Michael K. Ng

Conjugate Gradient Methods for Toeplitz Systems

In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of

Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems

n

Automated variable weighting in k-means type clustering

-by-

A fuzzy k-modes algorithm for clustering categorical data

n

An Entropy Weighting k-Means Algorithm for Subspace Clustering of High-Dimensional Sparse Data

Toeplitz systems is reduced to

A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions

O(n \log n)

Analysis of Half-Quadratic Minimization Methods for Signal and Image Recovery

operations as compared to

A New Total Variation Method for Multiplicative Noise Removal

O(n \log ^2 n)

Finding the Largest Eigenvalue of a Nonnegative Tensor

operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.

An optimization algorithm for clustering using weighted dissimilarity measures

We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of a three-dimensional convection-diffusion equation is used to illustrate the advantages of our methods.