Mathematical Science Seminar series S1 No.9-11

Events

Speakers

  • Professor Zhong-zhi Bai
  • Professor Zhigang Jia
  • Professor Yang Cao

Session 1

Speaker

  • Professor Zhong-zhi Bai
  • Chinese Academy of Sciences

Time

  • 15:00-15:50

Title

  • On Regularized HSS Iteration Method for Saddle-Point Linear Systems

Abstract

We report results on the regularized Hermitian and skew-Hermitian splitting method for the solution of large, sparse linear systems in saddle-point form. This method can be used as a stationary iterative solver or as a preconditioner for Krylov subspace iteration methods. We present unconditional convergence of the stationary iteration and examine the spectral property of the corresponding preconditioned matrix. Inexact variants are also discussed. Numerical results show that optimal convergence behavior can be achieved when using the proposed preconditioners to accelerate the convergence rates of the Krylov subspace
iteration methods.

Session 2

Speaker

  • Professor Zhigang Jia
  • Jiangsu Normal University

Time

  • 16:00-16:50

Title

  • Relaxed 2-D principal component analysis by $L_p$-norm for face recognition

Abstract

A relaxed two dimensional principal component analysis (R2DPCA) by Lp-norm is proposed for face recognition. The label information (if known) of training samples is applied to calculate the relaxation vector, which indicates the contribution of each subsets to the covariance matrix. The computed projection axes are accompanied with a weighting vector, which highly increase the face recognition rate. A restarted alternating direction search method of choosing optimal parameters is also presented to save operation time of solving large scale problems. Our numerical experiments demonstrate the feasibility of our proposed algorithms.

Session 3

Speaker

  • Professor Yang Cao
  • Nantong University

Time

  • 17:00-17:50

Title

  • Block triangular preconditioners based on ST decomposition for generalized saddle point problems

Abstract

In this talk, the symmetric-triangular decomposition is further studied to construct a class of block triangular preconditioners for generalized saddle point problems such that the preconditioned generalized saddle point matrices are symmetric and positive definite. Then the (preconditioned) conjugate gradient iterative method can be used. Three specific preconditioners are studied in detail. Eigen-properties of the corresponding preconditioned generalized saddle point matrices are studied. In particular, upper bounds on the condition number of the preconditioned matrices are analyzed. Finally, numerical experiments of a model Stokes equation are given to illustrate the efficiency of the new proposed preconditioners.