Showing papers on "Homotopy analysis method published in 1997"
TL;DR: In this paper, the homotopy analysis method was further improved by introducing a non-zero parameter into the traditional way of constructing a homhotopy, which can converge even in the whole region η ϵ [0, + ∞].
Abstract: In this paper, the non-linear approximate technique called Homotopy Analysis Method proposed by Liao is further improved by introducing a non-zero parameter into the traditional way of constructing a homotopy. The 2D viscous laminar flow over an infinite flat-plain governed by the non-linear differential equation f′''(η) + f(η)f″(η) 2 = 0 with boundary conditions f(0) = f′(0) = 0, f′(+ ∞) = 1 is used as an example to describe its basic ideas. As a result, a family of approximations is obtained for the above-mentioned problem, which is much more general than the power series given by Blasius [Z. Math. Phys. 36, 1(1908)] and can converge even in the whole region η ϵ [0, + ∞). Moreover, the Blasius' solution is only a special case of ours. We also obtain the second-derivative of f(η) at η = 0, i.e. f″(0) = 0.33206, which is exactly the same as the numerical result given by Howarth [Proc. Roy. Soc. London A164, 547 (1938)].
354 citations
TL;DR: In this article, the problem of finding all isolated solutions to polynomial equations in n unknown unknowns is studied. But their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems.
Abstract: Let P(x) = 0 be a system of n polynomial equations in n unknowns Denoting P = (p1,…, pn), we want to find all isolated solutions offor x = (x1,…,xn) This problem is very common in many fields of science and engineering, such as formula construction, geometric intersection problems, inverse kinematics, power flow problems with PQ-specified bases, computation of equilibrium states, etc Elimination theory-based methods, most notably the Buchberger algorithm (Buchberger 1985) for constructing Grobner bases, are the classical approach to solving (11), but their reliance on symbolic manipulation makes those methods seem somewhat unsuitable for all but small problems
205 citations
TL;DR: In this paper, the basic ideas of a new kind of analytical technique, namely the Homotopy Analysis Method (HAM), are briefly described, and a simple but typical example is used to illustrate the validity and the great potential of the HAM.
143 citations
Book•
11 Apr 1997
TL;DR: Abstract Homotopy Theory Case Studies Exact Sequences ElementaryHomotopy Coherence Abstract Simple Homotology Theory Additive Simple HomOTopy Theories.
Abstract: Homotopy Theory Case Studies Exact Sequences Elementary Homotopy Coherence Simple Homotopy Theory Additive Simple Homotopy Theories.
100 citations
TL;DR: In this paper, a new numerical approach for strongly nonlinear problems is proposed and applied to solve, as an example, a non-linear heat transfer problem, i.e., microwave heating of an unit plate, so as to verify its validity and great potential.
Abstract: In this paper, the Homotopy Analysis Method (HAM) proposed by Liao (1992a, 1992b, 1992c, 1992e, 1995a, 1997a) is greatly improved by introducing a nonzero variable ℏ. Based on the HAM, a new numerical approach for strongly non-linear problems is proposed and applied to solve, as an example, a non-linear heat transfer problem, i.e. microwave heating of an unit plate, so as to verify its validity and great potential. Our numerical experiments show that, by the proposed approach, iteration is not absolutely necessary for solving non-linear problems. This fact may deepen our understanding about numerical techniques for non-linear problems and widen our field of vision. Moreover, the basic ideas proposed in this paper may afford us a great possibility to greatly improve our current numerical techniques.
76 citations
TL;DR: In this paper, a probability-one homotopy continuation algorithm for mixed structured singular value controller synthesis using general xed-structure multipliers has been proposed, which can be initialized using an arbitrary (admissible) multiplier and a stabilizing compensator.
Abstract: SUMMARY Continuation algorithms that avoid multiplier{controller iteration have been developed earlier for xedarchitecture, mixed structured singular value controller synthesis. These algorithms have only been formulated for the special case of Popov multipliers and rely on an ad hoc initialization scheme. In addition, the algorithms have not used the prediction capabilities obtained by computing the Jacobian matrix of the continuation (or homotopy) map, and have assumed that the homotopy zero curve is monotonic. This paper develops probability-one homotopy algorithms based on the use of general xed-structure multipliers. These algorithms can be initialized using an arbitrary (admissible) multiplier and a stabilizing compensator. In addition, as with all probability-one algorithms, the homotopy zero curve is not assumed to be monotonic and prediction is accomplished by using the homotopy Jacobian matrix. This approach also appears to have some advantages over the bilinear matrix inequality (BMI) approaches resulting from extensions of the LMI framework for robustness analysis.
