122233994
An efficient method for evaluating the integral of a class of highly oscillatory functions
Highly oscillatory integrals require special techniques for their effective evaluation. Various studies have been conducted to find computational methods for evaluating such integrals. In this paper we present an efficient numerical method to evaluate a class of generalised Fourier integrals (on a line or a square) with integrands of the form f(x)e^i^k^g^(^x^), under the assumption that in the domain of integration, both f and g are sufficiently smooth and that g does not have any stationary/critical points. Numerical analysis and results are given to illustrate the effectiveness of our method for computing generalised Fourier integrals.
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.
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.
205413619
An efficient preconditioned iterative solver for solving a coupled fluid structure interaction problem
Modelling the interaction of an acoustic field in a fluid and an elastic structure submerged in the fluid leads to a system of complex linear equations with a complicated sparsity structure and, for higher wavenumbers and adequate modelling, the systems are very large. Direct methods are not practical. Preconditioned iterative methods, which are suitable for single operator equations, are not immediately applicable to the coupled case. This article proposes a block diagonal preconditioner of the sparse approximate inverse (SPAI) type that can accelerate the convergence of Krylov iterative solvers for the coupled system. Moreover, the proposed preconditioner can properly and implicitly scale the coupled matrix. Some numerical results are presented to demonstrate the effectiveness of the new method.
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.
24158595
A Fractal Watermark Solution for Product Data
Copyright and integrity protection over product data is becoming an urgent requirement, because in a Distributed Networked Manufacturing environment, enterprises have to interact with each other to share product data. In this paper, we introduce a novel watermarking solution for product data based on the idea of fractal, which classifies product data watermarks into three tiers: fractal watermark, tree-structure watermark, and tree-node watermark. It embeds watermarks not only in value-carrying nodes but also in product structures that "glue" all the nodes together by embedding multiple structure watermarks in product data. Simulation results show that the proposed technique is robust against various forms of malicious attacks of product data, and naturally immune from structure transformation attacks and subtractive (sub-tree) attacks. Moreover, the solution can protect tree-structure data whose knowledge is mainly embodied in the tree structure efficiently. It is very easy to implement and popularize.
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.
13249966
Prediction of integral membrane protein type by collocated hydrophobic amino acid pairs
A computational model, IMP-TYPE, is proposed for the classification of five types of integral membrane proteins from protein sequence. The proposed model aims not only at providing accurate predictions but most importantly it incorporates interesting and transparent biological patterns. When contrasted with the best-performing existing models, IMP-TYPE reduces the error rates of these methods by 19 and 34% for two out-of-sample tests performed on benchmark datasets. Our empirical evaluations also show that the proposed method provides even bigger improvements, i.e., 29 and 45% error rate reductions, when predictions are performed for sequences that share low (40%) identity with sequences from the training dataset. We also show that IMP-TYPE can be used in a standalone mode, i.e., it duplicates significant majority of correct predictions provided by other leading methods, while providing additional correct predictions which are incorrectly classified by the other methods. Our method computes predictions using a Support Vector Machine classifier that takes feature-based encoded sequence as its input. The input feature set includes hydrophobic AA pairs, which were selected by utilizing a consensus of three feature selection algorithms. The hydrophobic residues that build up the AA pairs used by our method are shown to be associated with the formation of transmembrane helices in a few recent studies concerning integral membrane proteins. Our study also indicates that Met and Phe display a certain degree of hydrophobicity, which may be more crucial than their polarity or aromaticity when they occur in the transmembrane segments. This conclusion is supported by a recent study on potential of mean force for membrane protein folding and a study of scales for membrane propensity of amino acids.
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.
35419934
MATLAB Simulation of Gradient-Based Neural Network for Online Matrix Inversion
This paper investigates the simulation of a gradient-based recurrent neural network for online solution of the matrix-inverse problem. Several important techniques are employed as follows to simulate such a neural system. 1) Kronecker product of matrices is introduced to transform a matrix-differential-equation (MDE) to a vector-differential-equation (VDE); i.e., finally, a standard ordinary-differential-equation (ODE) is obtained. 2) MATLAB routine "ode45" is introduced to solve the transformed initial-value ODE problem. 3) In addition to various implementation errors, different kinds of activation functions are simulated to show the characteristics of such a neural network. Simulation results substantiate the theoretical analysis and efficacy of the gradient-based neural network for online constant matrix inversion.