Yao-Lin Jiang

A generalization of the inexact parameterized Uzawa methods for saddle point problems

Abstract For large sparse saddle point problems, Bai and Wang recently studied a class of parameterized inexact Uzawa methods (see Z.-Z. Bai, Z.-Q. Wang, On paramaterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900–2932). In this paper, we generalize these methods and propose a class of generalized inexact parameterized iterative schemes for solving the saddle point problems. We derive conditions for guaranteeing the convergence of these iterative methods. With different choices of the parameter matrices, the generalized iterative methods lead to a series of existing and new iterative methods including the classical Uzawa method, the inexact Uzawa method, the GSOR method and the GIAOR method.

Time Domain Model Order Reduction of General Orthogonal Polynomials for Linear Input-Output Systems

For a class of large linear input-output systems, we present a new model order reduction algorithm based on general orthogonal polynomials in the time domain. The main idea of the algorithm is first to expand the unknown state variables in the space spanned by orthogonal polynomials, then the coefficient terms of polynomial expansion are calculated by a recurrence formula. The basic procedure is to use the coefficient terms to generate a projection matrix. Many classic methods with orthogonal polynomials are special cases of the general approach. The proposed approach has a good computational efficiency and preserves the stability and passivity under certain condition. Numerical experiments are reported to verify the theoretical analysis.

On Time-Domain Simulation of Lossless Transmission Lines with Nonlinear Terminations

A time-domain approach is presented to solve nonlinear circuits with lossless transmission lines. Mathematically, the circuits are described by a special kind of nonlinear differential-algebraic equations (DAEs) with multiple constant delays. In order to directly compute these delay systems in time-domain, decoupling by waveform relaxation (WR) is applied to the systems. For the relaxation-based method we provide a new convergence proof. Numerical experiments are given to illustrate the novel approach.

Waveform relaxation methods for fractional differential equations with the Caputo derivatives

In this paper, we report new waveform relaxation methods for fractional differential equations with the Caputo derivatives. The convergence properties of the waveform relaxation methods are studied under linear and nonlinear conditions for the right-hand side of equations. Some convergent splittings are given. This is the first time for studying the waveform relaxation methods for fractional differential equations in references.

Computing periodic solutions of linear differential-algebraic equations by waveform relaxation

We propose an algorithm, which is based on the waveform relaxation (WR) approach, to compute the periodic solutions of a linear system described by differential-algebraic equations. For this kind of two-point boundary problems, we derive an analytic expression of the spectral set for the periodic WR operator. We show that the periodic WR algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1. Numerical examples, where discrete waveforms are computed with a backward-difference formula, further illustrate the correctness of the theoretical work in this paper.

Improving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation

We study the convergence performance of relaxation-based algorithms for circuit simulation in the time domain. The circuits are modeled by linear integral-differential-algebraic equations. We show that in theory, convergence depends only on the spectral properties of certain matrices when splitting is applied to the circuit matrices to set up the waveform relaxation solution of a circuit. A new decoupling technique is derived, which speeds up the convergence of relaxation-based algorithms. In function spaces a Krylovs subspace method, namely the waveform generalized minimal residual algorithm, is also presented in the paper. Numerical examples are given to illustrate how judicious splitting and how Krylovs method can help improve convergence in some situations.

A Note on the Spectra and Pseudospectra of Waveform Relaxation Operators for Linear Differential-Algebraic Equations

In this note, we derived the expressions of the spectra and pseudospectra of waveform relaxation operators for general linear differential-algebraic equations of index one. The expressions are generalizations of those of Lumsdaine and Wu (SIAM J. Sci. Comput., 18 (1997), pp. 286--304) and the derivation here is nontrivial. Numerical experiments are given to verify the theoretical results.

Application of General Orthogonal Polynomials to Fast Simulation of Nonlinear Descriptor Systems Through Piecewise-Linear Approximation

In this letter, we report an approach combining piecewise-linear (PWL) approximation with general orthogonal polynomials to efficiently simulate large nonlinear descriptor systems in time-domain. The main idea of this approach is first to approximate a nonlinear function by a piecewise-linear representation. Then, using the recursive formulae of general orthogonal polynomials, orthonormal bases can be produced for fast simulation of the PWL model. The effectiveness of our approach is demonstrated on two nonlinear circuit models.

Convergence analysis of waveform relaxation for nonlinear differential-algebraic equations of index one

We give a new and simple convergence theorem on the waveform relaxation (WR) solution for a system of nonlinear differential-algebraic equations of index one. We show that if the norms of certain matrices derived from the Jacobians of the system functions are less than one, then the WR solution converges. The new sufficient condition includes previously reported conditions as special cases. Examples are given to confirm the theoretical analysis.

Model reduction of bilinear systems based on Laguerre series expansion

Abstract We present a model reduction method for bilinear systems based on the Laguerre series expansion of the kernels resulting from the Volterra representation theory. By employing a two-sided projection, the reduced order system preserves a desired number of Laguerre coefficients, thereby approximating the original system faithfully. Furthermore, the relationship between the proposed Laguerre-based methods and the moment matching methods is studied, which reveals that these two approaches are equivalent under some specific conditions.