Analysis of the cell boundary element methods for convection dominated convection-diffusion equations

We introduce two kinds of the cell boundary element (CBE) methods for convection dominated convection-diffusion equations: one is the CBE method with the exact bubble function and the other with inexact bubble functions. The main focus of this paper is on inexact bubble CBE methods. For inexact bubble CBE methods we introduce a family of numerical methods depending on two parameters, one for control of interior layers and the other for outflow boundary layers. Stability and convergence analysis are provided and numerical tests for inexact bubble CBEs with various choices of parameters are presented.     

A circular splitting search algorithm for systems of complex equations

The results of Moore and Jones on finding safe starting regions for iterative methods for real functions are extended to finding safe starting discs for iteration methods for complex functions in one or more variables. The methods is based on the splitting of circular discs. A simple numerical example is given.     

Methods of aggregation

We study a class of methods for accelerating the convergence of iterative methods for solving linear systems. The methods proceed by replacing the given linear system with a derived one of smaller size, the aggregated system. The solution of the latter is used to accelerate the original iterative process. The construction of the aggregated system as well as the passage of information between it and the original system depends on one or more approximations of the solution of the latter. A number of variants are introduced, estimates of the acceleration are obtained, and numerical experiments are performed. The theory and computations show the methods to be effective.     

On the numerical treatment of nonconvex energy problems of mechanics

The present paper presents three numerical methods devised for the solution of hemivariational inequality problems. The theory of hemivariational inequalities appeared as a development of variational inequalities, namely an extension foregoing the assumption of convexity that is essentially connected to the latter. The methods that follow partly constitute extensions of methods applied for the numerical solution of variational inequalities. All three of them actually use the solution of a central convex subproblem as their kernel. The use of well established techniques for the solution of the convex subproblems makes up an effective, reliable and versatile family of numerical algorithms for large scale problems. The first one is based on the decomposition of the contigent cone of the (super)-potential of the problem into convex components. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. The third one is based on the fact that nonconvexity in mechanics is closely related to irreversible effects that affect the Hessian matrix of the respective (super)-potential. All three methods are applied to solve the same problem and the obtained results are compared.     

Fault detection based on fractional order models: Application to diagnosis of thermal systems

The aim of this paper is to propose diagnosis methods based on fractional order models and to validate their efficiency to detect faults occurring in thermal systems. Indeed, it is first shown that fractional operator allows to derive in a straightforward way fractional models for thermal phenomena. In order to apply classical diagnosis methods, such models could be approximated by integer order models, but at the expense of much higher involved parameters and reduced precision. Thus, two diagnosis methods initially developed for integer order models are here extended to handle fractional order models. The first one is the generalized dynamic parity space method and the second one is the Luenberger diagnosis observer. Proposed methods are then applied to a single-input multi-output thermal testing bench and demonstrate the methods efficiency for detecting faults affecting thermal systems.     

Numerical simulation of stochastic PDEs for excitable media

We discuss the numerical solution of a number of stochastic perturbations of the Barkley model of excitable media, widely used in the study of spiral waves. Two numerical methods are considered for solving this equation, one based on Barkley's original formulation and one based on spectral methods. It is found to be beneficial to modify the nonlinearity describing the reaction kinetics. An efficient method of approximating the Wiener process is presented. The effectiveness of the methods depends on the stochastic PDE under consideration.     

Composition methods in the presence of small parameters

We derive numerical methods for arbitrary small perturbations of exactly solvable differential equations. The methods, based in one instance on Gaussian quadrature, are symplectic if the system is Hamiltonian and are asymptotically more accurate than previously known methods.     

On the final steps of Newton and higher order methods

The Newton method is one of the most used methods for solving nonlinear system of equations when the Jacobian matrix is nonsingular. The method converges to a solution with Q-order two for initial points sufficiently close to the solution. The method of Halley and the method of Chebyshev are among methods that have local and cubic rate of convergence. Combining these methods with a backtracking and curvilinear strategy for unconstrained optimization problems these methods have been shown to be globally convergent. The backtracking forces a strict decrease of the function of the unconstrained optimization problem. It is shown that no damping of the step in the backtracking routine is needed close to a strict local minimizer and the global method behaves as a local method. The local behavior for the unconstrained optimization problem is investigated by considering problems with two unknowns and it is shown that there are no significant differences in the region where the global method turn into a local method for second and third order methods. Further, the final steps to reach a predefined tolerance are investigated. It is shown that the region where the higher order methods terminate in one or two iteration is significantly larger than the corresponding region for Newton’s method.     

Some variants of Hansen-Patrick method with third and fourth order convergence

Based on Hansen-Patrick method [E. Hansen, M. Patrick, A family of root finding methods, Numer. Math. 27 (1977) 257-269], we derive a two-parameter family of methods for solving nonlinear equations. All the methods of the family have third-order convergence, except one which has the fourth-order convergence. In terms of computational cost, all these methods require evaluations of one function, one first derivative and one second derivative per iteration. Numerical examples are given to support that the methods thus obtained are competitive with other robust methods of similar kind. Moreover, it is shown by way of illustration that the present methods, particularly fourth-order method, are very effective in high precision computations.     

Fast algorithm for geometrical coding of digital images

A fast algorithm for geometrical coding is presented in the paper. It allows one to perform the image analysis by methods of differential geometry rather than widespread methods based on gradient and Hessian calculations. Nowadays, one of the most important aspects in image analysis is the speed of computations. The algorithm for geometrical coding is comparable or faster than traditional algorithms of image analysis and in many cases it gives better results. Several examples of algorithm working results for contour detection are presented.     

