Convergence Of Iterative Methods Research Articles

Об одной вычислительной реализации блочного метода Гаусса-Зейделя для нормальных систем уравнений

Статья посвящена модификации блочного варианта метода Гаусса-Зейделя для нормальных систем уравнений, который является одним из достаточно эффективных методов решения, в общем случае переопределенных, систем линейных алгебраических уравнений большой размерности. Основным недостатком методов, основанных на нормальных системах уравнений, является тот факт, что число обусловленности нормальной системы равно квадрату числа обусловленности исходной задачи. Этот факт отрицательно влияет на скорость сходимости итерационных методов, основанных на нормальных системах уравнений. Для повышения скорости сходимости итерационных методов, основанных на нормальных системах уравнений, при решении плохо обусловленных задач в настоящее время используются различные варианты предобуславливателей, позволяющие снизить число обусловленности исходной системы уравнений. Однако универсального предобуславливателя для всех задач не существует. Одним из эффективных подходов, позволяющих повысить скорость сходимости итерационного метода Гаусса-Зейделя для нормальных систем уравнений, является использование его блочного варианта. Недостатком блочного метода Гаусса-Зейделя для нормальных систем является тот факт, что на каждой итерации необходимо вычислять псевдообратную матрицу. Известно, что нахождение псевдообратной матрицы является достаточно сложной вычислительной процедурой. В настоящей работе предлагается процедуру псевдообращения матрицы заменить на задачу решения нормальных систем уравнений методом Холецкого. Нормальные уравнения, возникающие на каждой итерации метода Гаусса-Зейделя, имеют сравнительно невысокую размерность по сравнению с исходной системой. Приводятся результаты вычислительных экспериментов, демонстрирующие эффективность предлагаемого подхода.

Numerical Homogenization of Elliptic Multiscale Problems by Subspace Decomposition

Numerical homogenization tries to approximate solutions of elliptic partial differential equations with strongly oscillating coefficients by the solution of localized problems over small subregions. We develop and analyze a rapidly convergent iterative method for numerical homogenization that shares this feature with existing approaches and is modeled after the Schwarz method. The method is highly parallelizable and of lower computational complexity than comparable methods that as ours do not make explicit or implicit use of a scale separation.

WIDECARS spectra fitting in a premixed ethylene–air flame

WIDECARS measures temperature and mole fractions of most of the major species in ethylene–air flames. One of the issues in implementing this technique is fitting the experimental spectra to theory to obtain flame conditions (temperature, species mole fractions). Individual spectra contain many species resonances, and theory is slow to compute. Libraries of precalculated spectra can be used, but a library of sufficient density for accurate interpolation is large given the many variables. A new fitting algorithm is presented which utilizes a less‐dense library, and additional spectra are calculated during fitting to maintain accuracy. The iterative convergence method converts the problem of minimizing fit error, which converges slowly, to a zero finding problem, which converges reliably, rapidly, and accurately to best fit. Various practical fitting issues, such as the effects of dye laser mode noise and variability, phase‐matching efficiency, and shifts of the spectrum on the spectrometer are addressed. The technique is demonstrated in the analysis of experimental measurements in an equivalence ratio 2.1 ethylene–air flame above the surface of a McKenna burner. Precision errors because of experimental and fitting effects are discussed. Copyright © 2015 John Wiley & Sons, Ltd.

Relationships between different types of initial conditions for simultaneous root finding methods

The construction of initial conditions of an iterative method is one of the most important problems in solving nonlinear equations. In this paper, we obtain relationships between different types of initial conditions that guarantee the convergence of iterative methods for simultaneously finding all zeros of a polynomial. In particular, we show that any local convergence theorem for a simultaneous method can be converted into a convergence theorem with computationally verifiable initial conditions which is of practical importance. Thus, we propose a new approach for obtaining semilocal convergence results for simultaneous methods via local convergence results.

Convergence for iterative methods on Banach spaces of a convergence structure with applications to fractional calculus

We present a semilocal convergence for some iterative methods on a Banach space with a convergence structure to locate zeros of operators which are not necessarily Frechet-differentiable as in earlier studies such as (Argyros in J Approx Theory Appl 9(1):1–10, 1993; Argyros in Southwest J Pure Appl Math 1:32–38, 1995; Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008; Meyer in Numer Funct Anal Optim 13(5 and 6):463–473, 1992). This way we expand the applicability of these methods. If the operator involved is Frechet-differentiable one approach leads to more precise error estimates on the distances involved than before (Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008; Meyer in Numer Funct Anal Optim 13(5 and 6):463–473, 1992) and under the same hypotheses. Special cases are presented and some examples from fractional calculus.

