Use Of Iterative Methods Research Articles

Nonconvex process optimization

Difficulties associated with nonconvexity in successive quadratic programming (SQP) methods are studied. It is shown that projected indefiniteness of the Hessian matrix of the Lagrangian function can (i) place restrictions on the order in which inequalities can be added or deleted from the active set, (ii) generate redundant active sets whose resolution is nontrivial, (iii) give rise to quadratic programming (QP) subproblems that have multiple Kuhn-Tucker points, and (iv) produce nondescent directions in the SQP method that can lead to failure. Related issues concerned with the use of feasible or infeasible starting points for the iterative quadratic programs, forcing positive definiteness to ensure convexity, and using iterative methods to solve the linear Kuhn-Tucker conditions associated with the QP subproblems are also studied. A new active set strategy that (i) monitors projected indefiniteness to guide the addition of constraints to the active set, (ii) permits line searching for negative values of the line search parameter, and (iii) does not necessarily delete active constraints with incorrect Kuhn-Tucker multipliers is proposed. Constraint redundancy is circumvented using an algorithm that identifies all nontrivial redundant subsets of smallest size and determines which, if any, exchanges are justified. Nondescent in the NLP's is resolved using a linear programming (LP)-based trust region method that guarantees descent regardless of merit function. It is also shown that there is no justification for using feasible starting points at the QP level of the calculations, that forcing positive definiteness to ensure convexity can cause termination at undesired solutions, and that the use of iterative methods to solve the linear Kuhn-Tucker equations for the QP's can cause a deterioration in numerical performance. Many small chemical process examples are used to highlight difficulties so that geometric illustrations can be used while heat exchange network design and distillation operations examples are used to show that these same difficulties carry over to larger problems.

Fast Iterative Solution of Carrier Continuity Equations for Three-Dimensional Device Simulation

In this paper the use of iterative methods for the solution of the carrier continuity equations in three-dimensional semiconductor device simulators is summarized. An overview of the derivation of the linear systems from the basic stationary semiconductor device equations is given and the algebraic properties of the nonsymmetric coefficient matrices are discussed. Results from the following classes of iterative methods are presented: The classical conjugate gradient (CG), the symmetrized conjugate gradient (SCG), the generalized minimum residual (GMRES), and the conjugate gradient squared (CGS) method. Preconditioners of incomplete factorization type with partial fill-in are considered. High performance implementations for these algorithms on vector, concurrent, and vector–concurrent computers are presented.

Quasi-TEM analysis of the generalized microstrip line by using FFT and iterative methods

A quasi-TEM analysis of a generic microstrip line is carried out by the combined use of iterative methods and FFT. Asymptotic extraction techniques are used in the determination of the Green's function, allowing reduction of memory storage requirements and CPU time. Several iterative algorithms are compared for solving the final system of convolution linear equations. Numerical results are presented and compared with published data.

A general numerical method for the solution of gravity wave problems. Part 1: 2D steep gravity waves in shallow water

AbstractIn this paper, 2D steep gravity waves in shallow water are used to introduce and examine a new kind of numerical method for the solution of non‐linear problems called the finite process method (FPM). On the basis of the velocity potential function and the FPM, a numerical method for 2D non‐linear gravity waves in shallow water is described which can be applied to solve 3D problems, e.g. the wave resistance of a ship moving in deep or shallow water. The convergence is examined and a comparison with the results of other authors is made.The FPM can successfully avoid the use of iterative methods and therefore can overcome the disadvantages and limitations of such methods. In contrast to iterative methods, the FPM is insensitive to the selection of the initial solution and the number of unknowns. The basic idea of the FPM can be used to solve other non‐linear problems. Its disadvantage is that much more CPU time is needed to obtain a sufficiently accurate result.

