Showing papers on "Homotopy analysis method published in 1993"

Artificial parameter homotopy methods for the DC operating point problem

Abstract: Efficient and robust computation of one or more of the operating points of a nonlinear circuit is a necessary first step in a circuit simulator. The application of globally convergent probability-one homotopy methods to various systems of nonlinear equations that arise in circuit simulation is discussed. The coercivity conditions required for such methods are established using concepts from circuit theory. The theoretical claims of global convergence for such methods are substantiated by experiments with a collection of examples that have proved difficult for commercial simulation packages that do not use homotopy methods. Moreover, by careful design of the homotopy equations, the performance of the homotopy methods can be made quite reasonable. An extension to the steady-state problem in the time domain is also discussed. >

Homotopy methods for the solution of general modified algebraic Riccati equations

Abstract: In the paper, the authors present a method for the numerical solution of modified algebraic Riccati equations, ATQ + QA − QΣQ + V + F(Q) = 0, where the perturbation term F is low rank. They use a continuous homotopy approach with discrete corrections at each homotopy step. Their algorithm requires the solution of several perturbed Lyapunov equations during each iteration; thus they also address the rapid solution of sets of perturbed Lyapunov equations.

Multi-parameter homotopy methods for finding DC operating points of nonlinear circuits

Abstract: The authors introduce real and complex multi-parameter homotopy methods for finding DC solutions of nonlinear circuits. They show, using arguments from algebraic topology and circuit and polynomial examples, that multi-parameter homotopy methods can avoid bifurcation points and folds along solution paths, and find multiple solutions with relative ease. These concepts are illustrated on a third-order polynomial with a cusp, an object with folds and a bifurcation point, and on two circuit examples. Simple, isolated solution points in a compact region are assumed. >

A Homotopy Algorithm for Maximum Entropy Design

Abstract: Maximum entropy design is a generalization of LQG that was developed to enable the synthesis of robust control laws for flexible structures. The method was developed by Hyland and motivated by insights gained from Statistical Energy Analysis. Maximum entropy design has been used successfully in control design for ground-based structural testbeds and certain benchmark problems. The maximum entropy design equations consist of two Riccati equations coupled to two Lyapunov equations. When the uncertainty is zero the equations decouple and the Riccati equations become the standard LQG regulator and estimator equations. A previous homotopy algorithm to solve the coupled equations relies on an iterative scheme that exhibits slow convergence properties as the uncertainty level is increased. This paper develops a new homotopy algorithm that does not suffer from this defect and in fact has quadratic convergence rates along the homotopy curve.

Fast adaptive polynomial root tracking using a homotopy continuation method

Abstract: A fast adaptive method for tracking the roots of a time-varying complex domain polynomial is derived. The approach uses the method of homotopy continuation and is efficient from both mathematical and implementation standpoints. The method is globally convergent and tracks all roots simultaneously. An example that verifies the accurate tracking ability of the algorithm is presented. Applications which could benefit from this method are also discussed. >

Optimal homotopy methods for solving nonlinear systems

Abstract: This paper deals with the monotone homotopy methods for solving the system of nonlinear equations. In Sect. 2 the homotopy with a distancemonotone homotopy path is discussed and it is proved that a homotopy with a given structure has a distance monotone homotopy path under some regular conditions. Then two structure-variable homotopy algorithms called local straighten-up method and global straighten-up method are developed to approximate the optimal homotopy, and an adaptive step-size control strategy for these algorithms is proposed respectively. A new method for analyzing the convergence is presented, which is based on the geometrical properties of the algorithm. Then the convergence of the algorithm is proved under certain regular conditions. Finally two numerical examples are given to illustrate the effectiveness of the above two algorithms.

Some computational aspects of the stability analysis of nonlinear space trusses

Abstract: A hybrid computational method is presented that seeks to combine efficiency and accuracy of the quasi-Newton method in the direct energy minimization with the robust modified homotopy method that is capable to follow post-buckling path very accurately.

A numerical homotopy method and investigations of a spring-mass system

Abstract: A numerical technique is developed to determine the behavior of periodic solutions to highly nonlinear non-autonomous systems of ordinary differential equations. The method is based on shooting in conjunction with a probability one homotopy method and an implementation of the topological index. It is shown that solutions may be characterized a priori in terms of an index and this is developed into a powerful numerical and investigative tool. This method is used to investigate the periodic solutions of a nonlinear fourth order system of differential equations. These equations describe the motion of a forced mechanical oscillator and are extremely difficult to evaluate numerically. Solutions are presented which could not be found using local methods. These include flip, saddle node and symmetry breaking pitchfork bifurcations.

A homotopy algorithm for the combined H-2/H-infinity model reduction problem

Abstract: The authors attempt to make significant progress in developing novel stable, globally convergent numerical algorithms for solving the optimality conditions for H-2/H-infinity order reduction given in W. M. Haddad (1989). The approach taken is based on the construction of probability-one homotopy maps. Homotopy algorithms based on two formulations - input normal form; and Ly, Bryson, and Cannon's (1985) 2 /spl times/ 2 block parametrization - are developed and compared. Numerical results are presented. >

A continuation homotopy method for the inverse problem of operator identification and its application

Abstract: A new widly convergent method for solving the problem of operator identification is illustrated. Numerical simulations are carried out to test the feasibility and to study the general characteristics of the technique without the real measurement data. This technique is a direct application of the continuation homotopy method for solving nonlinear systems of equations. It is found that this method does give excellent results in solving the inverse problem of the elliptic differential equations.