysis system, and preliminary experiments are presented using its 3D‐Var version. Comparison of the Lanczos‐based CG and the BiCG shows that the algorithms converge at the same rate and to the same solution. Despite the additional computational cost, the importance of the re‐orthogonalization step is also shown to be fundamental to any of these CG algorithms. Furthermore, when using the Hessian eigenvectors for preconditioning, the BiCG behaviour is shown to be comparable to that of the Lanczos‐CG algorithm. Both schemes construct the same approximation of the Hessian with the same number of eigenvectors, and benefit in the same way from the reduction of the condition number. The efficiency, computational cost, and stability of the three algorithms are discussed. Copyright © 2012 Royal Meteorological Society