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An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method on each subdomain is solved by a sparse direct solver. The solution of the interface linear system in the domain decomposition framework is accelerated by a Krylov subspace method. We study the spectral radius of the iteration matrix of the Schwarz method for the LORM problems, and thus propose an optimized parameter for the transmission condition, which is different from that for the classical electromagnetic problems. The numerical results show that the proposed method is effective. [ABSTRACT FROM AUTHOR]

In this paper, we present a new block preconditioner for solving the saddle point linear systems. The proposed method is developed from an augmented reformulation of the saddle point problem into a new linear system with an almost block triangular coefficient matrix. Theoretical results are derived on the eigenvalue distribution of the preconditioned matrix, and an efficient algorithmic implementation is developed and presented. Several numerical examples are reported to support the theoretical findings and to illustrate the favourable convergence properties of the proposed preconditioner, also compared to other popular solvers for saddle point problems. [ABSTRACT FROM AUTHOR]

This article presents an additive-state-decomposition-based repetitive control (RC) method for a class of nonlinear systems with nonintegral-chain form and multiple mismatched disturbances. The key idea of this method is to decompose the original tracking problem for an uncertain nonlinear system into two tractable subproblems, namely RC problem for a primary linear time-invariant (LTI) periodic system and robust stabilization problem for a nonlinear secondary system. By taking the equivalent total effect of the multiple disturbances as one matched total disturbance, an improved linear extended state observer (ILESO) is constructed to estimate and compensate for the adverse effect of the multiple disturbances. The method has three significant features: First, it builds a bridge between the existing RC methods for LTI systems and nonlinear systems and, thus, broadens the application scope of RC; second, besides the known information of the model being used to the design of ILESO, the periodic nonlinearity in the plant is extracted to be treated by RC, which enhances both the transient and steady-state performances; and third, the ILESO-based control scheme and the RC law are designed separately, which is flexible for system design. The stability criterion and design procedure are presented in detail. Finally, numerical and experimental results illustrate the validity and superiorities of the method. [ABSTRACT FROM AUTHOR]

This paper presents an output‐feedback repetitive control method for a class of systems with nonlinearities, unknown state‐dependent uncertainties, and external disturbances. An additive‐state‐decomposition approach is used to decompose the system under study into a primary linear periodic system in charge of repetitive‐control tracking task and a secondary linearized system in charge of robustly stabilization task. Such a treatment not only significantly simplifies system analysis and controller design, but also provides an effective method to cope with the intrinsic constraints between tracking and disturbance rejection. Another attractive point lies in two aspects. On one hand, a periodic term from the nonlinearity in the plant is extracted to be treated in the primary system and the known plant information is used to design controller parameters, which greatly improves the disturbance rejection ability of the system. On the other hand, repetitive control gain, feedback control gain, and the observer gain are optimized through an optimization algorithm by introducing an index to evaluate the overall system performance. System stability criteria and design procedure are developed. Finally, a case study on the tracking control of a chuck‐piece system demonstrates the validity of the proposed method. Comparisons with some existing methods show that the proposed method achieves better performances in both signal tracking and disturbance rejection. [ABSTRACT FROM AUTHOR]

This study provides an overview and in‐depth analysis of recent advances in stability of linear systems with time‐varying delays. First, recent developments of a delay convex analysis approach, a reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, in‐depth analysis of the Bessel–Legendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a time‐varying delay, where information on the bounds of the time‐varying delay and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the time‐varying delay by employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some existing results. Finally, several challenging issues are pointed out to direct the near future research. [ABSTRACT FROM AUTHOR]

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method (Heyouni, 1996) based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough elapsed CPU time to converge than the earlier established restarted shifted FOM, the weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recently popular application of handling time fractional differential equations. [ABSTRACT FROM AUTHOR]

Landweber method is one of the classical iterative methods for solving linear discrete ill-posed problems. However, Landweber method generally converges very slowly. In this paper, we present the vector extrapolation based Landweber method, which exhibits fast and stable convergence behavior. Moreover, a restarted version of the vector extrapolation based Landweber method is proposed for practical considerations. Numerical results are given to illustrate the benefits of the vector extrapolation based Landweber method. [ABSTRACT FROM AUTHOR]