On the Use of Iteration Methods for Approximating the Natural Logarithm

On the Use of Iteration Methods for Approximating the Natural Logarithm

On the Use of Iteration Methods for Approximating the Natural Logarithm

On the Use of Iteration Methods for Approximating the Natural Logarithm

Iterative Methods for Solving Bordered Systems with Applications to Continuation Methods

Consider the linear system \[ M = \left[ {\begin{array}{*{20}c} A & b {c^T } & d \end{array} } \right], \] where A may be nearly singular but the vectors b and c are such that M is nonsingular. If A is large and sparse, use of iterative methods seems attractive. In this paper, a number of conjugate gradient like methods are considered. Estimates of eigenvalues of M based on those of A are derived. A primary issue is the exploitation of special properties of A, e.g. symmetry, in these algorithms. Often, a good preconditioning for A is available and we show various ways of exploiting it. Results of numerical experiments derived from the use of a continuation method for solving a nonlinear elliptic problem are presented and some general conclusions concerning the relative performance of these algorithms are drawn.

The use of iterative methods for solving large sparse PDE-related linear systems

A preliminary report is given on the ITPACK project for developing adaptive iterative algorithms and software for solving large sparse PDE-related linear systems of equations. These iterative procedures are implemented in a collection of Fortran subroutines.

An Integral Equation Approach to Three-Dimensional Modelling

Summary An integral equation method is used to derive the electromagnetic response of a three-dimensional heterogeneity in a three-layer medium. The method consists of replacing the heterogeneity with point dipole scattering currents. The kernel of the integral equation is a tensor Green's function which is derived for the three-layer case. It is shown how the use of iteration methods and Simpson's rule integration might significantly reduce computing requirements. The problem of initial estimates is discussed. The method is extended to include the case of a simultaneous conductivity and permeability anomaly.

Computer Analysis of Electron Paramagnetic Resonance Spectra

Algebraic methods that are useful in the reduction of EPR spectra to the magnetic parameters in the phenomenological Hamiltonian are summarized and programs presently available to accomplish the necessary computations are described. Among the topics discussed are (i) the calculation of the spectrum of the complete spin Hamiltonian for single-crystal experiments, with the principal axis system; (ii) the transformation of the Hamiltonian to the magnetic quantization axes, which is convenient for perturbation theory; (iii) the use of iteration methods to determine the parameters by a least-squares technique; (iv) the detailed fitting of EPR spectra of polycrystalline or glassy-state magnetic sites; (v) the correlation methods in the analysis of solution spectra; (vi) a novel integral transformation to improve the resolution; and (vii) the calculation of the dipolar sum for line width studies.

An investigation into the use of iteration methods for the analysis of axially symmetric and sheet beam electrode shapes with an emitting surface

The techniques for analyzing an axially symmetric or two-dimensional electrode system with an emitting surface have been studied with a computer program. The program is very easy to use and performs in one pass the steps which have usually previously been followed on analog and digital devices separately. The program alternately solves Poisson's equation by a relaxation technique, traces trajectories, recomputes the space-charge distribution and again performs a relaxation process. This procedure continues until satisfactory convergence has been achieved. The ease of use of the computer program has allowed the investigation of different methods of speeding up the convergence. The results are illustrated with two specific examples.

The Use of Iterative Methods for Finding the Latent Roots and Vectors of Matrices

The Use of Iterative Methods for Finding the Latent Roots and Vectors of Matrices

The use of iterative methods for finding the latent roots and vectors of matrices

The use of iterative methods for finding the latent roots and vectors of matrices

Free tidal oscillations in a rotating square sea

This subject has been studied by many investigators, but solutions have not been adequately illustrated numerically, and some of the solutions have been limited to slow rotations. The use of iteration methods for a square sea indicated the possibility of covering a wider range of rotations and of exploring the use of such numerical methods in tidal problems where the interactions of tides in two phases offer unusual complications. In the case of a square sea these interactions only needed special consideration at the boundary. The investigation has been extended to throw light on the use of the energy equation to specify free oscillations, commonly referred to as Rayleigh’s Principle, but adapted to rotating seas, and simpler solutions of well-known period formulae have been obtained.