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.
In this paper we provide an overview about an orthonormal (multi) wavelet-based method for isometric immersion
of smooth n-variate m-dimensional vector fields onto fractal curves and surfaces. This method was proposed in
an earlier publication by two of the authors, with the purpose of extending the applicability of emerging GPU-programming
to rich diversity of multidimensional problems. Here we propose (in Section 3) several directions
for upgrading the method, with respective new applications.
This article is a systematic overview of compression, smoothing and denoising techniques based on shrinkage of
wavelet coefficients, and proposes an advanced technique for generating enhanced composite
wavelet shrinkage strategies.