The preconditioned Gauss–Seidel method faster than the SOR method

In recent years, a number of preconditioners have been applied to linear systems [A.D. Gunawardena, S.K. Jain, L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154–156 (1991) 123–143; T. Kohno, H. Kotakemori, H. Niki, M. Usui, Improving modified Gauss–Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997) 113–123; H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+_Smax)$(I+S_{\mathit{max}})$, J. Comput. Appl. Math. 145 (2002) 373–378; H. Kotakemori, H. Niki, N. Okamoto, Accelerated iteration method for Z$Z$-matrices, J. Comput. Appl. Math. 75 (1996) 87–97; M. Usui, H. Niki, T.Kohno, Adaptive Gauss-Seidel method for linear systems, Internat. J. Comput. Math. 51(1994)119–125 [10]]. Since these preconditioners are constructed from the elements of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the n$n$th row of matrix A. In the present paper, in order to deal with this drawback, we propose a new preconditioner. In addition, the convergence and comparison theorems of the proposed method are established. Simple numerical examples are also given, and we show that the convergence rate of the proposed method is better than that of the optimum SOR.     

An hybrid method that improves the accessibility of Steffensen’s method

Steffensen’s method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen’s method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen’s method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.     

Multi-direction-based Nelder–Mead method

The original Nelder–Mead (NM) method tends to be used to optimize low-dimensional functions. This article provides a modified NM that has the capability of large-scale optimization. The modification of NM is characterized by (a) working with a population of points, (b) mining multiple search directions through two strategies – point-grouping and variable-centroid multi-direction (VCMD), thus giving rise to VCMD plus grouping (VCMD_g) and (c) introducing random coefficients into NM and performing mutation on best-points, producing a random NM (NM_r). The combination of NM_r and VCMD_g, NM_r-VCMD_g, is just the modified NM in this work.     

Integral analysis method – IAM

This paper presents the theoretical foundations of the new integral analysis method (IAM), and its application to a facility location problem. This methodology integrates the cardinal and ordinal criteria of combinatorial stochastic optimization problems in four stages: definition of the problem, cardinal analysis, ordinal analysis and integration analysis. The method uses the concepts of stochastic multicriteria acceptability analysis (SMAA), Monte Carlo simulation, optimization techniques and elements of probability. The proposed method (IAM) was used to determine optimal locations for the retail stores of a Colombian coffee marketing company.     

The Farrell–Hsiang method revisited

We present a sufficient condition for groups to satisfy the Farrell–Jones Conjecture in algebraic K-theory and L-theory. The condition is formulated in terms of finite quotients of the group in question and is motivated by work of Farrell–Hsiang.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

The ξ-compensator method in hydrodynamics

Some aspects of the Hamiltonian formalism for the dynamics of an inhomogeneous incompressible fluid relating to methods of calculating the Poisson brackets in definite problems are considered. As an example, the Poisson bracket of local velocities is calculated. The computational technique is based on the method of -compensators. It is shown that a previously established expression for this Poisson bracket can indeed be obtained from the general expression for {F, G} by the -compensator method.I. V. Kurchatov Institute of Atomic Energy, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 440–451, June, 1992.     

An extended Hamilton — Jacobi method

We develop a new method for solving Hamilton??s canonical differential equations. The method is based on the search for invariant vortex manifolds of special type. In the case of Lagrangian (potential) manifolds, we arrive at the classical Hamilton ?? Jacobi method.     

On an Aitken–Newton type method

We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is $\sqrt[5]{6},$ which is higher than the efficiency index of the Aitken or Newton methods. Monotone sequences are obtained for initial approximations farther from the solution, if they satisfy the Fourier condition and the nonlinear mapping satisfies monotony and convexity assumptions on the domain.     

Quadrature-free spline method for two-dimensional Navier-Stokes equation

In this paper, a quadrature-free scheme of spline method for two-dimensional Navier- Stokes equation is derived, which can dramatically improve the efficiency of spline method for fluid problems proposed by Lai and Wenston（2004）. Additionally, the explicit formulation for boundary condition with up to second order derivatives is presented. The numerical simulations on several benchmark problems show that the scheme is very efficient.     

Stein’s method on Wiener chaos

We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esséen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.     

Economical cascadic multigrid method (ECMG)

In this paper,an economical cascadic multigrid method is proposed.Compared with the usual cascadic multigrid method developed by Bornemann and Deuflhard,the new one requires less iterations on each level,especially on the coarser grids.Many operations can be saved in the new cascadic multigrid algorithms.The main ingredient is the control of the iteration numbers on the each level to preserve the accuracy without over iterations.The theoretical justification is based on the observations that the error reduction rate of an iteration scheme in terms of the smoothing property is no longer accurate while the iteration number is big enough.A new formulae of the error reduction rate is employed in our new algorithm.Numerical experiments are reported to support our theory.     

Convergence of the Lavrent’ev method

The convergence of the Lavrent’ev method, which is a well-known regularization method for integral equations of the first kind, is analyzed as applied to equations with arbitrary linear bounded operators. A theorem concerning necessary and sufficient conditions for this convergence is proved. It is shown that these conditions are satisfied for two classes of integral equations that do not possess the properties required by the classical Lavrent’ev method.     

Equivariant isospectrality and Sunada’s method

We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada’s method. We also observe that if a good orbifold ${\mathcal{O}}We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada’s method. We also observe that if a good orbifold O{\mathcal{O}} and a smooth manifold M are isospectral, then they cannot admit non-trivial finite Riemannian covers M₁ ?O{M_1 \to\mathcal{O}} and M ₂ → M where M ₁ and M ₂ are isospectral manifolds.     

Newton’s method on Lie groups

The purpose of this paper is to study the strong convergence of fixed points for a family of demi-continuous pseudo-contractions by hybrid projection algorithms in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others.