Yair Shapira

Upper bounds for convergence rates of acceleration methods with initial iterations

GMRES(n,k), a version of GMRES for the solution of large sparse linear systems, is introduced. A cycle of GMRES(n,k) consists of n Richardson iterations followed by k iterations of GMRES. Such cycles can be repeated until convergence is achieved. The advantage in this approach is in the opportunity to use moderate k, which results in time and memory saving. Because the number of inner products among the vectors of iteration is about k2/2, using a moderate k is particularly attractive on message-passing parallel architectures, where inner products require expensive global communication. The present analysis provides tight upper bounds for the convergence rates of GMRES(n,k) for problems with diagonalizable coefficient matrices whose spectra lie in an ellipse in 0. The advantage of GMRES(n,k) over GMRES(k) is illustrated numerically.

Applications in Image Processing

In this chapter, we illustrate the usefulness of matrix-based multigrid in the field of image processing. This field is particularly suitable for uniform, rectangular grids because images are often stored in 2-d uniform arrays of pixels. We introduce algorithms for denoising noisy grayscale (noncolor) as well as color images. These algorithms are based on nonlinear elliptic PDEs. These PDEs are linearized by the fixed point iteration, and the linear elliptic PDE is discretized by the finite volume discretization method. The linear system obtained is solved by the AutoMUG(0) method. The numerical results are illustrated in images.

Towards Automatic Multigrid Algorithms for SPD, Nonsymmetric and Indefinite Problems

A new multigrid algorithm is constructed for the solution of linear systems of equations which arise from the discretization of elliptic PDEs. It is defined in terms of the difference scheme on the fine grid only, and no rediscretization of the PDE is required. Numerical experiments show that this algorithm gives high convergence rates for several classes of problems: symmetric, nonsymmetric and problems with discontinuous coefficients, nonuniform grids, and nonrectangular domains. When supplemented with an acceleration method, good convergence is achieved also for pure convection problems and indefinite Helmholtz equations.

Multigrid methods for 3-D definite and indefinite problems

A multigrid method for the solution of certain finite difference and finite volume schemes for elliptic PDEs is introduced. A parallelizable version of it, suitable for two-level and multi-level analysis, is also defined and serves as a theoretical tool for deriving suitable implementations for the main version. For indefinite equations, this analysis provides a prediction of a suitable mesh size for the coarsest grid used. Numerical experiments show the efficiency of the method for 3-D diffusion problems with discontinuous coefficients and highly indefinite Helmholtz equations.

Multigrid for Locally Refined Meshes

A multilevel method for the solution of finite element schemes on locally refined meshes is introduced. For isotropic diffusion problems, the condition number of the two-level method is bounded independently of the mesh size and the discontinuities in the diffusion coefficient. The curves of discontinuity need not be aligned with the coarse mesh. Indeed, numerical applications with 10 levels of local refinement yield a rapid convergence of the corresponding 10-level, multigrid V-cycle and other multigrid cycles which are more suitable for parallelism even when the discontinuities are invisible on most of the coarse meshes.

Feature selection for multiple binary classification problems

Abstract The present study proposes an unsupervised method for selection of feature subsets, which retain sufficient information for classification purposes. Multiple alternative physically feasible partitions can be dealt with, using the present method. The method is based on an alternative approach to data representation, in which the axes are the data points instead of the features (a transpose projection). Under this representation, coherent features are located in the vicinity of each other, and hence can be clustered, while noisy features are pointed out and eliminated. The method bypasses the “curse of dimensionality” and demonstrates good results in particular in small data sets.

Preconditioning Spectral Element Schemes for Definite and Indefinite Problems

Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution.

COLORING UPDATE METHODS

For linear update methods (such as SOR), a coloring method is introduced for which the multicolor iteration matrix has the same spectrum as the original iteration matrix. It applies to general linear systems, not necessarily arising from PDEs. When the iteration matrices are nonsingular, it is shown that they are similar to each other.

A Domain Decomposition Approach

In this chapter, we introduce a domain decomposition two-grid iterative method for solving sparse linear systems that arise from the discretization of elliptic PDEs on general unstructured grids that do not necessarily arise from local refinement. The coarse grid consists of the vertices of the subdomains in the domain decomposition. Assuming that the coefficient matrix is SPD and diagonally dominant, we supply an upper bound for the condition number of the V(0,0) cycle.

A geometrical fuzzy clustering-based solution to glottal wave estimation

Accurate estimation of the glottal waveform (GW) is required for purposes such as natural speech synthesis, speaker recognition, physiological speech processing, etc. Most methods available for GW estimation are based on inverse filtering of the speech signal through the vocal tract, and they all suffer from inaccuracies due to incorrect assumptions. The method for GW estimation developed in the present study is based on fuzzy clustering of quasi-linear geometrical substructures, represented within the signal shifts hyperspace. Algorithms for estimation of the driving function to the vocal tract are presented and evaluated on simulated and real data. Comparison of the fuzzy clustering-based method with the PSIAIF and Wong’s closed-phase algorithms shows that the present method is superior with respect to both the GW estimation and determination of GW event time instants.