Hou-Biao Li

Chebyshev-type methods and preconditioning techniques

Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners.

Novel Broadband Reflectarray Antenna with Compound-Cross-Loop Elements for Millimeter-wave Application

A novel broadband millimeter-wave reflectarray antenna composed of compound-cross-loop elements of variable lengths is proposed. Compared to the conventional single-layer reflectarray elements, the compound-cross-loop elements can realize much larger phase variation range from 0° to 465°, leading to broader bandwidth. Using this technique, a 15°-beam-steering reflectarray operating at 30 GHz is designed. The computed results demonstrate the agreement of the main beam steering with the design requirement, and a 1-dB gain bandwidth close to 25.17% is obtained. The validity of the obtained results is verified by comparing the ones generated by Ansoft High Frequency Structure Simulator (HFSS) with those produced by Ansoft Designer. The antenna is useful for millimeter-wave applications.

On some subclasses of P‐matrices

A matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row averages is called a B-matrix by J. M. Pena in References (SIAM J. Matrix Anal. Appl. 2001; 22:1027–1037) and (Numer. Math. 2003; 95:337–345). In this paper, it is generalized to more extended matrices—MB-matrices, which is proved to be a subclass of the class of P-matrices. Subsequently, we establish relationships between defined and some already known subclasses of P-matrices (see, References SIAM J. Matrix Anal. Appl. 2000; 21:67–78; Linear Algebra Appl. 2004; 393:353–364; Linear Algebra Appl. 1995; 225:117–123). As an application, some subclasses of P-matrices are used to localize the real eigenvalues of a real matrix. Copyright

An efficient Chebyshev-tau method for solving the space fractional diffusion equations

Fractional diffusion equations (FDEs) have recently been paid much attention. Finding accurate and efficient methods for solving FDEs has become an active research undertaking. In this paper, an efficient method based on the shifted Chebyshev-tau idea is presented to solve an initial-boundary value problem for the FDEs. The method is derived by expanding the required approximate solution as the elements of shifted Chebyshev polynomials. Using the operational matrix of the fractional derivative, the problem can be reduced to a set of linear algebraic equations. From a computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and only a small number of shifted Chebyshev polynomials is needed.

Scattering Analysis of Large-Scale Periodic Structures Using the Sub-Entire Domain Basis Function Method and Characteristic Function Method

In this paper, the sub-entire domain basis function method and characteristic function (CF) method are used to analyze scattering of large-scale periodic structures. The former can dramatically reduce the number of unknows. To reduce the time for impedance matrix generation, the CF method can be used when the distance between the source function and the testing function positions is big enough. Finally, some simulation examples are given to validate the method.

Strang-type preconditioners for solving fractional diffusion equations by boundary value methods

The finite difference scheme with the shifted Grunwald formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, the boundary value method (BVM) was developed as a popular algorithm for solving the large systems of ODEs. This method requires the solutions of one or more nonsymmetric and large-scale linear systems. In this paper, the GMRES method with the block circulant preconditioner is proposed to solve relevant linear systems. Some conclusions about the convergence analysis and spectrum of the preconditioned matrices are also drawn if the diffusion coefficients are constant. Finally, extensive numerical experiments are reported to show the performance of our method for solving the fractional diffusion equations.

Quasi-Minimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations

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.

BiCR-type methods for families of shifted linear systems

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.

On some new approximate factorization methods for block tridiagonal matrices suitable for vector and parallel processors

In this paper, to obtain an efficient parallel algorithm to solve sparse block-tridiagonal linear systems, stair matrices are used to construct some parallel polynomial approximate inverse preconditioners. These preconditioners are suitable when the desired goal is to maximize parallelism. Moreover, some theoretical results concerning these preconditioners are presented and how to construct preconditioners effectively for any nonsingular block tridiagonal H-matrices is also described. In addition, the validity of these preconditioners is illustrated with some numerical experiments arising from the second order elliptic partial differential equations and oil reservoir simulations.

On k-step CSCS-based polynomial preconditioners for Toeplitz linear systems with application to fractional diffusion equations

Abstract The implicit finite difference scheme with the shifted Gruwald 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.