Vertex-based marching algorithms for finding multidimensional geometric intersections

This article is a survey of the current state of the art in vertex-based marching algorithms for solving systems of nonlinear equations and solving multidimensional intersection problems. It addresses also ongoing research and future work on the topic. Among the new topics discussed here for the first time is the problem of characterizing the type of singularities of piecewise affine manifolds, which are the numerical approximations to the solution manifolds, as generated by the most advanced of the considered vertex-based algorithms: the Marching-Simplex algorithm. Several approaches are proposed for solving this problem, all of which are related to modifications, extensions and generalizations of the Morse lemma in differential topology.