Showing papers on "Homotopy analysis method published in 1995"

Reduced-order dynamic compensation using the Hyland-Bernstein optimal projection equations

Abstract: Gradient-based homotopy algorithms have previously been developed for synthesizing H/sub 2/ optimal reduced-order dynamic compensators. These algorithms are made efficient and avoid high-order singularities along the homotopy path by constraining the controller realization to a minimal parameter basis. However the resultant homotopy algorithms sometimes experience numerical ill-conditioning or failure due to the minimal parameterization constraint. This paper presents a new homotopy algorithm which is based on solving the optimal projection equations, a set of coupled Riccati and Lyapunov equations that characterize the optimal reduced-order dynamic compensator. Path following in the proposed algorithm is accomplished using a predictor/corrector scheme that computes the prediction and correction steps by efficiently solving a set of four Lyapunov equations coupled by relatively low rank linear operators. The algorithm does not suffer from ill-conditioning due to constraining the controller basis and often exhibits better numerical properties than the gradient-based homotopy algorithms.

Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators

Abstract: GivenA1, the discrete approximation of a linear self-adjoint partial differential operator, the smallest few eigenvalues and eigenvectors ofA1 are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operatorA0 with known eigenvalues and eigenvectors, define the homotopyH(t)=(1−t)A0+tA1, 0≤t≤1. If the eigenvectors ofH(t0) are known, then they are used to determine the eigenpairs ofH(t0+dt) via the Rayleigh quotient iteration, for some value ofdt. This is repeated untilt becomes 1, when the solution to the original problem is found. A fundamental problem is the selection of the step sizedt. A simple criterion to selectdt is given. It is shown that the iterative solver used to find the eigenvector at each step can be stabilized by applying a low-rank perturbation to the relevant matrix. By carrying out a small part of the calculation in higher precision, it is demonstrated that eigenvectors corresponding to clustered eigenvalues can be computed to high accuracy. Some numerical results for the Schrodinger eigenvalue problem are given. This algorithm will also be used to compute the bifurcation point of a parametrized partial differential equation.

Parameter embedding methods for finding dc operating points: formulation and implementation

Abstract: We address the calculation of dc operating points of nonlinear circuits by using parameter embedding methods, and we show that the usefulness of these methods depends on the type of a circuit's descriptive equations. We discuss various approaches to embedding a parameter into nonlinear equations that describe bipolar and MOS transistor circuits. Embedding algorithms were implemented in an industrial circuit simulator. We demonstrated that homotopy methods can be used as an alternative to the NewtonRaphson-type solvers and that they can be successfully applied to solving nonlinear circuit equations as well as to calculating dc operating points and transfer curves of nonlinear transistor circuits.

Topology and geometry of single hidden layer network, least squares weight solutions

Abstract: In this paper the topological and geometric properties of the weight solutions for multilayer perceptron (MLP) networks under the MSE error criterion are characterized. The characterization is obtained by analyzing a homotopy from linear to nonlinear networks in which the hidden node function is slowly transformed from a linear to the final sigmoidal nonlinearity. Two different geometric perspectives for this optimization process are developed. The generic topology of the nonlinear MLP weight solutions is described and related to the geometric interpretations, error surfaces, and homotopy paths, both analytically and using carefully constructed examples. These results illustrate that although the natural homotopy provides a practically valuable heuristic for training, it suffers from a number of theoretical and practical difficulties. The linear system is a bifurcation point of the homotopy equations, and solution paths are therefore generically discontinuous. Bifurcations and infinite solutions further occur for data sets that are not of measure zero. These results weaken the guarantees on global convergence and exhaustive behavior normally associated with homotopy methods. However, the analyses presented provide a clear understanding of the relationship between linear and nonlinear perceptron networks, and thus a firm foundation for development of more powerful training methods. The geometric perspectives and generic topological results describing the nature of the solutions are further generally applicable to network analysis and algorithm evaluation.

A universal constant for the convergence of Newton's method and an application to the classical homotopy method

Abstract: We give a new theorem concerning the convergence of Newton's method to compute an approximate zero of a system of equations. In this result, the constanth0=0.162434... appears, which plays a fundamental role in the localization of “good” initial points for the Newton iteration. We apply it to the determination of an appropriate discretization of the time interval in the classical homotopy method.

The homotopy method applied to the symmetric eigenproblem

Abstract: 1995 v Acknowledgements It is my strong desire to thank all the people who have supported the present work. Above all, I am indebted to Prof. W. Gander, my supervisor, for giving me the freedom to explore new ideas and for providing a good productive environment. His algorithmic spirit and his appreciation for numerical details has become for me an ideal to strive for. Also his conndence in the ultimate success of our work was very encouraging. I thank Dr. P. Arbenz, co-examiner, who initiated me into the eld of the algebraic eigenvalue problem and who has originated the subject of this thesis. I had many fruitful discussions with him. Special thanks goes to Prof. R. Jeltsch who spontaneously accepted to be co-examiner. My thank also goes to Prof. T. Y. Li who invited me to Michigan State University in spring 1994. He gave me valuable insight into his research on the homotopy method. I am grateful to the numerous people in the Institute for Scientiic Computing who have helped me by frequent discussions held in an open and stimulating atmosphere. To my mother, who has always supported me, I would like to dedicate this thesis. vi Abstract Abstract The homotopy method has been used in the past for nding zeros of nonlinear functions. Recently this method has been given new attention and it is being used for the computation of matrix eigenvalues and eigenvectors because of its natural parallelism. Early algorithms based on the homotopy approach for solving the symmetric eigen-problem suuer either from insuucient orthogonality between computed eigenvalues or low eeciency. A new algorithm presented in this thesis overcomes these problems by using deeation of close eigenpairs and maintains a high degree of parallelism by applying the divide-and-conquer principle. The selection of appropriate starting matrices is discussed too. A numerical comparison of a sequential implementation of the new algorithm shows that it is competitive with other well-established algorithms both in speed and accuracy. Algorithms based on the homotopy method are excellent candidates for multicom-puters, because of there natural parallelism. It is examined in this thesis how the algorithm can be implemented on distributed-memory multicomputer. The experimental performance results obtained on an Intel Paragon are very satisfying.

Sequential homotopy-based computation of multiple solutions to nonlinear equations

Abstract: Homotopy methods have achieved significant success in solving systems of nonlinear equations for which the number of solutions are known and the homotopy paths are bounded We present a two-stage homotopy process which does not require a-priori knowledge of the number of solutions to a system of nonlinear equations This approach makes use of compact manifolds to find solutions sequentially along disconnected homotopy paths The procedure is tested on two standard optimization and neural network benchmark problems

On the numerical imbedding method

Abstract: WhenF(x) = 0 is a system of nonlinear equations, we have established the parametrizied Homotopy algorithm for solvingF(x) = 0 and some theorems for algorithm have been obtained