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.
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.
12724602
New wavelet preconditioner for solving boundary integral equations over nonsmooth boundaries
Matrices obtained by wavelet discretisation of partial differential equations or boundary integral equations (BIEs) are typically sparse with a finger-like sparsity pattern, in contrast to matrices obtained by traditional single scale discretisation, which are dense. In some cases diagonal preconditioning is sufficient, but there are effective preconditioners that can be used when this is not the case. Here, we consider the particular case of BIEs whose boundary has a geometrical singularity, and propose a new preconditioning strategy based on permutations of the unknowns. The strategy's performance is analysed and compared with related techniques for the double layer equation for Laplace's equation. *E-mail: stuarth@liverpool.ac.uk. Support of a UK EPSRC grant (GR/R22315) is gratefully acknowledged.
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.

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.
61248473
Efficient parallelization of the iterative solution of a coupled fluid–structure interaction problem: Research Articles
In this paper we consider the parallelization of the generation and iterative solution of coupled linear systems modelling the interaction of an acoustic field in a fluid medium with an elastic structure immersed in the fluid. The particular case studied is that of a hollow steel sphere in water. The aim of the work is to speed up the generation and solution of the systems. We describe the methods used, which involve special sparse storage arrangements and a novel application of a sparse approximate inverse preconditioning technique, and present results showing that the methods are very effective in terms of speeding up the generation and iterative solution of the systems. Copyright © 2007 John Wiley & Sons, Ltd.
14996476
Global exponential convergence and stability of gradient-based neural network for online matrix inversion
Wang proposed a gradient-based neural network (GNN) to solve online matrix-inverses. Global asymptotical convergence was shown for such a neural network when applied to inverting nonsingular matrices. As compared to the previously-presented asymptotical convergence, this paper investigates more desirable properties of the gradient-based neural network; e.g., global exponential convergence for nonsingular matrix inversion, and global stability even for the singular-matrix case. Illustrative simulation results further demonstrate the theoretical analysis of gradient-based neural network for online matrix inversion.
3106142
MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Online Solution of Linear Time-Varying Matrix Equation AXB-C=0
Different from gradient neural networks (GNN), a special kind ofrecurrent neural networks has been proposed recently by Zhanget alfor solving online linear matrix equations withtime-varying coefficients. Such recurrent neural networks, designedbased on a matrix-valued error-function, could achieve globalexponential convergence when solving online time-varying problemsin comparison with gradient neural networks. This paperinvestigates the MATLAB simulation of Zhang neural networks (ZNN)for real-time solution of linear time-varying matrix equationAXB- C= 0. Gradient neural networks aresimulated and compared as well. Simulation results substantiate thetheoretical analysis and efficacy of ZNN on linear time-varyingmatrix equation solving.
1262399
Efficient Holistic Twig Join Algorithm on XML Documents with Optimization Rules and Index
Efficiently finding all occurrences of a twig pattern is a core operation of XML query processing. In this paper, we first propose a generic twig join algorithm TJFGeneric which only scans the leaf query nodes using a cursor interface to return elements. And then we present TJFGeneric+ algorithm incorporating two optimization rules into TJFGeneric to explore potential benefit of disk read and CPU cost. In particular, available indices (such as B+-tree) are applied to skip the elements that do not participate in final results. Experimental results on various datasets indicate that our algorithms outperform previous approaches significantly.
5764451
Regularized Boost for Semi-Supervised Learning
Semi-supervised inductive learning concerns how to learn a decision rule from a data set containing both labeled and unlabeled data. Several boosting algorithms have been extended to semi-supervised learning with various strategies. To our knowledge, however, none of them takes local smoothness constraints among data into account during ensemble learning. In this paper, we introduce a local smoothness regularizer to semi-supervised boosting algorithms based on the universal optimization framework of margin cost functionals. Our regularizer is applicable to existing semi-supervised boosting algorithms to improve their generalization and speed up their training. Comparative results on synthetic, benchmark and real world tasks demonstrate the effectiveness of our local smoothness regularizer. We discuss relevant issues and relate our regularizer to previous work.
6117831
MATLAB Simulink modeling and simulation of LVI-based primal-dual neural network for solving linear and quadratic programs
In view of parallel-processing nature and circuit-implementation convenience, recurrent neural networks are often employed to solve optimization problems. Recently, a primal-dual neural network based on linear variational inequalities (LVI) was developed by Zhang et al. for the online solution of linear-programming (LP) and quadratic-programming (QP) problems simultaneously subject to equality, inequality and bound constraints. For the final purpose of field programmable gate array (FPGA) and application-specific integrated circuit (ASIC) realization, we investigate in this paper the MATLAB Simulink modeling and simulative verification of such an LVI-based primal-dual neural network (LVI-PDNN). By using click-and-drag mouse operations in MATLAB Simulink environment, we could quickly model and simulate complicated dynamic systems. Modeling and simulative results substantiate the theoretical analysis and efficacy of the LVI-PDNN for solving online the linear and quadratic programs.