Mathematics

This paper is concerned with the development of the Newton-Arithmetic Mean method for large systems of nonlinear equations with block-partitioned jacobian matrix. This method is well suited for implementation on a parallel computer; its degree of decomposition is very high. The convergence of the method is analysed for the class of systems whose jacobian matrix satisfies an affine invariant Lipschitz condition. An estimation of the radius of the attraction ball is given. Special attention is reserved to the case of weakly nonlinear systems. A numerical example highlights some peculiar properties of the method.

Given n pairwise distinct and arbitrarily spaced points Pi in a domain ~D of the x-y plane and n real numbersf., consider the problem of computing a bivariate functionf(x, y) of class C I in ~ whose values in Pi are exactly j~, i = 1,..., n, and whose first or second order partial derivatives satisfy appropriate equality and inequality constraints on a given set ofp points Qt in (D.

This paper concerns with the problem of solving optimal control problems by means of nonlinear programming methods. The technique employed to obtain a mathematical programming problem from an optimal control problem is explained and the Newton interior-point method, chosen for its solution, is presented with special regard to the choice of the involved parameters. An analysis of the behaviour of the method is reported on four optimal control problems, related to improving water quality in an aeration process and to the study of diffusion convection processes.

In this paper we consider a two-stage iterative method for solving weakly nonlinear systems generated by the discretization of semilinear elliptic boundary value problems. This method is well suited for implementation on parallel computers. Theorems about the convergence and the monotone convergence of the method are proved. An application of the method for solving real practical problems related to the study of reaction-diffusion processes and of interacting populations is described.

In this paper we consider a parametrized system of weakly nonlinear equations which corresponds to a nonlinear elliptic boundary-value problem with zero source, homogeneous boundary conditions and a positive parameter in the linear term. Positive solutions of this system are of interest to us. A characterization of this positive solution is given. Such a solution is determined by the Modified Newton-Arithmetic Mean method. This method is well suited for implementation on parallel computers. A theorem about the monotone convergence of the method is proved. An application of the method for solving a real practical problem related to the study of interacting populations is described.

This paper deals with the solution of nonlinear programming problems arising from elliptic control problems by an interior point scheme. At each step of the scheme, we have to solve a large scale symmetric and indefinite system; inner iterative solvers, with adaptive stopping rule, can be used in order to avoid unnecessary inner iterations, especially when the current outer iterate is far from the solution.

This paper concerns with the numerical evaluation of the variable projection method for quadratic programming problems in three data analysis applications. The three applications give rise to large-scale quadratic programs with remarkable differences in the Hessian definition and/or in the structure of the constraints Keywords: Large-scale quadratic programming, variable projection method, support vector machines, image restoration problem, bivariate interpolation problem.

A backward error analysis of the direct elimination method for linear equality constrained least squares problems is presented. It is proved that the solution computed by the method is the exact solution of a perturbed problem and bounds for data perturbations are given. The numerical stability of the the method is related to the way in which the constraints are used to eliminate variables and these theoretical conclusions are confirmed by a numerical example.

In this paper we consider the application of additive operator splitting methods for solving a finite difference nonlinear system of the form F (u) = (I − τ A(u))u − w = 0 generated by the discretization of two dimensional diffusion-convection problems with Neumann boundary conditions. Existence and uniqueness of a solution of this system has been proved under standard assumptions on the matrix A(u) and the source term w. Using the fact that the matrix A(u) can be decomposed into different splittings, we develop a nonlinear Arithmetic Mean method and a two-stage iterative method (a fixed-point-Arithmetic Mean method) for solving the system above. The convergence of these methods has been analyzed. Numerical experiments show that the fixed-point-Arithmetic Mean method is rapidly convergent when the diffusion coefficient is weakly nonlinear.

SUNTO -L'applicazione di noti metodi che utilizzano funzioni di tipo blending per la costruzione di funzioni bivariate C 1 per l'interpolazione di dati, richiede la conoscenza delle derivate parziali del primo ordine ai vertici di una triangolazione sottostante.

