Dipartimento di Matematica - Sapienza Università di Roma

Seminario di Modellistica Differenziale Numerica    

     Keywords: Computational Fluid Dynamics, Differential Games, Front propagation, Hamilton-Jacobi equations, Image processing, Material Science, Optimal control, Sand piles      Keywords: Computational Fluid Dynamics, Differential Games, Front propagation, Hamilton-Jacobi equations, Image processing, Material Science, Optimal control, Sand piles      Keywords: Computational Fluid Dynamics, Differential Games, Front propagation, Hamilton-Jacobi equations, Image processing, Material Science, Optimal control, Sand piles      Keywords: Computational Fluid Dynamics, Differential Games, Front propagation, Hamilton-Jacobi equations, Image processing, Material Science, Optimal control, Sand piles


Calendario degli incontri a.a. 2017-2018

Martedì 20 marzo 2018, ore 15.00, Aula di Consiglio

M. De Santis
DIAG, Sapienza
Active-set Algorithmic Frameworks for Sparse Optimization

Abstract: In the context of sparse optimization problems, being able to quickly identify the active set (i.e. the subset of zero components in an optimal solution) is becoming a crucial task as it can considerably reduce the complexity of the problem. We describe an active set estimate that tries to quickly identify as many active variables as possible at a given point, while guaranteeing that some approximate optimality conditions are satisfied. A relevant feature of the estimate is that it gives a reduction of the objective function when setting to zero all those variables estimated active. This enables to easily embed the active set estimate into a given globally converging algorithmic framework. Some numerical results are reported to show the effectiveness of the approach.
The presented results are related to joint works with Andrea Cristofari (Università di Padova), Stefano Lucidi (Sapienza, Università di Roma), Francesco Rinaldi (Università di Padova).

Martedì 20 marzo 2018, ore 16.00, Aula di Consiglio

Maya Briani
Runge-Kutta Discontinous Galerkin methods for shallow-water equations on a canals network

Abstract: We shall consider the one-dimensional Saint-Venant equations for the numerical simulation of shallow water flows on a canals network. We assume that no source term is present and we deal with the delicate problem of defining and simulating the dynamic at the canals junction. After revisiting the basics of Runge-Kutta (RK) discontinuous Galerkin (DG) on a single canal, we describe the numerical coupling condition at the junction according to the two main used approaches, which impose the equality of water height and the continuity of energy (besides the conservation of flow) between incoming and outgoing canals. We conclude giving some numerical tests on one-to-two and two-to-one junctions.

Martedì 13 marzo 2018, ore 15.00, Aula di Consiglio

Silvia Falletta
Politecnico di Torino
Metodi FEM-BEM di tipo spazio-tempo per la risoluzione di problemi di propagazione di onde in domini illimitati

Abstract: In questo seminario presenterò problemi di propagazione di onde in domini illimitati bi e tridimensionali. Per la risoluzione di tali problemi mediante metodi FEM, considererò una condizione non riflettente basata su un'equazione integrale di tipo spazio-tempo, discretizzata in tempo mediante formule di quadratura di convoluzione di Lubich e in spazio mediante un metodo di Galerkin o di collocazione. Oltre a risultare numericamente stabile e a mostrare un'ottimale accuratezza, la condizione NRBC ha la proprietà di poter essere applicata a contorni artificiali di forma generica; inoltre, dal punto di vista numerico, risulta competitiva con ben note (ma meno accurate) condizioni NRBC di tipo locale. L'approccio proposto è stato applicato a diversi problemi di propagazione di onde: onde acustiche generate da sorgenti localizzate lontano dall'ostacolo, scatterate da uno o più ostacoli fissi o in movimento, onde sismiche che si propagano in mezzi isotropi. I risultati numerici ottenuti dimostrano l'efficienza e l’ accuratezza del metodo proposto.

