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

Numerical Methods for Image Registration

“…An important method to save computational work and to deal with non-linearities is to use a multilevel continuation. Multilevel continuation is well established for optimization problems and systems of non-linear equations; see, e.g., [1,2]. However, in image registration it has an additional advantage.…”

“…There are two main approaches for the discretization of the registration problem (1). In the first so-called optimize-discretize approach one forms the objective function, then differentiates to obtain the continuous EulerLagrange equations, which are finally discretized and solved numerically; see, e.g., [14,6,17].…”

Fredholm Mappings and Banach Manifolds

“…One way of solving the equation (1) u(x) = y for any fixed y ∈ Y, is to embed (1) in a continuum of problems (2) H(x, t) = y, (0 ≤ t ≤ 1), which is solved when t = 0. When t = 1, problem (2) becomes (1). If it is possible to continue the solution for all t ∈ [0, 1], then (1) is solved.…”

Solving Fixed-Point Problems with Inequality and Equality Constraints via a Non-Interior Point Homotopy Path-Following Method

“…The determination of a constructive proof of the fixed-point theorem and therefore finding a fixed point became an attractive topic. The homotopy method, as a globally convergent algorithm, is a powerful tool in handling fixed-point problems (e.g., [5][6][7][8][9][10] and the references therein). The general Brouwer fixed-point theorem states that if a bounded closed subset in is diffeomorphic to the closed unit ball, then any continuous self-mapping in it has a fixed point.…”