The focus of my disciplinary research includes numerical analysis, global optimization, differential equations, and numerical linear algebra. I am curious about problems that have both theoretical and numerical/applied challenges, and require to design new techniques that need analysis and validation.
On the other hand, being a teaching professor at Emory has brought to my attention some of the issues students have in learning mathematics at the college level. Since the Spring Semester 2019, I am working with Emory dance professor Lori Teague on a research project that entails the use of movement to actively learn mathematics, to fascinate people that are not into the sciences, and to foster a sense of community in the attempt to unify the sciences and the arts.
Research Interests
Mathematics through Movement
with Lori Teague, Emory University, USA
Mathematics through Movement began as a collaborative experiment between myself and dance professor Lori Teague in Spring 2019.
We are interested in a transdisciplinary project -- what is in both disciplines, across the different disciplines and beyond the disciplines. Its goal is the comprehension of the actual world, whose imperative is the unification of knowledge.
Our experience in the last three years can be defined as an attempt to discover the deep connections between mathematics and movement studies, a raw experiment. Although we did not formalize assessment strategies on the impact of our experiments during the mathematics learning journey, we did gain feedback from the students that illustrates how movement influences their learning.
In addition, the initiatives and funding that supported this project reflect Emory University’s interest and support. Emory is an optimal place to turn our experiment into a noteworthy innovation in mathematics education.
At the same time, we believe that movement studies will also benefit from the pedagogical and creative aspects of this project.
Our choice is to adopt a student-directed learning environment, to observe/analyze the evolution of movements that were transpiring, to process the analysis and rational interpretation of all students.
​
Click here for an article on our story.
Double Descent and Color Intermittent Diffusion for Unconstrained Global Optimization
with Luca Dieci & Haomin Zhou, Georgia Institute of Technology, Atlanta, USA
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
with Nicola Guglielmi, Gran Sasso Science Institute, L'Aquila, Italy
​The pseudospectra of a matrix A is defined as the set of all the eigenvalues of the perturbed matrix. Unlike the spectrum, the pseudospectrum depends on the chosen norm.
​
Stability radii -- 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.
Existing techniques compute this quantity accurately, but the cost is multiple SVDs of order $n$, which makes the method suitable for moderately sized problems. We presented a novel approach based on a two-level iteration, which is able to solve large problems quickly.
​
Polynomial Stability -- Since the method above can be applied to different structures, we can also study the robustness of a stable polynomial by reformulating the problem considering the associated companion matrix and introducing a structured pseudospectrum, according to the perturbations applied to the polynomial coefficients. This gives us the distance of a stable polynomial to the set of the unstable ones.
Stability of gyroscopic systems -- The two-level technique used to approximate the stability radius can be generalized to study the properties of second order systems. In particular, a second order mechanical system called a gyroscopic system. A necessary condition for a real gyroscopic system to be stable is that all its eigenvalues lie on the imaginary axis (marginal stability). The eigenvalues can leave the axis only as a result of a collision, called strong interaction. Therefore, given a stable system, one would like to find the maximal perturbation such that the system remains stable, that is minimizing the distance among the eigenvalues until coalescence. In this case it is really important for the problem to maintain the original structure.
​
Defectivity measures -- Let A be either a complex or real matrix with all distinct eigenvalues. We worked on the computation of both the unstructured and the real-structured (if the matrix is real) distance of A 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.
We developed a method that couples a system of differential equations on a low rank manifold, computing the pseudoeigenvalue of A which is closest to be defective. The methodology can be extended to a structured matrix, where it is required that the distance is computed within some manifold defining the structure of the matrix.
Self-similarity and Fractal Image Compression
with Francesco Leonetti, University of L'Aquila, Italy
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
Collaborators