Martedì 6 marzo 2018, ore 12.00, Aula di Consiglio

Adriano Festa
Università di Rouen
A model for a network of conveyor belts with discontinuous speed and capacity

Abstract: We introduce a macroscopic model for a network of conveyor belts with various speeds and capacities. In a different way from traffic flow models, the product densities are forced to move with a constant velocity unless they reach a maximal capacity and start to queue. This kind of dynamics is governed by scalar conservation laws consisting of a discontinuous flux function. We define appropriate coupling conditions to get well-posed solutions at intersections and provide a detailed description of the solution. Some numerical simulations are presented to illustrate and confirm the theoretical results for different network configurations.
Joint work with M. Pfirsching and S. Goettlich.

Martedì 20 febbraio 2018, ore 15.00, Aula di Consiglio

Giulio Paolucci
Dottorato, SAPIENZA Università di Roma
Adaptive Filtered Schemes for first order Hamilton-Jacobi equations

Abstract: The accurate numerical solution of Hamilton-Jacobi equations is a challenging topic of growing importance in many fields of application but due to the lack of regularity of viscosity solutions the construction of high-order methods can be rather difficult. We will consider a class of “filtered” schemes for first order evolutive Hamilton-Jacobi equations. These schemes, already proposed in the literature, are based on a mixture of a high-order (possibly unstable) scheme and a monotone scheme, according to a filter function F and a coupling parameter epsilon. This construction allows to have a scheme which is high-order accurate where the solution is smooth and is monotone otherwise. This feature is crucial to prove that the scheme converges to the unique viscosity solutions. In this talk we will present an improvement of the classical filtered scheme, introducing an adaptive and automatic choice of the parameter epsilon at every iteration. To this end, we use a smoothness indicator in order to select the regions where we can compute the regularity threshold epsilon. Our smoothness indicator is based on some ideas developed for the construction of the WENO schemes, but other indicators with similar properties can be used. We present a convergence result and error estimates for the new scheme, the proofs are based on the properties of the scheme and on the properties of the indicators. We will illustrate also a number of numerical tests confirming that the adaptive filtered scheme is very efficient in many situations and improves previous results in the literature.
Joint work with Maurizio Falcone and Silvia Tozza.

Martedì 6 febbraio 2018, ore 15.00, Aula L

V. Roscani
Image denoising techniques for astronomical images

Abstract: The goal of image denoising algorithms is to reduce the contribution of the observational noise intrinsic to the images, preserving information (e.g. small sources, morphological details etc.). State-of-the-art mathematical algorithms have been developed and successfully applied in many fields such as medical imaging, video surveillance, satellite observations of the earth. Although numerous types of denoising algorithms exist, a detailed evaluation of their impact on astronomical images is currently lacking, with few examples of denoising applied to astronomical data found in the literature. Testing different algorithms is then fundamental to make a reasonable comparison, especially it is of interest to understand their performance with respect to faint and distant astronomical objects (e.g. galaxies in the early Universe) which are predominant in extragalactic images. I will show how some of these algorithms can enhance the analysis quality for this type of images and improve performances of future astronomical surveys.

Martedì 30 gennaio 2018, ore 15.00, Aula di Consiglio

Y. Queau
TU Munich
Depth map super-resolution from shading

Abstract: Super-resolution is one of the most classical inverse problems in computer vision. Given one low-resolution and possibly noisy input image, it aims at estimating a higher-resolution and denoised image. Efficiently solving this problem is crucial in low-cost 3D-sensing, since consumer depth sensors such as Microsoft's Kinect provide a very low resolution depth image which is prone to strong quantization and noise artifacts. On the other hand, such sensors usually also provide a companion RGB image which is typically of higher resolution and better quality. Single-view 3D-reconstruction could be achieved using solely these color clues, however this task is another ill-posed inverse problem, known as shape-from-shading. In this talk, I will show how these two ill-posed problems can be jointly solved within a variational framework, by using the low-resolution depth clues to disambiguate shape-from-shading or, symmetrically, using the high-resolution shading clues to disambiguate depth super-resolution. The numerical solving of the resulting non-convex variational problem using an augmented Lagrangian approach will be discussed, and real-world experimental results will be presented.

