The concept of well group in a special but important case captures homological properties of the zero set of a continuous map @math on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within @math distance r from f for a given r>0. The main drawback of the approach is that the computability of well groups was shown only when dim K=n or n=1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K<2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps @math with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.
Verification of zeros. An important topic in the interval computation community is the verification of the (non)existence of zeros of a given function @cite_3 . While the nonexistence can be often verified by interval arithmetic alone, a proof of existence requires additional methods which often include topological considerations. In the case of continuous maps @math , Miranda's or Borsuk's theorem can be used for zero verification @cite_22 @cite_15 , or the computation of the topological degree @cite_18 @cite_6 @cite_20 . Fulfilled assumptions of these tests not only yield a zero in @math but also a robust'' zero and a nontrivial @math th well group @math for some @math . Recently, topological degree has been used for simplification of vector fields @cite_5 .
@cite_18 @cite_22 @cite_3 @cite_6 @cite_5 @cite_15 @cite_20
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We show how interval arithmetic can be used in connection with Borsuk's theorem to computationally prove the existence of a solution of a system of nonlinear equations. It turns out that this new test, which can be checked computationally in several different ways, is more general than an existing test based on Miranda's theorem in the sense that it is successful for a larger set of situations. A numerical example is included. Preface Symbol index 1. Basic properties of interval arithmetic 2. Enclosures for the range of a function 3. Matrices and sublinear mappings 4. The solution of square linear systems of equations 5. Nonlinear systems of equations 6. Hull computation References Author index Subject index. In this note we give a new representation for closed sets under which the robust zero set of a function is computable. We call this representation the component cover representation. The computation of the zero set is based on topological index theory, the most powerful tool for finding robust solutions of equations. We show that the assumptions of the well-known Kantorovich theorem imply the assumptions of Miranda’s theorem, but not vice versa. In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to compute arbitrarily close piece-wise interval approximations. Even though this fragment is undecidable, we prove that there is a (possibly non-terminating) algorithm for checking satisfiability such that (1) whenever it terminates, it computes a correct answer, and (2) it always terminates when the input is robust. A formula is robust, if its satisfiability does not change under small perturbations. As a basic tool for our algorithm we use the notion of degree from the field of (differential) topology.