Mathematics of Computation

• Numerical optimization. Development, analysis and implementation of algorithms for constrained and unconstrained optimization,  nonlinear systems of equalities and inequalities, quadratic programming problems.  Applications to image reconstruction problems and industrial applications. Analysis and iterative solution of large-scale linear systems from optimization.  Updating strategies for algebraic preconditioners in the iterative solution of sequences of large-scale linear systems.

Involved person: Benedetta Morini

• 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

• 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

• Constructive Point-Free Mathematics. This research field, which is a branch of Mathematical Logic, is focused on methods and methodologies rather than a specific area in Mathematics. In fact, “point-free” means that a universe is not
  required to exist in advance in order to interpret a mathematical theory: for example, analysis is developed without presuming the existence of real and complex numbers, topology does not describe spaces as structured sets of points, etc. The structure which provides meaning to these theories focuses on the algebraic, computational, and logical relations among their statements, starting off from their axioms. Similarly, “constructive” means that an abstract algorithmic interpretation equips every proof in a theory, providing
  computational evidence and an abstract justification to the corresponding theorem. Specific themes of particular interest along this approach are: homotopy type theory, well quasi orders, category theory, Grothendieck’s topoi, constructive real analysis. Minor lines which are pursued alongside the main themes are: how to mechanise reasoning in this field within the so-called logical frameworks; the epistemological understanding of these results in the Philosophy of Information.

Involved person:  Marco Benini

• Topos theory. Toposes were originally introduced by Alexandre Grothendieck as purveyors of cohomology invariants useful in algebraic geometry (in particular in relation to Weil’s conjectures), but very soon their fruitfulness and prospective impact became apparent also in other fields of Mathematics. In fact, it was realized that a topos can be considered not only as “generalized space”, but also as a “mathematical universe” or as an object which embodies the “semantical content” of mathematical theories of a very general form. More recently, toposes have started being effectively used as sorts of “unifying bridges” making it possible to link different mathematical theories together, to generate and study dualities and equivalences, to transfer ideas and results from one mathematical field to another and to demonstrate new results within a given theory. My main research interest is to further develop this unification programme both at the theoretical and at the applied level, particularly in subjects such as duality theory, algebra, model theory, algebraic geometry and proof theory.