Tiexiang Li

On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems

The transmission eigenvalue problem, besides its critical role in inverse scattering problems, deserves special interest of its own due to the fact that the corresponding differential operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigenvalues and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmetric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.

Pole assignment for linear and quadratic systems with time-delay in control

SUMMARY We consider the pole assignment problems for time-invariant linear and quadratic control systems, with time-delay in the control. Closed-loop eigenvectors in X D Œx1, x2, ��� � are chosen from their corresponding invariant subspaces, possibly optimizing some robustness measure, and explicit expressions for the feedback matrices are given in terms of X. Condition of the problems is also investigated. Our approach extends the well-known Kautsky, Nichols, and Van Dooren algorithm. Consequently, the results are similar to those for systems without time-delay, except for the presence of the ‘secondary’ eigenvalues and the condition of the problems. Simple illustrative numerical examples are given. Copyright

Large-scale Stein and Lyapunov equations, Smith method, and applications

We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with

SOLVING LARGE-SCALE NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS BY DOUBLING ∗

A_k = A^{2^k}

A semiorthogonal generalized Arnoldi method and its variations for quadratic eigenvalue problems

not explicitly computed but in the recursive form

Robust Pole Assignment for Ordinary and Descriptor Systems via the Schur Form

A_k = A_{k-1}^{2}

A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems

, and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension n, the resulting algorithms are of an efficient O(n) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.

A modified Newton's method for rational Riccati equations arising in stochastic control

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation XCX − XD − AX + B =0 , withM ≡ (D, −C; −B, A) ∈ R(n1+n2)×(n1+n2) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A −1 u, A −� u, D −1 v ,a ndD −� v computable in O(n )c omplexity (withn =m ax{n1 ,n 2}), for some vectors u and v ,a ndB, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 392-412) is adapted, with the appropriate applications of the Sherman-Morrison- Woodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

Robust pole assignment via the Schur-Newton algorithms

In this paper, we are concerned with the computation of a few eigenpairs with smallest eigenvalues in absolute value of quadratic eigenvalue problems. We first develop a semiorthogonal generalized Arnoldi method where the name comes from the application of a pseudo inner product in the construction of a generalized Arnoldi reduction for a generalized eigenvalue problem. The method applies the Rayleigh– Ritz orthogonal projection technique on the quadratic eigenvalue problem. Consequently, it preserves the spectral properties of the original quadratic eigenvalue problem. Furthermore, we propose a refinement scheme to improve the accuracy of the Ritz vectors for the quadratic eigenvalue problem. Given shifts, we also show how to restart the method by implicitly updating the starting vector and constructing better projection subspace. We combine the ideas of the refinement and the restart by selecting shifts upon the information of refined Ritz vectors. Finally, an implicitly restarted refined semiorthogonal generalized Arnoldi method is developed. Numerical examples demonstrate that the implicitly restarted semiorthogonal generalized Arnoldi method with or without refinement has superior convergence behaviors than the implicitly restarted Arnoldi method applied to the linearized quadratic eigenvalue problem. Copyright

ARE-type iterations for rational Riccati equations arising in stochastic control

In Chu (Syst Control Lett 56:303–314, 2007), the pole assignment problem was considered for the control system \(\dot{x} = Ax + Bu\) with linear state-feedback \(u = Fx.\) An algorithm using the Schur form has been proposed, producing good suboptimal solutions which can be refined further using optimization. In this paper, the algorithm is improved, incorporating the minimization of the feedback gain \(\|F\|.\) It is also extended for the pole assignment of the descriptor system \(E\dot{x} = Ax + Bu\) with linear state- and derivative-feedback \(u = Fx - G\dot{x}.\) Newton refinement for the solutions is discussed and several illustrative numerical examples are presented.