Showing papers on "Homotopy analysis method published in 1989"

Modern homotopy methods in optimization

Abstract: Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely These new techniques have been successfully applied to solve Brouwer faced point problems, polynomial systems of equations, and discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements This paper summarizes the theory of globally convergent homotopy algorithms for unconstrained and constrained optimization, and gives some examples of actual application of homotopy techniques to engineering optimization problems

Homotopy algorithm for symmetric eigenvalue problems

Abstract: The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rerely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvalues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.

A geometrical interpretation of the without-exception feasibility of PL homotopy methods

Abstract: PL homotopy methods are effective methods to locate zerces (or fixed points) of highly nonlinear mappings. Due to the lexicographical system, the methods are feasible without exceptions. This paper presents a geometrical interpretation of the without-exception feasibility.

Solving nonlinear resistive networks by a homotopy method using a rectangular subdivision

Abstract: The authors present an efficient algorithm for solving bipolar transistor networks. Two types of formulation techniques are used for deriving a network equation, i.e., the topological formulation and the n-port formulation. The equation f(x)=0 is solved by a homotopy method, in which a homotopy h(x,t)=f(x)-(1-t)f(x/sup 0/) is introduced and the solution curve of h(x,t)=0 is traced from an obvious solution (x/sup 0/,0) to the solution (x*,1) which is sought. It is shown that the convergence of the algorithm is guaranteed by fairly mild conditions. A rectangular subdivision and an upper bounding technique of linear programming are used for tracing the solution curve. >

A simple homotopy for solving deficient polynomial systems

Abstract: Most systems of polynomials which arise in applications have fewer than the expected number of solutions. A simple homotopy is presented for finding all solutions of such a “deficient” system. Different from current homotopies used for such systems, only one parameter is needed to regularize the problem. Within some limits an arbitrary starting problem can be chosen, as long as its solution set is known.

On the solutions to polynomial systems obtained by homotopy methods

Abstract: A natural class of homotopy methods for solving polynomial systems is considered. It is shown that at least one solution from each connected component of the solution set is obtained. This generalizes the results of previous papers which concentrated on isolated solutions, i.e. connected components with one single point. The number of solution paths ending in a connected component is independent of the particular homotopy in use and defines in a natural way the multiplicity of the connected component. A few numerical experiments illustrate the obtained results.

Nonlinear digital signal processing using homotopy continuation methods

Abstract: Homotopy-based algorithms for solving the systems of nonlinear equations which arise in DSP when pole-zero models are used are proposed. Applications include autoregressive-moving average (ARMA) estimation for spectral estimation and system identification and optimal infinite-impulse-response (IIR) filter design. Nonlinear defining equations for the optimal system parameters result in a multimodal performance surface. Current numerical methods for solving these systems of nonlinear equations use gradient-based numerical methods which cannot distinguish between local and global minima or solve related suboptimal problems and which may not converge. Homotopy-based continuation methods are globally convergent numerical methods from which algorithms for finding the global minima of these nonlinear parameter estimation problems can be formed. A review is presented of homotopy continuation methods and digital signal processing using pole-zero structures, the homotopy functions proposed for solving these problems are discussed, and an illustrative example is provided. >

Parallel homotopy curve tracking on a hypercube

Abstract: An investigation is conducted to find good parallel algorithms for solving systems of nonlinear equations using probability-one homotopy methods. Particular attention is paid to algorithms for the hypercube. Methods for one of the most computationally expensive steps of the homotopy approach, the computation of the kernel of the Jacobian matrix of the homotopy map, are studied. General nonlinear systems of equations with small and dense Jacobian matrices are considered, however, polynomial systems are not, since their structure leads to different strategies for parallelism. The mathematics behind the homotopy algorithm is summarized and the use of orthogonal factorizations is discussed. Parallel algorithms for orthogonal factorizations and triangular system solving are described. Computational results are presented and discussed.

A numerical investigation of the Schaeffer homotopy in the problem of Taylor-Couette flows

Abstract: Numerical methods are used to study the 4,6-cell exchange process in the Taylor vortex problem, with particular reference to the homotopy devised by Schaeffer. The homotopy describes a transformation between two models, one incorporating periodic boundary conditions and so referring to flows in an infinite annulus, the other with realistic boundary conditions. Our calculations indicate that the former model is more complicated than previously suspected and lead to a better understanding of the consequences of Schaeffer's device.

Eigenproperties of large-scale structures by finite element partitioning and homotopy continuation

Abstract: Finite element partitioning (or substructuring) is employed to estimate the eigenproperties of large-scale structural systems. A homotopy equation is constructed and its solutions are characterized by a number of curves which connect the eigensolutions of the partitions with those of the complete system. A step-by-step tracing procedure is developed to follow these curves. At each step, prediction and correction are performed. The Rayleigh–Ritz procedure and the conjugate gradient method are used as predictor and corrector, respectively. Compared with the sole use of either the Rayleigh–Ritz or gradient methods, the proposed method is more reliable and more efficient for large-scale problems. Numerical implementation is well suited for supercomputers.