Weakly-supervised scene parsing with multiple contextual cues

Scene parsing, fully labeling an image with each region corresponding to a label, is one of the core problems of computer vision. Previous methods to this problem usually rely on patch-level models trained from well labeled data. In this paper, we propose a weakly-supervised scene parsing algorithm that semantically parses a collection of images with multi-label, which is guided by the top-down category models and bottom-up local patch contexts across images that closely related segments usually have similar labels. Images are segmented to patches on multi-level and the contextual relations of patches are discovered via sparse representation by ℓ1 minimization, based on which a graph is constructed. The multi-level spatial context of patches is also embedded in the graph, based on which image-level labels can be propagated to segments optimally. The contextual patch labeling process is formulated in an optimization framework and solved by a convergent iterative method. The category models are learned from the decomposed label representations of the image set and applied to the segments. Final labeling is obtained by combining all the information on pixel level. The effectiveness of the proposed method is demonstrated in experiments on two benchmark datasets and comparisons are taken.

Increasing the order of convergence for iterative methods to solve nonlinear systems

For solving nonlinear systems, we introduce a technique that improves the order of convergence of any given iterative method which uses the extended Newton iteration as a predictor. Based on a given iterative method of order $$p\ge 2$$pź2, a new method of order $$p+2$$p+2 is developed by introducing just one evaluation of the function. We obtain some new methods with higher order of convergence by applying this procedure to some known methods. Computational efficiency in the general form is discussed and comparisons are made between these new methods and the ones from which have been derived. We also perform several numerical tests to show the asymptotic behaviors which confirm the theoretical results.

New convergence proofs of modulus-based synchronous multisplitting iteration methods for linear complementarity problems

Modulus-based synchronous multisplitting iteration methods were recently proposed for solving linear complementarity problems. We give a much simpler approach to prove the convergence of these iteration methods when the system matrix is an H+-matrix. Moreover, this idea can also be applied to prove the convergence of two-step modulus-based synchronous multisplitting iteration methods, which avoids the construction of the irreducible system matrix and the introduction of the infinite norm.

Parallel Hybrid Iterative Methods for Variational Inequalities, Equilibrium Problems, and Common Fixed Point Problems

In this paper, we propose two strongly convergent parallel hybrid iterative methods for finding a common element of the set of fixed points of a family of asymptotically quasi ϕ-nonexpansive mappings, the set of solutions of variational inequalities and the set of solutions of equilibrium problems in uniformly smooth and 2-uniformly convex Banach spaces. A numerical experiment is given to verify the efficiency of the proposed parallel algorithms.

Mutual Embeddings

This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.

Special iterative methods for solution of the steady Convection-Diffusion-Reaction equation with dominant convection

Iterative methods based on skew-symmetric splitting of initial matrix, arising from central finite-difference approximation of steady convection-diffusion-reaction (CDR) equation in 2-D domain are considered. The property of obtained large sparse nonsymmetric linear system Au = f is investigated. A new class of triangular and product triangular skew-symmetric iterative methods is presented. Sufficient conditions of convergence for iterative methods of solution CDR with variable coefficient of reaction and dominant convection are obtained. The results of numerical experiments for the solution of a two-dimensional CDR equation are presented. The uniform grid, central differences for the first derivatives, natural ordering of points, Peclet numbers Pe = 103, 104, 105, variable coefficient of reaction and different velocity coefficients have been used.

A Novel Iterative Method for Polar Decomposition and Matrix Sign Function

We define and investigate a globally convergent iterative method possessing sixth order of convergence which is intended to calculate the polar decomposition and the matrix sign function. Some analysis of stability and computational complexity are brought forward. The behaviors of the proposed algorithms are illustrated by numerical experiments.

Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier–Stokes equations

Some finite element iterative methods related to viscosities are designed to solve numerically the steady 2D/3D Navier–Stokes equations. The two-level finite element iterative methods are designed to solve numerically the steady 2D/3D Navier–Stokes equations for a large viscosity ν such that a strong uniqueness condition holds. The two-level finite element iterative methods consist of using the Stokes, Newton and Oseen iterations of m times on a coarse mesh with mesh size H and computing the Stokes, Newton and Oseen correction of one time on a fine grid with mesh size h≪H. Moreover, the one-level Oseen finite element iterative method based on a fine mesh with a small mesh size is designed to solve numerically the steady 2D/3D Navier–Stokes equations for small viscosity ν such that a weak uniqueness condition holds. The uniform stability and convergence of these methods with respect to ν and mesh sizes h and H and iterative times m are provided.