Recent research has shown that one can obtain a less conservative stability criterion for a continuous-time linear system with a time-varying delay by introducing a Lyapunov-Krasovskii functional with a polynomial matrix on the time-varying delay. This paper aims at analysing the stability of discrete-time linear systems with time-varying delays by introducing a delay-square-dependent Lyapunov functional. A novel convex method is presented to formulate a less conservative stability criterion, which is demonstrated through numerical simulation. Moreover, it is also shown that, if the polynomial inequality method is employed, the resultant stability criterion is inapplicable due to its extremely high numerical complexity. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new extension of the conjugate residual (CR) method for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A -orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods. [ABSTRACT FROM AUTHOR]

In this study, we derive a new iterative algorithm (including its preconditioned version) which is a hybridized variant of the biconjugate A -orthogonal residual stabilized (BiCORSTAB) method and the generalized product-type solvers based on BiCOR (GPBiCOR) method. The proposed method, which is named GPBiCOR( m , ℓ ) similarly to the GPBiCG( m , ℓ ) method proposed by Fujino (2002), can be regarded as an extension of the BiCORSTAB2 method introduced by Zhao and Huang (2013). Inspired by Fujino’s idea for improving the BiCGSTAB2 method, in the established GPBiCOR( m , ℓ ) method the parameters computed by the BiCORSTAB method are chosen at successive m iteration steps, and afterwards the parameters of the GPBiCOR method are utilized in the subsequent ℓ iteration steps. Therefore, the proposed method can inherit the low computational cost of BiCORSTAB and the attractive convergence of GPBiCOR. Extensive numerical convergence results on selected real and complex matrices are shown to assess the performances of the proposed GPBiCOR( m , ℓ ) method, also against other popular non-Hermitian Krylov subspace methods. [ABSTRACT FROM AUTHOR]

The method of moments for the simplified scattering and radiation on thin wire models can lead to the large-scale linear systems with non-Hermitian Toeplitz-like coefficient matrices. Since such reduced linear systems are typically dense and ill conditioned, iterative solvers are rarely considered to deal with the discretized linear systems in these electromagnetic model simulations. However, utilizing the Toeplitz-like structure of coefficient matrices, we can use the fast Toeplitz matrix-vector product (MVP) to refresh iterative solvers for the discretized linear systems. Compared with the traditional solvers, the fast numerical method can reduce the computational cost fromtoand the storage fromtowithout using any lossy compression. Due to the Toeplitz-like structure of coefficient matrices, circulant preconditioners are considered to further accelerate the convergence of iterative solvers. Since the circulant matrices can be diagonalized via fast Fourier transforms (FFTs), the fast Toeplitz MVP is utilized to retain the total computational complexity of the proposed methods at, whereis the number of spatial grid nodes. Theoretical analyses of the spectra of the preconditioned linear systems are also preliminarily investigated. Finally, extensive numerical experiments involving both scattering and radiation issues are employed to demonstrate the effectiveness of the proposed methods in aspects of the number of iterations and CPU time elapsed. [ABSTRACT FROM PUBLISHER]

This paper is concerned with stability of linear discrete time-delay systems. Note that a tighter estimation on a finite-sum term appearing in the forward difference of some Lyapunov functional leads to a less conservative delay-dependent stability criterion. By using Abel lemma, a novel finite-sum inequality is established, which can provide a tighter estimation than the ones in the literature for the finite-sum term. Applying this Abel lemma-based finite-sum inequality, a stability criterion for linear discrete time-delay systems is derived. It is shown through numerical examples that the stability criterion can provide a larger admissible maximum upper bound than stability criteria using a Jensen-type inequality approach and a free-weighting matrix approach. [ABSTRACT FROM AUTHOR]

