会议时间：6月8日，星期六，下午 3:00—6:00

会议地点：数学楼 2-1会议室

会议报告：

1. 报告人：白中治，中国科学院

报告题目：Computing Eigenpairs of Hermitian Matrices in Perfect Krylov Subspaces

报告摘要：For computing the smallest eigenvalue and the corresponding eigenvector of a Hermitian matrix, by introducing a concept of perfect Krylov subspace we propose a class of perfect Krylov subspace methods. For these methods, we prove their local, semilocal and global convergence properties, and discuss their inexact implementations and preconditioning strategies. In addition, we use numerical experiments to demonstrate the convergence properties and exhibit the competitiveness of these methods with a few state-of-the art iteration methods when they are employed to solve large and sparse Hermitian eigenvalue problems.

2. 报告人：伍渝江，兰州大学

报告题目：Minimum Residual HSS Iteration Method for Non-Hermitian Positive Definite Complex Linear Systems

报告摘要：This talk will present a non-stationary iteration method, or a minimum residual Hermitian and skew-Hermitian (MRHSS) iteration method for solving non-Hermitian positive definite complex linear systems. Convergence analysis and numerical results will be also given to illustrate the efficiency of the MRHSS method.

3. 报告人：黄玉梅，兰州大学

报告题目：Weighted Nuclear Norm Minimization Based Regularization Method for Image Restoration

报告摘要：Regularization methods have been substantially applied in image restoration due to the ill-posedness of the image restoration problem. Different assumptions or priors on images are applied in the construction of image regularization methods. In recent years, low-rank matrix approximation has been successfully introduced in image denoising and significant denoising effects have been achieved. The computation of low-rank matrix minimization is a NP hard problem and it is often replaced with the matrix's weighted nuclear norm minimization.  Nonlocal image denoising methods assume that an image contains an extensive amount of self-similarity. Based on such assumption, in this talk, we develop a   model for image restoration by using   weighted nuclear norm to be the regularization term. An alternating iterative algorithm is designed to solve the proposed model and we also present the convergence analyses of the algorithm. Numerical experiments show that the proposed method can recover the images much better than the existing regularization methods in terms of both recovered quantities and visual qualities.

4. 报告人：吴钢，中国矿业大学

报告题目：Randomized GLRAM-type Algorithms for High Dimensionality Reduction and Image Reconstruction

报告摘要：High-dimensionality reduction techniques are very important tools in machine learning and data mining. The method of generalized low rank approximations of matrices (GLRAM) and its variations are popular for dimensionality reduction and image reconstruction, which are based on native two-dimensional matrix patterns. However, they often suffer from heavily computational overhead in practice, especially for data with high dimensionality. In order to reduce the computational complexities of these type of algorithms, we apply randomized singular value decomposition (RSVD) on them and propose three randomized GLRAM-type algorithms. Theoretical results are established to show the validity and rationality of our proposed algorithms.

First, we discuss the decaying property of singular values of the matrices during iterations of the GLRAM algorithm, and provide the target rank required in the RSVD process from a theoretical point of view. Second, we show the relationships between the reconstruction errors generated by the original GLRAM-type algorithms and the randomized GLRAM-type algorithms. Third, we shed light on the convergence of the randomized GLRAM algorithm.

Numerical experiments on some real-world data sets illustrate the superiority of our proposed algorithms over their original counterparts and some state-of-the-art algorithms, for image reconstruction and face recognition.