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The Convergence of krylov subspace Methods for Large Unsymmetric Linear Systems

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Acta Mathematica Sinica,English Series 1998年第4期14卷 507-518页

作者： Jia Zhongxiao, Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Department of Applied Mathematics Dalian University of Technology

The convergence problem of many krylov subspace methods, e.g., FOM, GCR, GMRES and QMR, for solving large unsymmetric (non-Hermitian) linear systems is considered in a unified way when the coefficient matrix A is defective and its spectrum lies in the open right (left) half plane. Related theoretical error bounds are established and some intrinsic relationships between the convergence speed and the spectrum of A are exposed. It is shown that these methods are likely to converge slowly once one of the three cases occurs: A is defective, the distribution of its spectrum is not favorable, or the Jordan basis of A is ill conditioned. In the proof, some properties on the higher order derivatives of Chebyshev polynomials in an ellipse in the complex plane are derived, one of which corrects a result that has been used extensively in the literature.

关键词： Unsymmetric linear systems Convergence krylov subspace The Chebyshev polynomials Defective Derivatives

Adaptive detectors in the krylov subspace

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Science China(Information Sciences) 2014年第10期57卷 117-127页

作者： LIU WeiJian XIE WenChong LI RongFeng WANG ZeTao WANG YongLiang College of Electronic Science and Engineering National University of Defense Technology Key Research Lab Wuhan Radar Academy

The validity of the application of the krylov subspace techniques in adaptive filtering and detection is investigated. A new verification of the equivalence of two well-known methods in the krylov subspace, namely the multistage Wiener filters(MWF) and the auxiliary-vector filtering(AVF), is given in this paper. The MWF and AVF are incorporated into two well-known detectors, namely, the adaptive matched filter(AMF) and Kelly’s generalized likelihood ratio test(GLRT) including their diagonally loaded versions, which form new *** to the conventional AMF, GLRT, and their diagonally loaded versions as well as the reduced-rank AMF and GLRT, the probabilities of detection(PDs) of the new detectors are improved especially when the sample support is low. More importantly, the new detectors are robust of the rank selection of the clutter subspace compared to the reduced-rank AMF and GLRT. These new detectors all possess asymptotic constant false alarm rate(CFAR) property.

关键词： adaptive matched filter(AMF) auxiliary-vector filtering(AVF) diagonal loading generalized likelihood ratio test(GLRT) krylov subspace multistage Wiener filters(MWF)

DOA estimation of high-dimensional signals based on krylov subspace and weighted l_(1)-norm

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Journal of Systems Engineering and Electronics 2024年第3期35卷 532-540,F0002页

作者： YANG Zeqi LIU Yiheng ZHANG Hua MA Shuai CHANG Kai LIU Ning LYU Xiaode Aerospace Information Research Institute Chinese Academy of SciencesBeijing 100094China National Key Laboratory of Microwave Imaging Technology Beijing 100190China School of Electronic Electrical and Communication Engineering University of Chinese Academy of SciencesBeijing 100049China Northern Institute of Electronic Equipment Beijing 100191China

With the extensive application of large-scale array antennas,the increasing number of array elements leads to the increasing dimension of received signals,making it difficult to meet the real-time requirement of direction of arrival(DOA)estimation due to the computational complexity of *** subspace algorithms require estimation of the covariance matrix,which has high computational complexity and is prone to producing spurious *** order to reduce the computational complexity of DOA estimation algorithms and improve their estimation accuracy under large array elements,this paper proposes a DOA estimation method based on krylov subspace and weighted l_(1)-*** method uses the multistage Wiener filter(MSWF)iteration to solve the basis of the krylov subspace as an estimate of the signal subspace,further uses the measurement matrix to reduce the dimensionality of the signal subspace observation,constructs a weighted matrix,and combines the sparse reconstruction to establish a convex optimization function based on the residual sum of squares and weighted l_(1)-norm to solve the target *** results show that the proposed method has high resolution under large array conditions,effectively suppresses spurious peaks,reduces computational complexity,and has good robustness for low signal to noise ratio(SNR)environment.

关键词： direction of arrival(DOA) compressed sensing(CS) krylov subspace l_(1)-norm dimensionality reduction

Adaptive coherence estimator based on the krylov subspace technique for airborne radar

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Journal of Systems Engineering and Electronics 2015年第4期26卷 705-712页

作者： Weijian Liu Wenchong Xie Haibo Tong Honglin Wang Cui Zhou Yongliang Wang College of Electronic Science and Engineering National University of Defense Technology Air Force Early Warning Academy

A novel adaptive detector for airborne radar space-time adaptive detection （STAD） in partially homogeneous environments is proposed. The novel detector combines the numerically stable krylov subspace technique and diagonal loading technique, and it uses the framework of the adaptive coherence estimator （ACE）. It can effectively detect a target with low sample support. Compared with its natural competitors, the novel detector has higher proba- bility of detection （PD）, especially when the number of the training data is low. Moreover, it is shown to be practically constant false alarm rate （CFAR）.

关键词： airborne radar space-time adaptive detection (STAD),constant false alarm rate (CFAR) adaptive coherence estimator(ACE) krylov subspace numerical stability partially homoge-neous environments.

