Title: The Complete Manifold of Planar Triangular Meshes

Abstract: In this talk, we will consider two-dimensional shapes represented by triangular meshes of a given connectivity. We will show that the collection of admissible configurations representable by such meshes forms a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We will detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. We will end with some numerical experiments which show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.

Title: Data-driven methods for spline model design

Abstract: Free-form data fitting plays an important role in Computer Aided Geometric Design (CAGD) and Geometric Modelling. In particular, spline weighted least square approximations are ubiquitous in applications when trying to reconstruct an unknown function from its observations.

In this talk, we address two pre-processing steps necessary to fit a given point cloud with splines. Firstly, we discuss the computation of suitable point parameterization providing a data-driven strategy in order to improve the model accuracy. Subsequently, we discuss the role of weight functions both to express weighted spline least squares approximations as a convex combination of spline interpolates, and to reduce the computational cost of the fitting procedure.

Title: Optimal actuator design for vibration control

Abstract: We present an approach to optimal actuator design for vibration control based on shape and topology optimisation techniques. Given a PDE governing the vibration dynamics, we determine the best actuator shape/location for a given initial condition or a set of initial conditions not exceeding a chosen norm. We compute shape and topological sensitivities of the corresponding cost functionals. A numerical realization of the optimality condition is developed for the actuator shape using a level-set method for topological derivatives. A numerical example illustrating the design of actuator for the Euler-Bernoulli beam model is provided.

Title: Explicit resultant matrices for families of structured polynomial systems

Abstract: Generalized resultants characterize the roots of a polynomial system, similarly to the classical Sylvester resultant of two univariate polynomials. However, the problem of finding determinantal (ie. square and free of extraneous factors) matrices for the multivariate resultant does not have a general solution. In fact such formulas are known only for certain shapes of the Newton polytopes of the equations. In this talk we give an overview of different known constructions and we focus on a method that employs the Weyman complex, that can be used to obtain determinantal formulations for the multivariate resultant of structured systems, that is, systems of polynomial equations with a Newton polytopes of special shape. We also show how these matrices can be applied for the numerical computation of the roots of a polynomial system, as well as for solving the multiparameter eigenvalue problem.