Showing papers on "Homotopy analysis method published in 1991"

The homotopy model: a generalized model for smooth surface generation from cross sectional data

Abstract: A generalized model, called the homotopy model, is presented to reconstruct surfaces from cross-sectional data of objects using a homotopy to generate surfaces connecting consecutive contours. The homotopy model consists of continuous toroidal graph representation and homotopic generation of surfaces from the representation. It is shown that the homotopy model includes triangulation as a special case and generates smooth parametric surfaces from contour-line definitions using homotopy. The model can be applied to contours represented by parametric curves as well as linear line segments. First, a heuristic method that finds the optimal path on the toroidal graph is presented. Then the toroidal graph is expanded to a continuous version. Finally, homotopy is used for reconstructing parametric surfaces from the toroidal graph representation. A loft surface is also a special case of homotopy, a straight-line homotopy. Homotopy that corresponds to the cardinal spline surface is also introduced. Three-dimensional surface reconstruction of human auditory surface reconstruction of human auditory ossicles illustrates the advantages of the homotopy model over the others.

Homotopy approach to optimal, linear quadratic, fixed architecture compensation

Abstract: Optimal linear quadratic Gaussian compensators with constrained architecture are a sensible way to generate good multivariable feedback systems meeting strict implementation requirements. The optimality conditions obtained from the constrained linear quadratic Gaussian are a set of highly coupled matrix equations that cannot be solved algebraically except when the compensator is centralized and full order. An alternative to the use of general parameter optimization methods for solving the problem is to use homotopy. The benefit of the method is that it uses the solution to a simplified problem as a starting point and the final solution is then obtained by solving a simple differential equation. This paper investigates the convergence properties and the limitation of such an approach and sheds some light on the nature and the number of solutions of the constrained linear quadratic Gaussian problem. It also demonstrates the usefulness of homotopy on an example of an optimal decentralized compensator.

Homotopy Approaches to the H2 Reduced Order Model Problem

Abstract: The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. The algorithms proposed herein utilize probability-one homotopy theory as the main tool. It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system. With a carefully chosen initial system all the systems along the homotopy path will be asymptotically stable, controllable and observable. One method, which solves the matrix equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix. An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given. Several strategies for choosing the homotopy maps and the starting points (initial systems) are discussed and compared, in the context of some reduced order model problems from the literature. Numerical results are included for ten test problems, of sizes 2 through 17.

The Homotopy Principle and Algorithms for Linear Programming

Abstract: Linear programming techniques are formulated within the unifying framework of the homotopy principle and Newton’s method. Key strategies that determine the effectiveness of an implementation are considered in detail. A complexity analysis is developed for an elevator predictor, Newton corrector algorithm started at an arbitrary interior primal-dual feasible point. This analysis is based on a fundamental theorem of Smale [“Algorithms for solving equations,” Proc. Internat. Congress of Mathematicians, Universityof California, Berkeley, CA, 1986], in the form given by Renegar and Shub [Report No. 807, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1988].

Homotopy continuation method for the numerical solutions of generalised symmetric eigenvalue problems

Abstract: We consider a generalised symmetric eigenvalue problem Ax = XMx , where A and M are real n by n symmetric matrices such that M is positive semidefinite. The purpose of this paper is to develop- an algorithm based on the homotopy methods in [9, 11) to compute all eigenpairs, or a specified number of eigenvalues, in any part of the spectrum of the eigenvalue problem Ax = XMx . We obtain a special Kronecker structure of the pencil A - XM, and give an algorithm to compute the number of eigenvalues in a prescribed interval. With this information, we can locate the lost eigenpair by using the homotopy algorithm when multiple arrivals occur. The homotopy maintains the structures of the matrices A and M (if any), and the homotopy curves are n disjoint smooth curves. This method can be used to find all/some isolated eigenpairs for large sparse A and M on SIMD machines.

Solution of ill-conditioned load flow equation by homotopy continuation method

Abstract: A homotopy continuation method is used to solve ill-conditioned load flow equations. It presents the predictor step effective for following the path of homotopy functions formulated from the load flow equation. The aim is to solve the ill-conditioned load flow equation, the solution of which has long been discussed. Using the homotopy method described, it is concluded that the ill-conditioned load flow equation discussed has no solution. >

Note on unit tangent vector computation for homotopy curve tracking on a hypercube

Abstract: Probability-one homotopy methods are a class of methods for solving nonlinear systems of equations that are globally convergent from an arbitrary starting point. The essence of all such algorithms is the construction of an appropriate homotopy map @r"a(@l, x) and subsequent tracking of some smooth curve @c in the zero set of the homotopy map. Tracking a homotopy curve involves finding the unit tangent vector at different points along the zero curve, which amounts to calculating the kernel of the n x (n + 1) Jacobian matrix D@r"a(@l, x). While computing the tangent vector is just one part of the curve tracking algorithm, it can require a significant percentage of the total tracking time. This note presents computational results showing the performance of several different parallel orthogonal factorization/triangular system solving algorithms for the tangent vector computation on a hypercube.

