Verification of invertibility of complicated functions over large domains

Abstract

A new method to decide the invertibility of a given high-dimensional function over a domain is presented. The problem arises in the field of verified solution of differential algebraic equations (DAEs) related to the need to perform projections of certain constraint manifolds over large domains. The question of invertibility is reduced to a verified linear algebra problem involving first partials of the function under consideration. Different from conventional approaches, the elements of the resulting matrices are Taylor models for the derivatives of the functions. The linear algebra problem is solved based on Taylor model methods, and it will be shown the method is able to decide invertibility with a conciseness that often goes substantially beyond what can be obtained with other interval methods. The theory of the approach is presented. Comparisons with three other interval-based methods are performed for practical examples, illustrating the applicability of the new method.