Colloquium/Seminar

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Event(s) on October 2010


  • Monday, 18th October, 2010

    Title: CMIV Colloquium: Semi-Convergence and Relaxation Parameters for a Class of SIRT Algorithms
    Speaker: Prof. Per Christian Hansen, Department of Informatics and Mathematical Modelling, Technical University of Denmark, Denmark
    Time/Place: 11:30  -  12:30
    FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
    Abstract: Large-scale discretizations of ill-posed problems (such as imaging problems in tomography) call for the use of iterative methods, because direct factorization methods are infeasible. In particular, there is an interest in regularizing iterations, where the iteration vector can be considered as a regularized solution, with the iteration index playing the role of the regularizing parameter. Initially the iteration vector approaches a regularized solution, while continuing the iteration often leads to iteration vectors corrupted by noise. This work focuses on a class of non-stationary iteration methods, often referred to as Simultaneous Iterative Reconstruction Techniques (SIRT), including Landweber and Cimmino iteration. These methods incorporate a relaxation parameter, and the convergence rate of the initial iterations dependson the choice of this parameter. In principle, one can use a fixed parameter, but usually a good value is not known a priori. An attractive alternative is to choose the relaxation parameter automatically in each iteration, in such a way that fast semi-convergence is obtained. We study the semi-convergence for the SIRT methods, and we use our insight to propose two new methods for adaptively choosing the re-laxation parameter. Based on a careful analysis of the semi-convergence behavior of these methods, we propose two new techniques to specify the relaxation parameters during the iterations, so as to control the propagated noise component of the error. The advantage of using this strategy for noisy and ill-conditioned problems is demonstratedwith an example from tomography. In addition, we present a MATLAB package "AIR Tools" with implementations of these Algebraic Iterative Reconstruction methods. For each method, a number of strategies are available for choosing the relaxation parameter and the stopping rule. The package also includes three test problems.


  • Tuesday, 26th October, 2010

    Title: The meshless local Petrov-Galerkin method
    Speaker: Mr. Ahmad SHIRZADI, Department of Mathematics, Imam Khomeini International University, Iran
    Time/Place: 10:30  -  11:30
    FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist University
    Abstract: The meshless local Petrov-Galerkin (MLPG) approach, first proposed by Atluri and Zhu 1998, is a truly numerical meshless method for solving boundary value problems. The method uses the local weak form and shape functions from the moving least squares (MLS) approximation. The main advantage of this method over the widely used finite element method and a variety of meshless methods, is that it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. In this talk, the basis and frame of the MLPG method will be explained. By the 2-D Poisson’s equation, we will illustrate the numerical implementation of the method.

 

 


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