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Ph.D. in mathematics. Postdoctoral researcher at Gran Sasso Science Institute.
I obtained my Ph.D. in Mathematics from Università di Trento and Université Côte d'Azur. I am a postdoctoral researcher at Gran Sasso Science Institute. I formerly was an Istituto Nazionale di Alta Matematica fellow. My research focuses on discrete differential geometry applied to finite element methods and interpolation theory. In June 2026, I obtained the Italian habilitation to associate professorship.
About me
About me
I was born in Bologna in 1992. I obtained both my Bachelor and my Master in Mathematics from Univeristà di Bologna; I spent a semester at Emory University to write my master thesis. I obtained a Ph.D in Mathematics from Università di Trento in a co-tutelle programme with Université Côte d'Azur, in Nice, France.
Bearing the beauty. Rifugio Torre di Pisa, 2023
Research interests
Research interests
An example of histopolation. Taken from Polynomial interpolation of function averages on interval segments, SIAM Journal on Numerical Analysis.
Histopolation theory
Histopolation theory
Histopolation is the process of reconstructing an unknown function from some diffused samples or measurements, typically expressed as integrals over compact sets. The stability and the quality of this project are controlled by the generalized Lebesgue constant, a quantity that bounds (and in some cases equals) the norm of the histopolation operator.
Histopolation theory has many applications to real-life problems and other mathematical areas, such as numerical linear algebra and preconditioning, finite element methods, and finite volume schemes. Although the univariate case is now better understood, many questions are still open. In contrast, the multivariate framework is barely explored.
Some references: Bruni Bruno, L., Dell'Accio, F., Erb, W., Nudo, F., Polynomial histopolation on mock-Chebyshev segments; Bruni Bruno, L., Erb, W., Polynomial interpolation of function averages on interval segments
Conditioning (as a function of the arc-legnth) of the histopolation matrix for Chebyshev second family and segments of class (C2). Taken from the preprint On the conditioning of polynomial histopolation.
Conditioning and spectral features
Conditioning and spectral features
Even in one spatial dimension, histopolation matrices can be very dangerous to handle. This can be read in terms of their condition number, which essentially depends on the supports and the polynomial basis one chooses to work with.
In this context, the mean value theorem is not sufficient to extend well established results for nodal interpolation. However, the take-on message is similar: the monomial basis turns out to be intrinsically ill-conditioned, exhibiting an exponential growth of the condition number as a function of the polynomial degree; in contrast, a specific class of Chebyshev polynomials turns out to be extremely stable. Such analyses can be paired with numerical linear algebra techniques to optimize the conditioning in applications.
Some references: Bruni Bruno, L., Semplice, M., Serra-Capizzano, S., The numerical linear algebra of weights. Part 1: from the spectral analysis to conditioning and preconditioning in the one-dimensional Laplacian case; Bruni Bruno, L., Serra-Capizzano, S., On the conditioning of polynomial histopolation
A unisolvent family for degree 3 Whitney forms. Taken from my Ph.D. thesis Weights as degrees of freedom for high order Whitney finite elements.
Whitney forms and finite element exterior calculus
Whitney forms and finite element exterior calculus
Finite element exterior calculus is an elegant extension of the finite element method, leveraging on the richness of differential forms. In fact, operators in such a context are seen as acting on whole families of (usually polynomial-based) objects, which have a clear meaning in terms of integration.
The power of finite element exterior calculus relies on its generality and its (mostly) coordinate-free representation, and a renowned role in this theory is played by Whitney forms. This class of differential forms is based on a graceful geometrical duality with the degrees of freedom, historically termed weights. Many challenges regarding weights are ample: for instance, general unisolvence results and optimal distributions.
Some references: Alonso Rodríguez, A., Bruni Bruno, L., Rapetti, F., Towards nonuniform distributions of unisolvent weights for high-order Whitney edge elements; Bruni Bruno, L., Zampa, E., Unisolvent and minimal degrees of freedom for the second family of polynomial differential forms
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