BLOCK-SYMMETRIC AND BLOCK-LOWER-TRIANGULAR PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS＊

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained op- timization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigen- vectors of the corresponding preconditioned matrices are derived. Numerical implementa- tions show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.     

Rational extrapolation for the PageRank vector

An important problem in web search is to determine the importance of each page. From the mathematical point of view, this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1. Since this matrix is neither stochastic nor irreducible, the power method has convergence problems. So, the matrix is replaced by a convex combination, depending on a parameter , with a rank one matrix. Its left principal eigenvector now depends on , and it is the PageRank vector we are looking for. However, when is close to 1, the problem is ill-conditioned, and the power method converges slowly. So, the idea developed in this paper consists in computing the PageRank vector for several values of , and then to extrapolate them, by a conveniently chosen rational function, at a point near 1. The choice of this extrapolating function is based on the mathematical expression of the PageRank vector as a function of . Numerical experiments end the paper.

     

Using cross-product matrices to compute the SVD

This paper concerns accurate computation of the singular value decomposition (SVD) of an matrix . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of and take the square roots of the large eigenvalues of it as the large singular values of ; (b) form the Rayleigh quotient of with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.Supported in part by the National Science Foundation of China (No. 10471074).     

用奇异值分解方法计算具有重特征值矩阵的特征矢量

若当(Jordan)形是矩阵在相似条件下的一个标准形,在代数理论及其工程应用中都具有十分重要的意义．针对具有重特征值的矩阵,提出了一种运用奇异值分解方法计算它的特征矢量及若当形的算法．大量数值例子的计算结果表明,该算法在求解具有重特征值的矩阵的特征矢量及若当形上效果良好,优于商用软件MATLAB和MATHEMATICA．     

一类块三对角阵特征值的求解及应用

提出了求一类块三对角矩阵A的特征值和特征向量的方法,求得了该类矩阵的特征值和特征向量的表达式,并写出了用迭代法解该类方程组Au=f时迭代矩阵的特征值.     

一种求解大型稀疏对称矩阵(极端)特征值问题的有效算法

提出一种求解大型稀疏对称矩阵几个最大(最小)特征值和相应特征向量的迭代块DL(即Davidson-Lanczos)算法并且讨论了迭代块DL算法的收敛率     

On the computation of the main eigen-pairs of the continuous-time linear quadratic control problem

The degeneration of the eigenvalue equation of the discrete-time linear quadraticcontrol problem to the continuous-time one when△t→0?is given first.When thecontinuous-time n-dimensional eigenvalue equation,which has all the eigenvalues located inthe left half plane,has been reduced from the original2n-dimensional one,the present paperproposes that several of the eigenvalues nearest to the imaginary axis be obtained by thematrix transformation A_e=e~A.All the eigenvalues of A_e are in the unit circle,with theeigenvectors unchanged and the original eigenvaiues can be obtained by a logarithmoperation.And several of the eigenvalues of A_e nearest to the unit circle can be calculated bythe dual subspace iteration method.     

On the Analytical Approach of Codimension-Three Degenerate Bogdanov-Takens (B-T) Bifurcation in Satellite Dynamical System

In this paper, we have conducted parametric analysis on the dynamics of satellite complex system using bifurcation theory. At first, five equilibrium points $\mathcal{E}_{0,1,2,3,4}$ are symbolically computed in which $\mathcal{E}_{1,3}$ and $\mathcal{E}_{2,4}$ are symmetric. Then, several theorems are stated and proved for the existence of B-T bifurcation on all equilibrium points with the aid of generalized eigenvectors and practical formulae instead of linearizations. Moreover, a special case $\alpha_{2}=0$ is observed, which confirms all the discussed cases belong to a codimension-three bifurcation along with degeneracy conditions.