The important thing is not to stop questioning. Curiosity has its own reason for existing. [Albert Einstein]
The problem of finding the global minimizer of an objective function is one of the oldest in applied mathematics and it arises in different contexts.
The most reliable techniques to converge to a minimizer, the ones based on gradient descent flows, have the drawback of remaining trapped in the basins of attraction of the minimizers, which could just be local minimizers.
Our aim was to build up a new method which does not rely on the specific properties of the potential, as many existing algorithms do, and we also wanted to present methods and ideas that can be taught in a numerical optimization course.
We use a combination of new techniques, namely a double-descent method to search for minima and an intermittent colored diffusion technique to escape the basins of attraction of critical points.
Computation of pseudospectral matrix distances
The distance to instability problem, that is the computation of the distance of a stable matrix to the set of unstable matrices, has received a great deal of attention in the control theory literature since the mid-1980s; it belongs to the subfield known as robust control. The usual motivation behind such studies is to understand the behavior of the dynamical system x' = Ax. If A is a stable matrix one would like to know whether its stability is robust in the sense that it will be unaffected by small perturbations.
We developed fast methods for structured and unstructured problems, generalized eigenvalue problems, such as those arising from gyroscopic systems, and polynomial stability.
We also worked on the computation of both the unstructured and the real-structured distance of a matrix from the set of defective matrices, that is the set of those matrices with at least one multiple eigenvalue with algebraic multiplicity larger than its geometric multiplicity.
Fractal Image Compression
The actual need for mass information storage grows with the proliferation of image data. The images can be processed by compression techniques, which allow to reduce the amount of data necessary to their digital representation. Among these techniques, we find the fractal image compression, a method based on self-similarity. As not all the images are self-similar, we consider a weaker property: local self- similarity, which suggests a method of compression and reconstruction of the image which does not require the user's interaction. Moreover, we generalized those properties by using Lipschitz functions, obtaining the weak self-similarity, showing how an image that is not self-similar, like the disc, can be compressed and reconstructed as the union of a finite number of small copies of pieces of itself.
Work in Progress