The Homotopy Continuation Method: Numerically Implementable Topological Procedures

Abstract

Abstract. The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.In 1963, Hirsch [13] gave a short-and by now classic-proof of the Brouwer Fixed Point Theorem by a generic argument. This theorem is usually proved by some kind of degree argument and degree genetically counts inverse images of points. Hirsch directly looked at the inverse image of a generic point. The idea that one could replace degree arguments by looking at the inverse images of points of maps was made the theme of a book by J. Milnor [24] and later by V. Guillemin and A. Pollack [12, especially Chapters 2,3] and Hirsch [14, especially Chapter 5]. We offer these books as general references for transversality and Sard's theorem.Without knowledge of Hirsch's paper, H. Scarf [31] in 1967 used much the same ideas to numerically approximate a Brouwer fixed point. One "follows" a "path" which leads from the boundary to some one or more of the fixed points. For some contemporary papers using similar methods, see [19], [23]. This has been called the Newton method. B. C. Eaves [8] in 1972 developed a slightly different approach to the same end. His idea was to homotope one of a set of standard maps to the map in question; by foUowing the fixed points of the changing maps, the fixed-point-set of the map in question could be located. It is this version we call the homotopy continuation method and

MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method

“…Obviously, one is always left with the option to solve this equation numerically, say using spectral methods. In the present work, like Hayat et al [19], we have decided to rely on the HAM to obtain an analytical solution with the advantage that such a solution, if it can be found, is valid even at large elasticity or magnetic numbers [21][22][23][24]. It is worth mentioning that the method has recently been applied to many fluid mechanics problems for Newtonian and non-Newtonian fluids alike [25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body

“…Homotopy [16] is a basic concept in topology [17], and some numerical techniques such as the continuation method [18] and the homotopy continuation method [19] were developed. Different from perturbation techniques [20], the homotopy analysis method does not depend upon any small or large parameters and thus is valid for most nonlinear problems in science and engineering.…”

A Series Solution of the Unsteady Von Kármán Swirling Viscous Flows

“…Based on the homotopy, some numerical techniques such as the continuation method [23] and the homotopy continuation method [24] were developed. There is a suite of FORTRAN subroutines in Netlib for solving nonlinear systems of equations by homotopy methods, called HOMPACK.…”