The implicit finite difference scheme with the shifted Grüwald formula for discretizing the fractional diffusion equations (FDEs) often results in the ill-conditioned non-Hermitian Toeplitz systems. In the present paper, we consider to solve such Toeplitz systems by exploiting the preconditioned GMRES method. A k -step polynomial preconditioner is designed based on the circulant and skew-circulant splitting (CSCS) iteration method proposed by Ng (2003). Theoretical and experimental results involving numerical solutions of FDEs demonstrate that the proposed k -step preconditioner is efficient to accelerate the GMRES solver for non-Hermitian Toeplitz systems. [ABSTRACT FROM AUTHOR]

The conjugate orthogonal conjugate gradient (COCG) method has been considered an attractive part of the Lanczos-type Krylov subspace method for solving complex symmetric linear systems. However, it is often faced with apparently irregular convergence behaviors in practical electromagnetic simulations. To avoid such a problem, the symmetric quasi-minimal residual (QMR) method has been developed. On the other hand, the conjugate A-orthogonal conjugate residual (COCR) method, which can be regarded as an extension of the conjugate residual method, also had been established. It shows that the COCR often gives smoother convergence behavior than the COCG method. The purpose of this paper is to apply the QMR approaches to the COCG and COCR to derive two new methods (including their preconditioned versions), and to report the benefits of the modified methods by some practical examples arising in electromagnetic simulations. [ABSTRACT FROM AUTHOR]

The shifted linear systems with non-Hermitian matrices often arise from the numerical solutions for time-dependent PDEs, computing the large-scale eigenvalue problems, control theory and so on. In the present paper, we develop two shifted variants of BiCR-type methods for solving such linear systems. These variants of BiCR-type methods take advantage of the shifted structure, so that the number of matrix–vector multiplications and the number of inner products are the same as a single linear system. Finally, extensive numerical examples are reported to illustrate the performance and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This study is concerned with the stability of a linear system with an interval time-varying delay. First, a new augmented Lyapunov-Krasovskii functional (LKF) is constructed, which includes three integral terms in the form of ∫t-ht(h - t + s)jẋTRj ẋ (s) ds (j = 1, 2, 3). Second, three novel integral inequalities are established to estimate the upper bounds of the integrals ∫ t t-h(h - t + s)jẋTRj ẋ(s) ds (j = 0, 1, 2) appearing in the derivative of the LKF. Third, a matrix-based quadratic convex approach is introduced to prove not only the negative definiteness of the derivative of the LKF along with the trajectory of the system, but also the positive definiteness of the LKF. Finally, a novel delay-derivative-dependent stability criterion is formulated. The effectiveness of the stability criterion is shown through two numerical examples. [ABSTRACT FROM AUTHOR]

SUMMARY This paper studies the stability of linear systems with interval time-varying delays. By constructing a new Lyapunov-Krasovskii functional, two delay-derivative-dependent stability criteria are formulated by incorporating with two different bounding techniques to estimate some integral terms appearing in the derivative of the Lyapunov-Krasovskii functional. The first stability criterion is derived by using a generalized integral inequality, and the second stability criterion is obtained by employing a reciprocally convex approach. When applying these two stability criteria to check the stability of a linear system with an interval time-varying delay, it is shown through some numerical examples that the first stability criterion can provide a larger upper bound of the time-varying delay than the second stability criterion. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This article concerns the stability and stabilisation of a linear system with both state and input delays. First, the combination of an augmented Lyapunov functional and the free-weighting-matrix technique yields a new delay-independent stability criterion that includes the widely used one as a special case. This criterion is then extended to a new delay-dependent stability criterion by employing an integral inequality. Based on that, a stabilisation approach to design a state feedback controller is presented that requires no parameter tuning, as is needed with some existing methods. Finally, numerical examples illustrate that the method is effective and is an improvement over existing ones. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the problem of delay-dependent robust filtering for uncertain linear systems with time-varying delay. Two cases of time-varying delays are fully considered; one is the time-varying delay being continuous uniformly bounded while the other is the time-varying delay being differentiable uniformly bounded with delay-derivative bounded by a constant. A stable linear filter is designed to ensure that the filtering error system is asymptotically stable with a prescribed level of noise attenuation. Based on a new integral inequality, delay-dependent sufficient conditions for the existence of such a filter are established in terms of linear matrix inequalities (LMIs). Through deriving these conditions, neither model transformation nor bounding technique for cross terms is employed. Moreover, the relationship between the sufficient conditions for the two cases of time-varying delay is disclosed. Finally, two examples are also given to illustrate the effectiveness of the proposed methodology. [Copyright &y& Elsevier]

