Event(s) on July 2023

• Monday, 3rd July, 2023

  Title:	Alternating Nonnegative Least Squares for Nonnegative Matrix Factorization
  Speaker:	Professor Delin Chu, Department of Mathematics, National University of Singapore
  Time/Place:	11:00  -  12:00 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	Nonnegative matrix factorization (NMF) is a prominent technique for data dimensionality reduction. In this talk, a framework called ARkNLS (Alternating Rank-k Nonnegativity constrained Least Squares) is proposed for computing NMF. First, a recursive formula for the solution of the rank-k nonnegativity-constrained least squares (NLS) is established. This recursive formula can be used to derive the closed-form solution for the Rank-k NLS problem for any positive integer k. As a result, each subproblem for an alternating rank-k nonnegative least squares framework can be obtained based on this closed form solution. Assuming that all matrices involved in rank-k NLS in the context of NMF computation are of full rank, two of the currently best NMF algorithms HALS (hierarchical alternating least squares) and ANLS-BPP (Alternating NLS based on Block Principal Pivoting) can be considered as special cases of ARkNLS. This talk is then focused on the framework with k=3, which leads to a new algorithm for NMF via the closed-form solution of the rank-3 NLS problem. Furthermore, a new strategy that efficiently overcomes the potential singularity problem in rank-3 NLS within the context of NMF computation is also presented. Extensive numerical comparisons using real and synthetic data sets demonstrate that the proposed algorithm provides state-of-the-art performance in terms of computational accuracy and cpu time

  
  

• Tuesday, 4th July, 2023

  Title:	Meshless methods for PDEs with non-smooth coefficients and for the elastic wave scattering by obstacles
  Speaker:	Dr. Siqing LI, College of Mathematics, Taiyuan University of Technology, China
  Time/Place:	14:30  -  15:30 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	In this talk, meshless methods are applied to two aspects: elliptic PDEs with non-smooth coefficients, and elastic wave obstacle scattering problems. In the first part, RBF-FD methods are used for second-order elliptic PDEs with non-smooth coefficients. Oversampling is used in regions where coefficients vary rapidly, and the numerical solutions are obtained via weighted least-squares RBF-FD methods. Furthermore, when discontinuities appear in the convection terms, the PDEs are rewritten in divergence form. Numerical examples demonstrate that our proposed methods improve both robustness and accuracy. In the second part, time-harmonic elastic wave obstacle scattering problems are considered. The perfectly matched layer (PML) technique is employed to truncate the unbounded physical domain into a bounded computational domain. Subsequently, the kernel-based collocation method is utilized to solve the corresponding Navier and Helmholtz equations. The numerical example with a circular obstacle is tested to verify effectiveness of the method. By considering geometric flexibility in collocation methods, problems with irregular obstacles are also addressed.

  
  

• Monday, 10th July, 2023

  Title:	A tau matrix approximation based preconditioning technique for space-fractional diffusion equation with variable coefficients
  Speaker:	Dr. Xuelei Lin, Harbin Institute of Technology, Shenzhen
  Time/Place:	11:00  -  12:00 FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
  Abstract:	In this talk, a tau matrix-approximation based preconditioner is proposed for the linear system arising from unsteady-state space fractional diffusion equation. The preconditioner is fast diagonalizable by sine transform. Theoretically, we show that GMRES solver for the preconditioned system has a linear convergence rate independent of the discretization parameters.

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