In this paper we consider the Newton-iterative method for solving weakly nonlinear finite difference systems of the form F (u) = Au + G(u) = 0, where the jacobian matrix G ′ (u) satisfies an affine invariant Lipschitz condition. We also consider a modification of the method for which we can improve the likelihood of convergence from initial approximations that may be outside the attraction ball of the Newton-iterative method. We analyse the convergence of this damped method in the framework of the line search strategy. Numerical experiments on a diffusion-convection problem show the effectiveness of the method.

This paper deals with the analysis and the solution of the Karush-Kuhn-Tucker (KKT) system that arises at each iteration of an Interior-Point (IP) method for minimizing a nonlinear function subject to equality and inequality constraints. This system is generally large and sparse and it can be reduced so that the coefficient matrix is still sparse, symmetric and indefinite, with size equal to the number of the primal variables and of the equality constraints. Instead of transforming this reduced system to a quasidefinite form by regularization techniques used in available codes on IP methods, under standard assumptions on the nonlinear problem, the system can be viewed as the optimality Lagrange conditions for a linear equality constrained quadratic programming problem, so that Hestenes multipliers' method can be applied. Numerical experiments on elliptic control problems with boundary and distributed control show the effectiveness of Hestenes scheme as inner solver for IP methods.

This paper concerns with the solution of optimal control problems by means of nonlinear programming methods. The direct transcription, by finite difference approximation, of the optimal control problem into a finite dimensional nonlinear programming problem is described. An iterative procedure for the solution of this nonlinear program is presented. An extensive numerical analysis of the behaviour of the method is reported on boundary control and distributed control problems with boundary conditions of Dirichlet or Neumann or mixed type.

This work concerns with the solution of optimal control problems by means of nonlinear programming methods. The control problem is transcribed into a finite dimensional nonlinear programming problem by finite difference approximation. An iterative procedure for the solution of this nonlinear program is presented. An extensive numerical analysis of the behaviour of the method is reported on boundary control and distributed control problems with boundary conditions of Dirichlet or Neumann or mixed type.

In this work we analyze the Newton Interior-Point method presented in [El-Bakry, Tapia et al., J.Optim. Theory Appl., 89, 1996] for solving constrained systems of nonlinear equations arising from the Karush-Kuhn-Tucker conditions for nonlinear programming problems (KKT systems). More specifically, we consider a variant of the Newton Interior-Point method for KKT systems in in which the possibility to adaptively modify the perturbation parameter and the accuracy of the solution of the perturbed Newton equation, when one is still far away from the solution at an early stage, will moderate the difficulty of solving the KKT system. Using the results in [Durazzi, J.Optim. Theory Appl., 104, 2000] and [Bellavia, J.Optim. Theory Appl., 96, 1998], it is possible to establish a global convergence theory for this modified method. The method proposed in this work can be used effectively for large-scale optimization problems with structured sparsity, which occur, for instance, from the discretization of optimal control problems with inequality constraints.

This report concerns with the numerical solution of nonlinear reaction diffusion equations at the steady state in a two dimensional bounded domain supplemented by suitable boundary conditions. When we use finite differences or finite element discretizations, the nonlinear diffusion equation subject to Dirichlet boundary conditions can be transcribed into a nonlinear system of algebraic equations. In the case of finite differences, the matrix that arises from the discretization of the diffusion (and/or convection) term satisfies properties of monotonicity. This report is divided into two parts (chapters): the first part deals with the solution of a weakly nonlinear reaction diffusion equation while in the second part, the solution of a strongly nonlinear reaction diffusion equation is computed by an iterative procedure that "lags" the diffusion term. This procedure is called Lagged Diffusivity Functional Iteration (LDFI)-procedure.

There exists a considerably body of literature on the development, analysis and implementation of multiplicative and additive operator splitting methods for solving large and sparse systems of finite difference equations arising from the discretization of partial differential equations. In this note, we will review the Newton-Arithmetic Mean and the Modified Newton-Arithmetic Mean methods for solving nonlinear and weakly nonlinear systems of difference equations arising from the discretization of diffusion-convection problems.

Domain Decomposition and Operator Splitting are powerful concepts used in Parallel Computation and Large Scale Scientific Computation for the design of effective numerical methods. In this note, we will review various additive operator splitting methods for solving on parallel computers a large class of semidiscrete nonlinear diffusion problems.