119485415
On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem
The paper presents a Galerkin numerical method for solving the hyper-singular boundary integral equations for the exterior Helmholtz problem in three dimensions with a Neumann's boundary condition. Previous work in the topic has often dealt with the collocation method with a piecewise constant approximation because high order collocation and Galerkin methods are not available due to the presence of a hypersingular integral operator. This paper proposes a high order Galerkin method by using singularity subtraction technique to reduce the hyper-singular operator to a weakly singular one. Moreover, we show here how to extend the previous work (J. Appl. Numer. Math. 36 (4) (2001) 475-489) on sparse preconditioners to the Galerkin case leading to fast convergence of two iterative solvers: the conjugate gradient normal method and the generalised minimal residual method. A comparison with the collocation method is also presented for the Helmholtz problem with several wavenumbers.
3879090
On A Recursive Schur Preconditioner For Iterative Solution Of A Class Of Dense Matrix Problems
There are currently several distinct preconditioning methods for dense matrices based on applying a wavelet transform to obtain a matrix with a large number of small entries. A sparse preconditioner for this transformed matrix can be formed by setting to zero entries that are assumed to be unimportant. The effectiveness of the preconditioner depends on retaining the most important entries and on ensuring that they are positioned conveniently within the transformed matrix. In this paper we present a new, recursive preconditioning strategy that takes into account more of the significant entries without greatly increasing cost and outperforms existing methods in certain cases.
25215706
On Two Variants of an Algebraic Wavelet Preconditioner
A recursive method of constructing preconditioning matrices for the nonsymmetric stiffness matrix in a wavelet basis is proposed for solving a class of integral and differential equations. It is based on a level-by-level application of the wavelet scales decoupling the different wavelet levels in a matrix form just as in the well-known nonstandard form. The result is a powerful iterative method with built-in preconditioning leading to two specific algebraic multilevel iteration algorithms: one with an exact Schur preconditioning and the other with an approximate Schur preconditioning. Numerical examples are presented to illustrate the efficiency of the new algorithms.
12726286
Fast Multilevel Algorithm for a Minimization Problem in Impulse Noise Removal
An effective 2-phase method for removing impulse noise was recently proposed. Its phase 1 identifies noisy pixel candidates by using median-type filters. Then in phase $2$, it restores only the noisy pixel candidates by some variational methods. The resulting method can handle salt-and-pepper noise and random-valued impulse noise at a level as high as 90% and 60%, respectively. The aim of this paper is to generalize a fast multilevel method for Gaussian denoising to solving the minimization problem arising in phase $2$ of the 2-phase method. The multilevel algorithm gives better images than standard optimization methods such as the Newton method or the conjugate gradient method. Also it can handle more general regularization functionals than the smooth ones previously considered. Supporting numerical experiments on two-dimensional gray-scale images are presented.
15785287
A Nonlinear Multigrid Method for Total Variation Minimization from Image Restoration
Abstract
Image restoration has been an active research topic and variational formulations are particularly effective in high quality recovery. Although there exist many modelling and theoretical results, available iterative solvers are not yet robust in solving such modeling equations. Recent attempts on developing optimisation multigrid methods have been based on first order conditions. Different from this idea, this paper proposes to use piecewise linear function spanned subspace correction to design a multilevel method for directly solving the total variation minimisation. Our method appears to be more robust than the primal-dual method (Chan et al., SIAM J. Sci. Comput. 20(6), 1964–1977, 1999) previously found reliable. Supporting numerical results are presented.

21847139
An Analysis of Sparse Approximate Inverse Preconditioners for Boundary Integral Equations
Preconditioning techniques for dense linear systems arising from singular boundary integral equations are described and analyzed. A particular class of approximate inverse based preconditioners related to the mesh neighbor methods is known to be efficient. This paper shows that it is an operator splitting preconditioner and clusters eigenvalues for the normal equation matrix thus ensuring a fast convergence of the conjugate gradient normal method. Clustering of the eigenvalues of the preconditioned matrix and fast convergence of the generalized minimal residual method are also observed. For the type of problems considered, we demonstrate a crucial connection between two essential features of eigenvalue clustering for a sparse preconditioner---approximate inversion for a small cluster radius and operator splitting for a small cluster size. Experimental results from several boundary integral equations are presented.