20 citations
12 Oct 1997
TL;DR: Based on the homotopy theory, a robust nonlinear system identification algorithm was proposed in this paper, which is more robust and easier to compute, has very good convergence property and compatibility with the traditional linear system identification.
Abstract: Based on the homotopy theory, this paper presents a robust nonlinear system identification algorithm. Theoretical analysis and experimental results show that, unlike the ordinary nonlinear system identification methods, the new method is more robust and easier to compute, has very good convergence property and compatibility with the traditional linear system identification.
4 citations
Dissertation•
05 Sep 1997
TL;DR: It is shown that the efficiency of linear system solution by the adaptive GMRES(k) algorithm depends on the change in problem difficulty when the problem is scaled, as concluded analytically and experimentally.
Abstract: Globally convergent homotopy methods are used to solve difficult nonlinear systems of equations by tracking the zero curve of a homotopy map. Homotopy curve tracking involves solving a sequence of linear systems, which often vary greatly in difficulty. In this research, a popular iterative solution tool, GMRES(k), is adapted to deal with the sequence of such systems. The proposed adaptive strategy of GMRES(k) allows tuning of the restart parameter k based on the GMRES convergence rate for the given problem. Adaptive GMRES(k) is shown to be superior to several other iterative techniques on analog circuit simulation problems and on postbuckling structural analysis problems. Developing parallel techniques for robust but expensive sequential computations, such as globally convergent homotopy methods, is important. The design of these techniques encompasses the functionality of the iterative method (adaptive GMRES(k)) implemented sequentially and is based on the results of a parallel performance analysis of several implementations. An implementation of adaptive GMRES(k) with Householder reflections in its orthogonalization phase is developed. It is shown that the efficiency of linear system solution by the adaptive GMRES(k) algorithm depends on the change in problem difficulty when the problem is scaled. In contrast, a standard GMRES(k) implementation using Householder reflections maintains a constant efficiency with increase in problem size and number of processors, as concluded analytically and experimentally. The supporting numerical results are obtained on three distributed memory homogeneous parallel architectures: CRAY T3E, Intel Paragon, and IBM SP2.
3 citations
TL;DR: This paper applies homotopy theory to Den Haan and Marcet's Parameterized Expectation Approach to show how homotopies improve the computing speed and global convergence properties of this algorithm.
Abstract: Many numerical methods have been developed in an attempt to find solutions to nonlinear rational expectations models. Because these algorithms are numerical in nature, they rely heavily on computing power and take sizeable cycles to solve. In this paper we present a numerical tool known as homotopy theory that can be applied to these methods. Homotopy theory reduces the computing time associated with an iterative algorithm by using a rational expectation problem with known solutions and transforming it into the problem at hand. If this transformation is performed slowly, homotopy theory also helps the global convergence properties of the numerical algorithm. We apply homotopy theory to Den Haan and Marcet‘s Parameterized Expectation Approach to show how homotopies improve the computing speed and global convergence properties of this algorithm.
3 citations
Journal Article•
2 citations
Journal Article•
TL;DR: In this article, the basic ideas of homotopy analysis method are described, and a new family of real functions, called approaching functions, is introduced, which is rigorously proved that the convergence region of the traditional Taylor series can be greatly enlarged by multiplying its each term by ap-proaching functions of this family one after another in order.
Abstract: In this paper, the basic ideas of a new nonlinear analytical technique, namely Homotopy Analysis Method, are described, and a new family of real functions, called approaching functions, is introduced. Some basic properties of this family of real functions are given. It is rigorously proved that the convergence region of the traditional Taylor series can be greatly enlarged by multiplying its each term by ap- proaching functions of this family one after another in order.
01 Jan 1997
TL;DR: The homotopy nonlinear modeling method (HNMM) is nearly as simple as the traditional linear modeling methods and thus can be used very extensively.
Abstract: Based on the homotopy theory, this paper presents a novel nonlinear modeling method for speech signals, the homotopy nonlinear modeling method (HNMM). Unlike the ordinary nonlinear modeling methods, the HNMM is robust and easy to compute, and has very good convergence property. Besides, the HNMM is nearly as simple as the traditional linear modeling methods and thus can be used very extensively. The validity of the HNMM has been proved by experimental results.
TL;DR: Under suitable conditions, it is proved that there exists a homotopy solution curve, which is unique and continuous, and it is shown that the solution curve is singlevalue-d with respect to the Homotopy parameter.
Abstract: In this paper, we present a class of embedding methods for nonsmooth equations. Under suitable conditions, we prove that there exists a homotopy solution curve, which is unique and continuous. We also prove that the solution curve is singlevalue-d with respect to the homotopy parameter. Then we construct an efficient algorithm for this class of equations and prove its convergence. Finally, we apply the algorithm to the nonlinear complementarity problem. The numerical results show that the algorithm is satisfacotry.