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Let A ∈ ℂN x N be a nonsingular complex matrix and b and c be complex vectors of length N. The goal of this paper is to investigate approaches for efficient approximations of the bilinear form c*A-1b. Equivalently, we wish to approximate the scalar value c*x, where x solves the linear system Ax = b. Here the matrix A can be very large or its elements can be too costly to compute so that A is not explicitly available and it is used only in the form of the matrix-vector product. Therefore a direct method is not an option. For A Hermitian positive definite, b*A-1b can be efficiently approximated as a by-product of the conjugate-gradient iterations, which is mathematically equivalent to the matching moment approximations computed via the Gauss-Christoffel quadrature. In this paper we propose a new method using the biconjugate gradient iterations which is applicable to the general complex case. The proposed approach will be compared with existing ones using analytic arguments and numerical experiments. [ABSTRACT FROM AUTHOR]

This paper concerns numerical methods for computing complex geometrical optics (CGO) solutions to the conductivity equation ∇ ⋅ σ∇u (⋅, k) = 0 in ℝ² for piecewise smooth conductivities σ, where k is a complex parameter. The key is to solve an R-linear singular integral equation defined in the unit disk. Recently, Astala et al. [Appl. Comput. Harmon. Anat., 29 (2010), pp. 2-17] proposed a complicated method for numerical computation of CGO solutions by solving a periodic version of the ℝ-linear integral equation in a rectangle containing the unit disk. In this paper, based on the fast algorithms in [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133-157] for singular integral transforms, we propose a simpler numerical method which solves the ℝ-linear integral equation in the unit disk directly. For the resulting ℝ-linear operator equation, a minimal residual iterative method is proposed. Numerical examples illustrate the accuracy and efficiency of the new method. [ABSTRACT FROM AUTHOR]

As a solution to the frequently reported problem of the high frequencies reconstructed sometimes many iterations after the low frequencies in tomography, we proposed in a recent paper [V. Israel-Jost, Ph. Choquet, and A. Constantinesco, Int. J. Biomed. Imaging, 2006, paper 34043.] a general adaptation that can apply to many algorithms: the incomplete backprojection. We stressed the fact that this type of frequency adapted (FA) algorithm works only when the linear problem of tomography to be solved involves deblurring, i.e., when the point spread functions (PSFs) associated with each voxel are large. Thus, depending on the desired level of detail, a parameter is set to achieve the reconstruction within a small number of iterations. The aim of this paper is to give a mathematical analysis of both the acceleration obtained and the convergence of an FA algorithm derived from the simultaneous algebraic reconstruction technique (SART) algorithm of Andersen and Kak, the so-called FA-SART algorithm. [ABSTRACT FROM AUTHOR]

We present two iterative algorithms for solving real nonsymmetric and complex nonHermitian linear systems of equations and that were developed from variants of the nonsymmetric Lanczos method. In this paper, we give the theoretical background of the two iterative methods and discuss their main computational aspects. Using a large number of numerical experiments, we analyze their convergence properties, and we also compare them with other popular nonsymmetric iterative solvers in use today. [ABSTRACT FROM AUTHOR]

Anisotropic diffusion methods are well known for giving good qualitative results for image denoising. This paper gives a review of the anisotropic diffusion methodology and its application to image denoising. We propose a fixed-point iteration using a multigrid solver to solve a regularized anisotropic diffusion equation, which is not only well-posed, but also has a nontrivial steady-state solution. A new regularization parameter-choice method (Brent-NCP), combining Brent's method and the normalized cumulative periodogram information of the misfit, is also introduced. We test our algorithm on several common test images with different noise levels. The experimental results demonstrate the effectiveness of the anisotropic diffusion with a multigrid approach and the broad applicability of the Brent-NCP parameter-choice algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study the use of alternating direction algorithms for several ℓ1-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis pursuit denoising problems of both unconstrained and constrained forms, and others. We present and investigate two classes of algorithms derived from either the primal or the dual form of ℓ1-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an ℓ1-problem into one having blockwise separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the augmented Lagrangian function of the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical experiments are performed, using randomized partial Walsh Hadamard sensing matrices, to demonstrate the versatility and effectiveness of the proposed approach. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points: (i) that algorithm speed should be evaluated relative to appropriate solution accuracy; and (ii) that when erroneous measurements possibly exist, the ℓ1-fidelity should generally be preferable to the ℓ2-fidelity. [ABSTRACT FROM AUTHOR]

In this paper two noniterative algorithms for locating small electromagnetic inclusions from the scattering amplitude at a fixed frequency are developed. In particular, a variety of numerical results is presented to highlight their potential and their limitations. [ABSTRACT FROM AUTHOR]

In this paper, we study some efficient numerical methods for parameter identifications in elliptic systems. The proposed numerical methods are conducted iteratively and each iteration involves only solving positive definite linear algebraic systems, although the original inverse problems are ill-posed and highly nonlinear. The positive definite systems can be naturally preconditioned with their corresponding block diagonal matrices. Numerical experiments are presented to illustrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper presents the results of a scalability study for a three-dimensional semi- coarsening multigrid solver on a distributed memory computer. In particular, we are interested in the scalability of the solver-how the solution time varies as both problem size and number of processors are increased. For an iterative linear solver, scalability involves both algorithmic issues and implementation issues. We examine the scalability of the solver theoretically by constructing a simple parallel model and experimentally by results obtained on an IBM SP. The results are compared with those obtained for other solvers on the same computer. [ABSTRACT FROM AUTHOR]

In this paper a recently proposed additive version of the algebraic multilevel iteration method for iterative solution of elliptic boundary value problems is studied. The method constructs a nearly optimal order parameter-free preconditioner, which is robust with respect to anisotropy and discontinuity of the problem coefficients. It uses a new strategy for approximating the blocks corresponding to ‘new’ basis functions on each discretization level. To cope with the difficulties arising from the anisotropy, the problem on the coarsest mesh is solved using a bordering technique with a special choice of bordering vectors. The aim is to find a parameter-free ‘black-box’ robust solver. The results are derived in the framework of a hierarchical basis, linear finite element discretization of an elliptic problem on arbitrary triangular meshes, and a hierarchical basis, bilinear finite element discretization on Cartesian meshes. A comparison of the method with some other iterative solution techniques is presented. Robustness and high efficiency of the proposed algorithm are demonstrated on several model-type problems.

The restarted GMRES algorithm proposed by Saad and Schultz [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869] is one of the most popular iterative methods for the solution of large linear systems of equations Ax = b with a nonsymmetric and sparse matrix. This algorithm is particularly attractive when a good preconditioner is available. The present paper describes two new methods for determining preconditioners from spectral information gathered by the Arnoldi process during iterations by the restarted GMRES algorithm. These methods seek to determine an invariant subspace of the matrix A associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved. [ABSTRACT FROM AUTHOR]