Reconstructing Images Based on the CG Method with krylov subspace for ECT

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Reconstructing Images Based on the CG Method with Krylov Sub...

GENERALIZED CONJUGATE A-ORTHOGONAL RESIDUAL SQUARED METHOD FOR COMPLEX NON-HERMITIAN LINEAR SYSTEMS

2011 2nd Asia-Pacific Conference on Wearable Computing Systems

作者： Ronghua ZHANG Tianjin Polytechnic University School of Electrical Engineering ＆ Automation

Electrical capacitance tomography（ECT） is considered to be one of the most promising process tomography techniques. The image reconstruction for ECT is an inverse problem to find the permittivity distribution of an object by measuring the electrical capacitances between sets of electrodes placed around its periphery. The conjugate gradient（CG） method is one of the most popular methods applied for image reconstruction, although its convergence rate is low. In this paper, an advanced version of the CG method, i.e. the projected CG method, is proposed to solve the inverse problem for ECT. The solution space is projected onto the krylov subspace and the ill-posed problem is solved by the CG method in a low-dimensional specific subspace. Simulated results using the projected CG method are presented and compared with the conventional CG method, indicating that the proposed method can reduce the computation time, and improve the resolution of reconstructed images.

关键词： electrical capacitance tomography(ECT) image reconstruction conjugate gradient method krylov subspace

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Journal of Computational Mathematics 2014年第3期32卷 248-265页

作者： Jianhua Zhang Hua Dai Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016 China Department of Mathematics Anhui Science and Technology University Fengyang 233100 China Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016 China

Recently numerous numerical experiments on realistic calculation have shown that the conjugate A-orthogonal residual squared （CORS） method is often competitive with other popular methods. However, the CORS method, like the CGS method, shows irreg- ular convergence, especially appears large intermediate residual norm, which may lead to worse approximate solutions and slower convergence rate. In this paper, we present a new product-type method for solving complex non-Hermitian linear systems based on the bicon- jugate A-orthogonal residual （BiCOR） method, where one of the polynomials is a BiCOR polynomial, and the other is a BiCOR polynomial with the same degree corresponding to different initial residual. Numerical examples are given to illustrate the effectiveness of the proposed method.

关键词： krylov subspace BiCOR method CORS method Complex non-Hermitianlinear systems.

Algorithm for transient growth of perturbations in channel Poiseuille flow

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Applied Mathematics and Mechanics(English Edition) 2017年第11期38卷 1635-1650页

作者： Jianlei ZHANG Gang DONG Yi LI National Key Laboratory of Transient Physics Nanjing University of Science and Technology School of Mathematics and Statistics University of Sheffield

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the Orr- Somrnerfeld and Squire （OSS） equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a re- orthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes （N-S） equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is pro- posed. The algorithm is then applied to a one-dimensional base flow （the plane Poiseuille flow） and a two-dimensional base flow （the plane Poiseuille flow with a low-speed streak） in a channel. For one-dimensional cases, the effects of the spanwise width of the chan- nel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is character- ized by high- and low-speed streaks and the corresponding streamwise vortical structures. The lift-up mechanism that induces the transient growth of perturbations is discussed. The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one- and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the cross- check method.

关键词： transient growth Poiseuille flow Arnoldi method krylov subspace adjoint equation

SOME PROPERTIES OF LSQR FOR LARGE SPARSE LINEAR LEAST SQUARES PROBLEMS

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Journal of Systems Science & Complexity 2010年第4期23卷 815-821页

作者： Zhongxiao JIA Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China. Department of Mathematical Sciences Tsinghua University

It is well-known that many krylov solvers for linear systems,eigenvalue problems,andsingular value decomposition problems have very simple and elegant formulas for residual *** not only allow us to further understand the methods theoretically but also can be usedas cheap stopping criteria without forming approximate solutions and residuals at each step beforeconvergence takes *** for large sparse linear least squares problems is based on the Lanczosbidiagonalization process and is a krylov ***,there has not yet been an analogouslyelegant formula for residual *** paper derives such kind of *** addition,the authorgets some other properties of LSQR and its mathematically equivalent CGLS.

关键词： CGLS krylov subspace lanczos bidiagonalization least squares lsqr normal equations.

DGMRES Method Augmented with Eigenvectors for Computing the Drazin-inverse Solution of Singular Linear Systems

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Acta Mathematicae Applicatae Sinica 2016年第2期32卷 549-558页

作者： Bin MENG College of Science Nanjing University of Aeronautics and Astronautics

The DGMRES method for solving Drazin-inverse solution of singular linear systems is generally used with restarting. But the restarting often slows down the convergence and DGMRES often stagnates. We show that adding some eigenvectors to the subspace can improve the convergence just like the method proposed by R. Morgan in [R. Morgan, A restarted GMRES method augmented with eigenvectors, SIAM J. Matrix Anal App1., 16： 1154-1171, 1995. We derive the implementation of this method and present some numerical examples to show the advantages of this method.

关键词： Drazin-inverse DGMRES krylov subspace iterative method eigenvector