Iterative methods for solving variational inequalities of the theory of soft shells

The convergence of iterative methods for solving variational inequalities with monotone-type operators in Banach spaces is studied. Such inequalities arise in the description of deformation processes of soft rotational network shells. Certain properties of these operators, such as coercivity, potentiality, bounded Lipschitz continuity, pseudomonotonicity, and inverse strong monotonicity, are determined. An iterative method for solving these variational inequalities is proposed, its convergence is investigated, and the boundedness of the iterative sequence is proved. Moreover, it is proved that any weakly convergent subsequence of the iterative sequence converges to a solution of the original variational inequality.

On iterative methods for some elliptic equations with nonlocal conditions

The iterative methods for the solution of the system of the difference equations derived from the elliptic equation with nonlocal conditions are considered. The case of the matrix of the difference equations system being the M-matrix is investigated. Main results for the convergence of the iterative methods are obtained considering the structure of the spectrum of the difference operators with nonlocal conditions. Furthermore, the case when the matrix of the system of difference equations has only positive eigenvalues was investigated. The survey of results on convergence of iterative methods for difference problem with nonlocal condition is also presented. 1The research was partially supported by the Research Council of Lithuania (grant No. MIP-051/2011). 2The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014).

Numerical investigation of the dynamics for low tension marine cables

This paper presents a numerical investigation into the dynamics of marine cables which are extensively used in offshore industry. In this numerical study, the Euler-Bernoulli beam model is adopted to develop the governing equations of the cable. Bending stiffness is considered to cope with the low tension problem in local area of towing cable, and thus a more accurate solution with the consideration of the axial elongation can be given. The derived strongly-coupled and nonlinear governing equations are solved by a second-order accurate, implicit, and large time step stable central finite difference method. The quadratically convergent Newton-Raphson iteration method is applied to solving the discrete nonlinear algebraic equations. Then a towed array sonar system (TASS) problem is studied. The numerical solutions agree reasonably well with the experimental data and the simulated results of the references. The specified program of the present paper shows great robustness with high efficiency.

Some Approximated Solutions For Operator Equations By Using Frames

In this paper we give some approximated solutions for an operator equation \(Lu=f\) where \(L: H\rightarrow H\) is a bounded and self adjoint operator on a separable Hilbert space \(H\). We use frames in order to precondition the linear equation so that convergence of iterative methods is improved. Also we find an exact solution associated to a frame and then we seek an approximated solution in a finite dimensional subspace of \(H\) that is generated by a finite frame sequence.

On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric jacobian matrices

Preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) method is an unconditionally convergent iterative method for solving large sparse complex symmetric systems of linear equations. By making use of the PMHSS iteration as the inner solver to approximately solve the Newton equations, we establish a modified Newton-PMHSS method for solving large systems of nonlinear equations. Motivated by the idea in Chen et al. (2014), we analyze the local convergence properties under the Holder continuous condition, which is weaker than the assumptions used in modified Newton-HSS method proposed by Wu and Chen (2013). Numerical results are given to confirm the effectiveness of our method.

On-surface radiation condition for multiple scattering of waves

The formulation of the on-surface radiation condition (OSRC) is extended to handle wave scattering problems in the presence of multiple obstacles. The new multiple-OSRC simultaneously accounts for the outgoing behavior of the wave fields, as well as, the multiple wave reflections between the obstacles. Like boundary integral equations (BIE), this method leads to a reduction in dimensionality (from volume to surface) of the discretization region. However, as opposed to BIE, the proposed technique leads to boundary integral equations with smooth kernels. Hence, these Fredholm integral equations can be handled accurately and robustly with standard numerical approaches without the need to remove singularities. Moreover, under weak scattering conditions, this approach renders a convergent iterative method which bypasses the need to solve single scattering problems at each iteration.Inherited from the original OSRC, the proposed multiple-OSRC is generally a crude approximate method. If accuracy is not satisfactory, this approach may serve as a good initial guess or as an inexpensive pre-conditioner for Krylov iterative solutions of BIE.

On the guaranteed convergence of a cubically convergent Weierstrass-like root-finding method

Initial conditions that provide guaranteed and fast convergence of the Weierstrass-like cubically convergent iterative method for the simultaneous determination of all simple zeros of a polynomial are considered. It is proved that this method is convergent under suitable conditions stated in the spirit of Smale's point estimation theory. The proposed convergence conditions are computationally verifiable since they depend only on initial approximations and the degree of a given polynomial, which is of practical importance.