Abstract: The integral-inequality method is a new way of tackling the delay-dependent stabilization problem for a linear system with time-varying state and input delays: . In this paper, a new integral inequality for quadratic terms is first established. Then, it is used to obtain a new state- and input-delay-dependent criterion that ensures the stability of the closed-loop system with a memoryless state feedback controller. Finally, some numerical examples are presented to demonstrate that control systems designed based on the criterion are effective, even though neither nor is stabilizable. [Copyright &y& Elsevier]

This paper is concerned with sufficient conditions for a class of matrix-valued polynomial inequalities on closed intervals and their application to H ∞ filtering for linear systems with time-varying delays. First, a class of higher degree matrix-valued polynomial inequalities are transformed into the first degree matrix-valued polynomial inequalities by introducing some slack matrices. As a result, a convex property is applied to derive sufficient conditions for a class of higher degree matrix-valued polynomial inequalities on closed intervals. Second, by choosing a Lyapunov–Krasovskii functional with a quadratic matrix-valued polynomial on a time-varying delay, and estimating its time-derivative as a third-degree matrix-valued polynomial on the time-varying delay, the proposed sufficient conditions are utilized to formulate a novel bounded real lemma on the existence of H ∞ filters for time-delay systems. Third, some algorithms to filter design are provided and discussed in detail. It is pointed out that, if the filter gain is derived through a generalized inverse matrix method, it may be a fake solution, which is demonstrated through a liquid monopropellant rocket motor with a pressure feeding system. [ABSTRACT FROM AUTHOR]

Multi-shifted linear systems with non-Hermitian coefficient matrices arise in numerical solutions of time-dependent partial/fractional differential equations (PDEs/FDEs), in control theory, PageRank problems, and other research fields. We derive efficient variants of the restarted Changing Minimal Residual method based on the cost-effective Hessenberg procedure (CMRH) for this problem class. Then, we introduce a flexible variant of the algorithm that allows to use variable preconditioning at each iteration to further accelerate the convergence of shifted CMRH. We analyse the performance of the new class of methods in the numerical solution of PDEs and FDEs, also against other multi-shifted Krylov subspace methods. [ABSTRACT FROM AUTHOR]

In this paper, fast numerical methods are established to solve a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by a weighted and shifted Grünwald formula in time and a fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing a preconditioned Krylov subspace method is developed to solve the Toeplitz linear system derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the resulting Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around 1, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other testing methods. [ABSTRACT FROM AUTHOR]

A block lower triangular Toeplitz system arising from the time–space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and the flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of O (N) with N being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to accelerate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the effectiveness of our two proposed preconditioners. • The block lower triangular Toeplitz system arising from time–space fractional diffusion equation is discussed. • A limited-memory block bi-diagonal Toeplitz (B2T) preconditioner is developed for the system. • A new skew-circulant preconditioner is designed to compute the inverse of the B2T preconditioner multiplying a vector. • Numerical examples are given to demonstrate the efficiency of our preconditioning strategy. [ABSTRACT FROM AUTHOR]

Abstract PageRank problem is the cornerstone of Google search engine and is usually stated as solving a huge linear system. Moreover, when the damping factor approaches 1, the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve. In this paper, we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets. In particular, the off-diagonal part of the block coefficient matrix can be compressed by a simple low-rank factorization, which can be beneficial for solving the PageRank problem. Hence, a matrix partition method is proposed to discover the special sets of rows for supporting the low-rank factorization. Then a preconditioner based on the low-rank factorization is proposed for solving difficult PageRank problems. Numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]