Liang Bao

Order reduction of bilinear MIMO dynamical systems using new block Krylov subspaces

In this paper we study numerical methods for the model-order reduction of large-scale bilinear multi-input multi-output systems. A new projection method is proposed. The projection subspace is the union of some new block Krylov subspaces. We show that the reduced-order bilinear system constructed by the new method can match a desired number of moments of multivariable transfer functions corresponding to the kernels of Volterra series representation of the original system. Some numerical examples are presented to illustrate the effectiveness of the proposed method.

Matrix Sign Function Methods for Solving Projected Generalized Continuous-Time Sylvester Equations

In this technical note, we investigate the numerical solution of the projected generalized Sylvester equations via a matrix sign function method. Such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. Unlike the classical matrix sign function iteration, we propose a modification of the matrix sign function method that converges quadratically for pencils of arbitrary index. Numerical experiments report the effectiveness of the modified method.

Model-order reduction of large-scale kth-order linear dynamical systems via a kth-order Arnoldi method

In this paper, we first introduce a kth-order Krylov subspace 𝒢 n (A j ; u) based on a square matrix sequence {A j } and a vector u. Then we present a kth-order Arnoldi procedure for generating an orthonormal basis of 𝒢 n (A j ; u). By applying the projection technique, we derive a structure-preserving kth-order Arnoldi method for reduced-order modelling of the large-scale kth-order linear dynamical system. Applications to polynomial eigenvalue problems are also included. Numerical experiments report the effectiveness of this method.

Convergence analysis of a variant of the Newton method for solving nonlinear equations

The paper presents a convergence analysis of a modified Newton method for solving nonlinear systems of equations. The convergence results show that this method converges cubically in the nonsingular case, and linearly with the rate 3/8 under some sufficient conditions when the Jacobian is singular at the root. The convergence theory is used to analyze the convergence behavior when the modified Newton method is applied to a nonsymmetric algebraic Riccati equation arising in transport theory. Numerical experiment confirms the theoretical results.

Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

In this paper, we introduce a generalized Krylov subspace

Original Article: Simpler GMRES with deflated restarting

{\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}

Perturbation analysis of the generalized Sylvester equation and the generalized Lyapunov equation

based on a square matrix sequence {Aj} and a vector sequence {uj}. Next we present a generalized Arnoldi procedure for generating an orthonormal basis of

Model-order reduction of kth order MIMO dynamical systems using block kth order Krylov subspaces

{\mathcal{G}_{m}(\mathbf{A};\mathbf{u})}

On the convergence rate of an iterative method for solving nonsymmetric algebraic Riccati equations

. By applying the projection and the refined technique, we derive a restarted generalized Arnoldi method and a restarted refined generalized Arnoldi method for solving a large-scale polynomial eigenvalue problem (PEP). These two methods are applied to solve the PEP directly. Hence they preserve essential structures and properties of the PEP. Furthermore, restarting reduces the storage requirements. Some theoretical results are presented. Numerical tests report the effectiveness of these methods.

Residual-Based Simpler Block GMRES for Nonsymmetric Linear Systems with Multiple Right-Hand Sides

In this paper we consider the simpler GMRES method augmented by approximate eigenvectors for solving nonsymmetric linear systems. We modify the augmented restarted simpler GMRES proposed by Boojhawon and Bhuruth to obtain a simpler GMRES with deflated restarting. Moreover, we also propose a residual-based simpler GMRES with deflated restarting, which is numerically more stable. The main advantage over the augmented version is that the simpler GMRES with deflated restarting requires less matrix-vector products per restart cycle. Some details of implementation are also considered. Numerical experiments show that the residual-based simpler GMRES with deflated restarting is effective.