Multiplier convergence in trust-region methods with application to convergence of decomposition methods for MPECs

We study piecewise decomposition methods for mathematical programs with equilibrium constraints (MPECs) for which all constraint functions are linear. At each iteration of a decomposition method, one step of a nonlinear programming scheme is applied to one piece of the MPEC to obtain the next iterate. Our goal is to understand global convergence to B-stationary points of these methods when the embedded nonlinear programming solver is a trust-region scheme, and the selection of pieces is determined using multipliers generated by solving the trust-region subproblem. To this end we study global convergence of a linear trust-region scheme for linearly-constrained NLPs that we call a trust-search method. The trust-search has two features that are critical to global convergence of decomposition methods for MPECs: a robustness property with respect to switching pieces, and a multiplier convergence result that appears to be quite new for trust-region methods. These combine to clarify and strengthen global convergence of decomposition methods without resorting either to additional conditions such as eventual inactivity of the trust-region constraint, or more complex methods that require a separate subproblem for multiplier estimation.      

A new general eighth-order family of iterative methods for solving nonlinear equations

In this work, we present a family of iterative methods for solving nonlinear equations. It is proved that these methods have convergence order 8. These methods require three evaluations of the function, and only use one evaluation of the first derivative per iteration. The efficiency of the method is tested on a number of numerical examples. On comparison with the eighth-order methods, the iterative methods in the new family behave either similarly or better for the test examples.     

Numerical solution of the elastic body-plate problem by nonoverlapping domain decomposition type techniques

The purpose of this paper is to provide two numerical methods for solving the elastic body-plate problem by nonoverlapping domain decomposition type techniques, based on the discretization method by Wang. The first one is similar to an older method, but here the corresponding Schur complement matrix is preconditioned by a specific preconditioner associated with the plate problem. The second one is a ``displacement-force' type Schwarz alternating method. At each iteration step of the two methods, either a pure body or a pure plate problem needs to be solved. It is shown that both methods have a convergence rate independent of the size of the finite element mesh.

     

Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

In this paper, we introduce two new numerical methods for solving a variational inequality problem involving a monotone and Lipschitz continuous operator in a Hilbert space. We describe how to incorporate a regularization term depending on a parameter in the projection method and then establish the strong convergence of the resulting iterative regularization projection methods. Unlike known hybrid methods, the strong convergence of the new methods comes from the regularization technique. The first method is designed to work in the case where the Lipschitz constant of cost operator is known, whereas the second one is more easily implemented without this requirement. The reason is because the second method has used a simple computable stepsize rule. The variable stepsizes are generated by the second method at each iteration and based on the previous iterates. These stepsizes are found with only one cheap computation without line-search procedure. Several numerical experiments are implemented to show the computational effectiveness of the new methods over existing methods.     

DISCRETE MINUS ONE NORM LEAST-SQUARES FOR THE STRESS FORMULATION OF LINEAR ELASTICITY WITH NUMERICAL RESULTS

This paper studies the discrete minus one norm least-squares methods for the stress formulation of pure displacement linear elasticity in two dimensions. The proposed leastsquares functional is defined as the sum of the L^2 -and H^-l -norms of the residual equations weighted appropriately. The minus one norm in the functional is replaced by the discrete minus one norm and then the discrete minus one norm least-squares methods are analyzed with various numerical results focusing on the finite element accuracy and multigrid convergence performances.     

Elimination of accuracy saturation in regularizing algorithms

A general approach to modifying well-known methods for solving linear ill-posed problems is proposed. The approach enables one to exclude possible “saturation of accuracy” in methods on classes of sourcewise representable solutions. As a result, the modified methods possess an optimal order of accuracy on each of these classes. Numerical examples for the solution of 1D and 2D inverse problems are presented.     

On new higher order families of simultaneous methods for finding polynomial zeros

Two one parameter families of iterative methods for the simultaneous determination of simple zeros of algebraic polynomials are presented. The construction of these families are based on a one parameter family of the third order for finding a single root of nonlinear equation f(x)=0. Some previously derived simultaneous methods can be obtained from the presented families as special cases. We prove that the local convergence of the proposed families is of the order four. Numerical results are included to demonstrate the convergence properties of considered methods.     

A family of numerical methods for diffusion and reaction–diffusion equations

A family of methods is developed for the numerical solution of second-order parabolic partial differential equations in one space dimension. The methods are second-, third-, or fourth-order accurate in time; five of them are seen to be L₀-stable in the sense of Gourlay and Morris, while the sixth is seen to be A₀-stable, The methods are tested on four problems from the literature, three diffusion problems and one reaction–diffusion problem.     

New efficient derivative free family of seventh-order methods for solving systems of nonlinear equations

We present a three-step two-parameter family of derivative free methods with seventh-order of convergence for solving systems of nonlinear equations numerically. The proposed methods require evaluation of two central divided differences and inversion of only one matrix per iteration. As a result, the proposed family is more efficient as compared with the existing methods of same order. Numerical examples show that the proposed methods produce approximations of greater accuracy and remarkably reduce the computational time for solving systems of nonlinear equations.     

New Conjugacy Conditions and Related Nonlinear Conjugate Gradient Methods

Conjugate gradient methods are a class of important methods for unconstrained optimization, especially when the dimension is large. This paper proposes a new conjugacy condition, which considers an inexact line search scheme but reduces to the old one if the line search is exact. Based on the new conjugacy condition, two nonlinear conjugate gradient methods are constructed. Convergence analysis for the two methods is provided. Our numerical results show that one of the methods is very efficient for the given test problems. Accepted 15 September 2000. Online publication 8 December 2000.