# Mathematics

• ##### Data approximation and numerical methods for curves and surfaces
• One of our primary interests is in the construction, the analysis and the applications of Pythagorean Hodograph curves (PH curves, for short). These are plane or space parametric polynomial/piecewise-polynomial curves (expressed in Bézier or B-spline representation) for which the norm of the hodograph, that is, its first derivative with respect to the parameter, is a polynomial/piecewise-polynomial function as well. This distinguishing feature implies exact arc-length computation and allows one to obtain rational offset curves, real-time CNC interpolators, exact rotation-minimizing frames, etc. Thanks to all these properties, PH curves have reached an established practical value in a variety of applications including digital motion control, path planning, robotics, animation, computer graphics, etc. Although PH curves have already been studied for several years, their investigation still offers several open problems whose solution requires a combination of concepts from algebra, geometry and numerical analysis.
• A second research topic of great interest deals with the construction of curves and surfaces approximating noisy data. More precisely, we are interested in developing new techniques for constructing piecewisely defined curves and surfaces, but also univariate and bivariate subdivision schemes, that allow users to obtain smoothing tools capable of capturing the trend of the noisy data by guaranteeing the optimal trade-off between the goodness of the fit and the roughness of the final curve/surface.
• Involved person: Lucia Romani
##### geometric structures on manifolds
• The classification of surfaces is, quoting V. Arnol’d, “a top-class mathematical achievement, comparable with the discovery of America or X-rays”. Building on this, the ultimate goal of geometry is the classification of smooth manifolds, a task which is however out of reach. A more modest goal is the study (and hopefully the classification) of manifolds with structure (Riemannian metrics, symplectic and almost contatc structures, almost complex structures…). The study of some of these structures is motivated by physics: for instance, the search for special (pseudo)Riemannian metrics is intimately connected with General Relativity and Supersymmetry. In this research project we consider nilmanifolds, certain homogeneous spaces of nilpotent Lie groups, which are particularly suitable for studying and constructing geometric structures.
• Involved person: Giovanni Bazzoni
##### Mathematics of Signal Processing
• This is an emerging research field at the intersection of Numerical Analysis and Signal Processing. Our focus is non-stationary signals, i.e. signals that change their features over time, like chirps in gravitational waves, whistles in audio signals, or multipaths in the global navigation satellite system (GNSS) data. Non-stationary signals represent most real-life measurements and prove to be hard to analyze using classical techniques based on Fourier and Wavelet Transform. For this reason, there is a need to develop new algorithms able to handle them. In our group, we work on the development, mathematical analysis, and possible acceleration of this kind of algorithm. In particular, we develop methods for signal decomposition and time-frequency analysis. We publish the results of our work in peer-reviewed international journals and make them freely available online to the international community. Furthermore, we work on their applications to Medicine, Engineering, Geophysics, Astrophysics, and Economics, in collaboration with international experts in these fields.
• Involved persons: Antonio Cicone, Carlo Garoni, Stefano Serra-Capizzano
##### NLSE in domains with spatial singularities
• In recent years, significant progress has been made in the rigorous analysis of dispersive nonlinear partial differential equations (PDEs) within domains where spatial singularities may be present. Our primary focus is on the Nonlinear Schrödinger Equation (NLSE), considered one of the most important dispersive PDEs. In our models, singularities arise from localized defects in the propagation medium or vertices of branching structures (networks or metric graphs). Our research focuses particularly on describing these defects through appropriate boundary conditions, specifically via perturbations of the self-adjoint Laplacian on sets of zero measure, also known as point interactions. All these models belong to the family of nonlinear dispersive Hamiltonian PDEs and share several features with the well-known NLSE in the entire Euclidean space. Notably, similar to the singularity-free case, they admit solitons or standing wave solutions that can exhibit different forms of stability or instability. These solutions are believed to play a crucial role in shaping the overall asymptotic dynamics, according to the famous soliton resolution conjecture. Open questions involve the complete characterization of solitary solutions and their stability properties, and the asymptotic properties of the dynamics.
• Involved person: Claudio Cacciapuoti
##### Numerical linear algebra
• We are mainly interested in linear systems and eigenvalues of large dimensions arising from the discretization of a differential and/or integral operator. The research is mainly focussed on: 1) Sequence of matrices and asymptotic spectral distribution by genelarized locally toeplitz (GLT) theory for several kind of discretizations (Isogeometric analysis, finite elements, etc.). 2) Fast iterative methods for linear system. In particular new preconditioning proposal and multigrid methods for structured matrices. Definition and convergence analysis, with the final aim of obtaining optimal methods for the involved large linear systems. 3) Eigenvalue computation of large matrices with (hidden) structure: asymptotic expansion formulae and extrapolation methods starting from the GLT symbol. 4) Fractional derivatives. Analysis of different numerical methods and proposal of fast iterative solvers.
• Involved persons: Marco Donatelli, Stefano Serra-Capizzano, Mariarosa Mazza
##### NUMERICAL optimization and machine learning
• Numerical solution of optimization problems by algorithms based on either deterministic or random models.
• Design and theoretical analysis of the algorithms and their application on various classes of problems including those arising from machine learning.
• Analysis and iterative solution of large-scale linear systems arising from optimization problems.
• Preconditioning and multigrid strategies to speed up the convergence of iterative methods for large-scale optimization problems.
• Application of optimization methods to image reconstruction, inverse problems, and industrial applications.
• Involved persons: Marco Donatelli, Benedetta Morini

##### Numerical solutions of PDEs
• Development, analysis and implementation of numerical methods for hyperbolic conservation laws, with particular emphasis on high order accurate methods, grid- and scheme-adaptivity, well-balancing.
• Involved person: Matteo Semplice
##### Regularization methods for ill-posed problems
• We mainly focus on new regularization methods for ill-posed problems with an interest in both theoretical and numerical aspects. We combine techniques of structured numerical linear algebra with numerical methods for nonlinear models arising from regularization models based on fractional derivatives or on special semi-norms including sparsity or non-negativity constraints of the solution. Multilevel and wavelets decompositions. Concerning the applications a particular attention is devoted to image deconvolution and to collaborations with physicians in optics.
• Involved persons: Marco Donatelli, Stefano Serra-Capizzano
##### Splines and CAGD methods
• Splines are smooth piecewise functions whose pieces are drawn from certain given (usually polynomial) spaces. Thanks to their representation in terms of the B-spline basis, they are at the core of CAGD (Computer Aided Geometric Design), the collection of mathematical tools to describe and manipulate curves, surfaces, volumes and grids. CAGD primitives have been recently adopted in IgA (Isogeometric Analysis), a computational approach that combines and extends engineering analysis with CAGD. The key idea of IgA is to use the CAGD representation tools in both the design and the analysis phase, providing a true design-through-analysis methodology. Our research addresses different aspects of splines and CAGD methods, ranging from theoretical issues (construction of spline spaces and bases, approximation schemes,…) to applications in the context of IgA (analysis-suitable descriptions of complex geometries, adaptive refinement strategies, spectral analysis and design of fast solvers,…)
• Involved persons: Hendrik Speleers, Carla Manni