5116790
Flexible parallelization of fast wavelet transforms
In this paper, we present a new parallel algorithm for fast wavelet transforms (FWT) of a matrix of arbitrary size using any given number of parallel processors. The main idea in achieving the optimal load balancing is through a complexity analysis and flops minimization. This makes parallel implementation of FWT feasible and efficient on distributed memory machines with only a small number of processors and on local area networks. The new algorithm is tested by numerical experiments.
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.
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.

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.
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.
33477464
Boundary-Based Feature Extraction and Recognition of Breast Tumors Using Support Vector Machine
14878951
The orienteering problem in the plane revisited
We consider the <i>orienteering problem</i>: Given a set <i>P</i> of <i>n</i> points in the plane, a starting point <i>r</i> ∈ <i>P</i>, and a length constraint <i>B</i>, one needs to find a tour starting at <i>r</i> that visits as many points of <i>P</i> as possible and of length not exceeding <i>B</i>. We present a (1−ε)-approximation algorithm for this problem that runs in <i>n</i><sup><i>O</i>(1/ε)</sup> time, and visits at least (1−ε)<i>k</i><sub>opt</sub> points of <i>P</i>, where <i>k</i><sub>opt</sub> is the number of points visited by the optimal solution. This is the first polynomial time approximation scheme (PTAS) for this problem. The algorithm also works in higher dimensions.
1589839
Optimization of the Sliding Window Size for Protein Structure Prediction
Sliding window based methods are relatively often applied in prediction of various aspects related to protein structure. Despite their wide spread use, researchers did not establish a standard related to the size of the window, i.e., window sizes ranging between 7 and 17 residues were used in the past. To this end, this paper performs a computational study based on a probabilistic approach that aims at finding an optimal sliding window size. The results shows that formation of helical structure can be affected by amino acids (AAs) that are up to 9 positions away in the sequence, while the formation of coils and strands can be affected by AAs that are up to 3 and 6 positions away, respectively. Overall, our results suggest that a sliding window with 19 residues is optimal for secondary structure prediction, while for a specific prediction tasks, such as prediction of p-strands, a smaller window size is sufficient. Finally, the 20 AAs are categorized into five groups based on their influence of formation of the secondary structure. The finding related to the optimal window size was confirmed based on an independent experimental study related to the prediction of secondary protein structure
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.
16722943
The Euclidean Orienteering Problem Revisited
We consider the rooted orienteering problem: Given a set $P$ of $n$ points in the plane, a starting point $r \in P$, and a length constraint $B$, one needs to find a path starting from $r$ that visits as many points of $P$ as possible and of length not exceeding $B$. We present a $(1-\varepsilon)$-approximation algorithm for this problem that runs in $n^{O(1/\varepsilon)}$ time; the computed path visits at least $ (1-\varepsilon)k_{\mathrm{opt}}$ points of $P$, where $k_{\mathrm{opt}}$ is the number of points visited by an optimal solution. This is the first polynomial time approximation scheme for this problem. The algorithm also works in higher dimensions.