Homotopy approach for solving constrained optimization problems

Abstract: A homotopy approach for solving constrained parameter optimization problems is examined. The first-order necessary conditions, with the complementarity conditions represented using a technique due to Mangasarian (1967) are solved. The equations are augmented to avoid singularities which occur when the active constraint changes. The Chow-Yorke (1978) algorithm is used to track the homotopy path leading to the solution to the desired problem at the terminal point. A simple example which illustrates the technique and an application to a fuel optimal orbital transfer problem are presented. >

Constructive analysis for infinite‐dimensional nonlinear systems—infinite‐dimensional version of homotopy method

Abstract: Related to the problem such as determination of the operating point for the nonlinear resistor circuit equation, remarkable progress has recently been observed in the global solution method for the finite-dimensional nonlinear equation, called the homotopy method. This paper attempts to extend the homo-topy method to the infinite dimensional nonlinear equation (functional equation) such as the equation for the semiconductor devices. First, two streams are pointed out as the theoretical frameworks for the homotopy method for the finite-dimensional equation, which are: (1) the characterization of the solution structure for the nonlinear equation; and (2) the tracing of the solution trajectory. It is then pointed out that the Fredholm operator equation can be considered as a class of infinite-dimensional equations, which can characterize the set of solutions as in item (1). The existence theorem for the solution in the finite-dimensional homotopy method is extended to the class of Fredholm operator equations. Then it is pointed out that the A-proper operator equation can be considered as a class of infinite-dimensional equations, for which the computational algorithm for the solution can be described as in item (2). Based on the constructive theorem for the existence of the solution in the finite-dimensional homotopy method, it is shown that the solution for the particular class of A-proper operator equations can always be determined by numerical calculation. Finally, an application to the controllability problem is discussed.

A homotopy method for solving optimal projection equations for the reduced order model problem

Abstract: An algorithm using probability-1 homotopy theory is proposed for solving the optimal projection equations for the reduced order model problem. There is a family of systems (the homotopy) which make a continuous transformation from some initial system to the final system. The process of solving each system involves computing the derivative of the Drazin inverse of a matrix and solving a number of Sylvester equations. The central theorem of the present works shows the validity of the whole process, i.e., determines the class of initial systems which certainly lead to the final system along a homotopy path. Another theorem shows that the differentiation of the Drazin inverse is justified, i.e., that the derivative of the Drazin inverse exists. Finally, it is shown how the optimal solution to the reduced order model problem can be easily computed from a solution to the modified Lyapunov equations. >

A monotonous homotopy method for solving the resistivities of the earth

Abstract: On the basis of the mathematical model of electrical log, we have proposed a monotonous homotopy method for determining the parameters of the earth. By this method, the problem of nonlinear system of equations is reduced to an initial value problem of ordinary differential equations. It is a widely convergent method. Numerical simulations show that this numerical algorithm is effective.

A piecewise‐linear homotopy method for nonlinear programming

Abstract: In nonlinear programming problems, an objective function f(x) is optimized (maximized or minimized) subject to some constraints. Such problems are also called constrained optimization problems. Most of the algorithms in nonlinear programming are classified into two categories: 1) transformation methods; and 2) projection methods. The homotopy methods, which are the subject of this paper, belong to the category of projection methods. The main feature of the homotopy methods compared with other projection methods is that they are good at global convergence (which is lacking in most of the projection methods) but are not good at convergence speed (which is the strong point of most of the projection methods). This paper discusses the homotopy methods in nonlinear programming and show that the piecewise-linear homotopy method using the Newton homotopy and polyhedral subdivision is very effective for solving nonlinear optimization problems. A new algorithm is proposed that exploits the partial separability and linearity of the Kuhn-Tucker equations (which appear in the nonlinear programming problems). By this exploitation, the computation efficiency is improved markedly compared with the conventional homotopy methods using simplicial subdivision. Moreover, the proposed algorithm converges quadratically, thus accurate solutions can be obtained rapidly. It is proved also that the proposed algorithm is globally convergent for the constrained convex optimization problems. Except for the shortcoming that the programming is complicated, the proposed algorithm has wellbalanced effectiveness.

Computational complexity of the homotopy method for calculating solutions of strongly monotonic resistive circuit equations

Abstract: A priori estimation is presented for a computational complexity of the homotopy method applied to a certain class of hybrid equations for nonlinear strongly monotonic resistive circuits. First, an explanation is given as to why a computational complexity of the homotopy method cannot be a priori estimated for calculating solutions of hybrid equations in general. In this paper, the homotopy algorithm is considered in which a numerical path-following algorithm is executed based on the simplified Newton method. Then by introducing Urabe's theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, it is shown that a computational complexity of the algorithm can be a priori estimated when applied to a certain class of hybrid equations for nonlinear strongly monotonic resistive circuits whose domains are bounded. This paper considers two types of path-following algorithms: one with a numerical error estimation in the domain of a nonlinear operator; and one with a numerical error estimation in the range of the operator.