16708333
MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Time-Varying Lyapunov Equation Solving
This paper presents a new kind of recurrent neural network proposed by Zhang et al.for solving online Lyapunov equation with time-varying coefficient matrices. Global exponential convergence could be achieved by such a recurrent neural network when solving the time-varying problems in comparison with gradient neural networks (GNN). MATLAB simulation of both neural networks for the real-time solution of time-varying Lyapunov equation is then investigated through several important techniques. Computer-simulation results substantiate the theoretical analysis and demonstrate the efficacy of such a Zhang neural network (ZNN) on time-varying Lyapunov equation solving, especially when using power-sigmoid activation functions.
28337697
Trends in Neural Computation
Hyperbolic Function Networks for Pattern Classification.- Variable Selection for the Linear Support Vector Machine.- Selecting Data for Fast Support Vector Machines Training.- Universal Approach to Study Delayed Dynamical Systems.- A Hippocampus-Neocortex Model for Chaotic Association.- Latent Attractors: A General Paradigm for Context-Dependent Neural Computation.- Learning Mechanisms in Networks of Spiking Neurons.- GTSOM: Game Theoretic Self-organizing Maps.- How to Generate Different Neural Networks.- A Gradient-Based Forward Greedy Algorithm for Space Gaussian Process Regression.- An Evolved Recurrent Neural Network and Its Application.- A Min-Max Modular Network with Gaussian-Zero-Crossing Function.- Combining Competitive Learning Networks of Various Representations for Sequential Data Clustering.- Modular Neural Networks and Their Applications in Biometrics.- Performance Analysis of Dynamic Cell Structures.- Short Term Electric Load Forecasting: A Tutorial.- Performance Improvement for Formation-Keeping Control Using a Neural Network HJI Approach.- A Robust Blind Neural Equalizer Based on Higher-Order Cumulants.- The Artificial Neural Network Applied to Servo Control System.- Robot Localization Using Vision.
1757195
Combining linear discriminant functions with neural networks for supervised learning
A novel supervised learning method is proposed by combining linear discriminant functions with neural networks. The proposed method results in a tree-structured hybrid architecture. Due to constructive learning, the binary tree hierarchical architecture is automatically generated by a controlled growing process for a specific supervised learning task. Unlike the classic decision tree, the linear discriminant functions are merely employed in the intermediate level of the tree for heuristically partitioning a large and complicated task into several smaller and simpler subtasks in the proposed method. These subtasks are dealt with by component neural networks at the leaves of the tree accordingly. For constructive learning, growing and credit-assignment algorithms are developed to serve for the hybrid architecture. The proposed architecture provides an efficient way to apply existing neural networks (e.g. multi-layered perceptron) for solving a large scale problem. We have already applied the proposed method to a universal approximation problem and several benchmark classification problems in order to evaluate its performance. Simulation results have shown that the proposed method yields better results and faster training in comparison with the multilayered perceptron.
22766407
Watermarking Abstract Tree-Structured Data
Digital watermarking techniques have been successfully applied in multimedia data such as image, audio, video, etc. With the rapid development of Internet, the requirement of copyright and integrity protection over semi-structured data, such as XML documents and complex hypertext content, is becoming more and more urgent. However, it is more challenging to apply the effective watermarking schemes into semi-structured data, because watermark for semi-structured data must have the dual protection feature toward both structure and content. In general, most semi-structured data can be represented as tree. Therefore, in this paper, we propose a novel watermark scheme for tree-structured data based on the value lying both in the tree structure and in the node content, which gives a comprehensive protection for both node content and structure of tree. The experiments on the robustness of the solution against various forms of attacks are conducted. The results highlight correctness and efficiency of our solution.
17533153
OTJFast: Processing Ordered XML Twig Join Fast
Thanks to standard specifications for Web services, more and more data is represented and exchanged in XML. Efficiently finding all occurrences of a twig pattern is a core operation of XML query processing. Recently, a large number of holistic twig join algorithms have been proposed, however ordered XML twig join has been ignored in most of them. In this paper we design a novel algorithm called OTJFast to process ordered XML twig joins efficiently which only need to access the elements of the leaf query nodes. Experimental results show the benefits of our algorithm in term of the size of intermediate results, the number of element reads and the execution time.