Dipartimento di Matematica

Abstract

The theory of flight developed by Kutta, Lanchester and Zhukovsky is mainly based on the idea that a cambered surface produces lift through its ability to generate a vortex about itself. Since fluid flows around an obstacle generate vortices which are difficult to locate and to describe, in this talk we analyze the Stokes and Navier-Stokes equations for the two-dimensional motion of a viscous fluid in the exterior of a fixed obstacle. Firstly, we discuss nonstandard boundary conditions for the Stokes equations on a smooth obstacle, allowing for the generation of turbulence over the leeward wall of the body. Secondly, after studying the connection between the appearance of lift and the unique solvability of the Navier-Stokes equations, we show some numerical results that compare the aerodynamic response of different non-smooth obstacles, as the inlet velocity is increased until reaching the critical Reynolds number. The talk accounts for results contained in two articles prepared in collaboration with Filippo Gazzola and Andrei Fursikov (Moscow State University).

Abstract

In the context of functional data analysis, probability density functions as non-negative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral constraint without loss of information. On the other hand, all these properties are a challenge when the densities need to be approximated with spline functions, including construction of the respective B-spline basis. The Bayes space methodology of density functions enables to express them as real functions in the standard L2 space using the clr transformation. The resulting functions possess the zero integral constraint. This is a key to propose a new B-spline basis, holding the same property, and consequently to build a new class of spline functions, called simplicial splines, which can approximate probability density functions in a consistent way. The contribution provides also construction of smoothing simplicial splines and possi!
ble orthonormalization of the B-spline basis which might be useful in some applications. Finally, a concise analysis using simplicial splines is demonstrated with anthropometric data for the case of simplicial functional principal component analysis.

Contact alessandra.menafoglio@polimi.it

Abstract

Since the pioneering works of A. Einstein, M. von Smoluchowski, and P. Langevin, stochastic processes and the related Fokker-Planck equations have been used to model irregular motion in different situation. These equations are also considered to be adequate for modelling the motion of one pedestrian and of a crowd of people under certain circumstances. Therefore they can play a central role in the design of control strategies for different purposes as well as in the analysis and calibration of microscopic models of pedestrian interaction. The purpose of this talk is to review some mathematical results in this field, which includes different modelling issues concerning the optimal control of a single pedestrian subject to perturbation, the development of a new framework for pedestrians' avoidance dynamics based on the formulation of a Nash game, and a new approach to the analysis of superresolution 2D images of moving molecules on a cell membrane to obtain quantitative results on the organization of the membrane and on the interaction between the molecules.

[1] M. Annunziato, A. Borzì, A Fokker-Planck control framework for stochastic systems, EMS Surveys In Mathematical Sciences, 5 (2018), 65-98.
[2] M. Annunziato, A. Borzì, A Fokker-Planck control framework for multidimensional stochastic
processes, Journal of Computational and Applied Mathematics, 237 (2013), 487-507.
[3] S. Roy, A. Annunziato, A. Borzì, C. Klingenberg, A Fokker-Planck approach to control collective motion, Computational Optimization and Applications, 69 (2018), 423-459.
[4] S. Roy, A. Annunziato, A. Borzì, A Fokker-Planck feedback control-constrained approach for modelling crowd motion, Journal of Computational and Theoretical Transport, 45 (2016), 442-458.
[5] S. Roy, A. Borzì, A. Habbal, Pedestrian motion modelled by Fokker-Planck Nash games, Royal Society open science, 4, (2017) 170648-17.

Contact: marco.verani@polimi.it

Abstract

The image quality of ground based astronomical telescopes suffers from turbulences in the atmosphere. Adaptive Optics (AO) systems use wavefront sensor measurements of incoming light from guide stars to determine an optimal shape of deformable mirrors (DM) such that the image of the scientific object is corrected after reflection on the DM(s). The operation of the AO systems of the new generation of large astronomical telescopes under construction requires new mathematical methods that achieve reconstruction in real time. In the talk we will in particular focus on methods for wavefront reconstruction from pyramid sensor data and an analysis of the atmospheric tomography operator.

Contact christian.vergara@polimi.it

Abstract

This talk outlines a novel numerical approach for accurate and computationally efficient integrations of PDEs governing all-scale atmospheric dynamics. Such PDEs are not easy to handle, due to a huge disparity of spatial and temporal scales as well as a wide range of propagation speeds of natural phenomena captured by the equations. Moreover, atmospheric dynamics constitutes only a small perturbation about dominant balances that result from the Earth gravity, rotation, composition of its atmosphere and the energy input by the solar radiation. Maintaining this mean equilibrium, while accurately resolving the perturbations, conditions the design of atmospheric models and subjects their numerical procedures to stringent stability, accuracy and efficiency requirements.
The novel Finite-Volume Module of the Integrated Forecasting System (IFS) at ECMWF (hereafter IFS-FVM) solves perturbation forms of the fully compressible Euler/Navier-Stokes equations under gravity and rotation using non-oscillatory forward-in-time semi-implicit time stepping and finite-volume spatial discretisation. The IFS-FVM complements the established semi-implicit semi-Lagrangian pseudo-spectral IFS (IFS-ST) with the all-scale deep-atmosphere formulation cast in a generalised height-based vertical coordinate, fully conservative and monotone advection, flexible horizontal meshing and a predominantly local communication footprint. Yet, both dynamical cores can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude-latitude coordinates and physics parametrisations, thus facilitating their synergetic relation.
The focus of the talk is on the mathematical/numerical formulation of the IFS-FVM with the emphasis on the design of semi-implicit integrators and the associated elliptic Helmholtz problem. Relevant benchmark results and comparisons with corresponding IFS-ST results attest that IFS-FVM offers highly competitive solution quality and computational performance.

Contact: luca.bonaventura@polimi.it