MANUELA MANETTA
Double Descent
and
Color
Intermittent Diffusion
for Unconstrained Global Optimization
The problem of finding the global minimum of an objective function is one of the oldest in applied mathematics and it arises in different contexts, and may involve all branches of science. Consider, for example, the problem of protein folding, that is, the problem of finding the equilibrium configurations of N atoms, assuming the forces between the atoms are given by the gradient of a 3N-dimensional potential energy function. The equilibrium configuration sought is given by the global minimum of the energy potential.
The most reliable techniques to converge to a minimum, the ones based on gradient descent flows, have the drawback of remaining trapped in the basins of attraction of the minima, which could just be local minima.
The main idea of our technique is to take advantage of the knowledge of the Hessian inertia, in order to explore the landscape going from a saddle point to a minimum and vice-versa. ​
​We use a combination of new techniques, namely, a double-descent method to search for minima and an intermittent colored diffusion to escape the basins of attraction of critical points, following an educated path.
The method has been tested on different problems, both standard test functions and real-world problems, to show that the exploration process of the algorithm is well designed and it can search the function landscape effectively
Computation
of pseudospectral matrix distances
The pseudospectra of a matrix A represents the set of eigenvalues of its perturbed forms, depending on the chosen norm.
Stability Radii: This measures the robustness of a stable matrix against small perturbations, a key topic in control theory. Existing methods are computationally expensive, but our two-level iterative approach efficiently handles large problems.
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Polynomial Stability: By reformulating stability as a structured pseudospectrum problem, we computed the robustness of stable polynomials, identifying their distance to instability.
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Stability of Gyroscopic Systems: For second-order mechanical systems, stability requires eigenvalues on the imaginary axis. Our method minimizes eigenvalue displacement under perturbations while preserving the system's structure.
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Defectivity Measures: We developed techniques to calculate the distance of a matrix from the set of defective matrices, extending this to structured matrices using differential equations on low-rank manifolds.
Self-similarity and
Fractal
Image Compression
Fractal image compression, based on self-similarity, addresses the growing need for efficient mass data storage. This method reduces the data required for digital representation by identifying and compressing self-similar structures within an image.
Recognizing that not all images exhibit strict self-similarity, we explored local self-similarity, enabling compression and reconstruction without user interaction. We further generalized these principles using Lipschitz functions to define weak self-similarity, allowing even non-self-similar images, such as a disc, to be represented as a finite union of smaller, self-referential pieces.
While this research is somewhat outdated in the face of modern image compression technologies, it remains a captivating and accessible way to introduce mathematical concepts like fractals, self-similarity, and transformation geometry to students. It serves as a beautiful example of how mathematics connects to real-world challenges and can inspire curiosity in schools.