Martedì 30 gennaio 2018, ore 16.00, Aula di Consiglio

Jean-Denis Durou
IRIT, Toulouse
Multi-view shape-from-shading

Abstract: A way to overcome the concave/convex ambiguity of shape-from-shading (SfS) is to use several images of an object taken from different viewpoints, instead of a single one. A computer vision pipeline combining the structure-from-motion and the multi-view stereo techniques can be used to get an initial 3D-shape of the object surface, which is then refined using SfS.
This is a part of Jean Melou's PhD thesis, which is co-supervised with Yvain Queau (TUM, Munich, Germany).

Martedì 23 gennaio 2018, ore 15.00, Aula di Consiglio

Margherita Porcelli
Università di Firenze
Preconditioning semismooth Newton methods for optimal control problems with L^1-sparsity and control constraints

Abstract: PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L^1-terms as well as additional inclusion of box control constraints require the use of semismooth Newton methods. We propose robust preconditioners for different formulations of the Newton's equation. With the inclusion of a line-search strategy and an inexact approach for the solution of the linear systems, the resulting semismooth Newton's method is reliable for practical problems. We present results on the theoretical analysis of the preconditioned matrix and numerical experiments that illustrate the robustness of the proposed scheme.
This is joint work with Valeria Simoncini and Martin Stoll.

Lunedì 15 gennaio 2018, ore 15.30, Aula di Consiglio

Cristina Campi
Università di Genova
Generalized Hough transform for segmentation of X-ray Computed Tomography images

Abstract: The talk presents a generalization of the Hough transform (HT) to special classes of curves: recently, algebraic geometry arguments have been applied to generalize the HT approach to algebraic curves. A computationally efficient generalization of the algorithm for line detection can be implemented in order to automatically recognize profiles described by algebraic curves like rational and elliptic curves. Applications concern segmentation of X-ray Computed Tomography images.

Martedì 9 gennaio 2018, ore 15.00, Aula di Consiglio

Elisabetta Carlini
SAPIENZA Università di Roma
Discretizzazione di alcune equazioni di Fokker-Plank-Kolmogorov non lineari e applicazioni

Abstract: In questo seminario presento uno schema numerico per un sistema di equazioni di Fokker-Planck-Kolmogorov, in cui la dipendenza dei coefficienti è non lineare e non locale rispetto alle incognite. Lo schema è di tipo Semi-Lagrangiano, preserva la non negatività della soluzione e conserva la massa. Si mostra un risultato di convergenza e si studia la sua applicabilità in due esempi. Il primo riguarda un modello che descrive due specie interagenti e il secondo riguarda un sistema di giochi a campo medio per due popolazioni.
Lavoro in collaborazione con J.F.Silva

Martedì 12 dicembre 2017, ore 15.00, Aula B

Raffaele D'Ambrosio
Università L'Aquila
Recent advances in structure-preserving numerical integration of differential problems: deterministic and stochastic aspects

Abstract: The talk presents an overview of selected results regarding recent achievements in the field of structure-preserving numerical integration of deterministic and stochastic differential equations. The proposed methodology leads to problem-oriented numerical schemes, able to accurately and efficiently reproduce typical properties of the continuous problem along the discrete dynamics. These features include, for instance, preservation of invariants for Hamiltonian problems, retaining the long term dynamics of stochastic oscillators, achieving exponential mean-square contractivity for nonlinear stochastic differential equations, retaining periodic wavefronts in reaction-diffusion problems, and so on. The presentation aims to show the benefits of merging a-priori achievable informations on the problem into the numerical scheme, with a significant gain in accuracy and efficiency with respect to the classical case of general purpose numerical methods. The presented results deals with the joint research described in a selection of recent papers in collaboration with E. Buckwar (Johannes Kepler University of Linz), J.C. Butcher (University of Auckland), L. Dieci (Georgia Institute of Technology), E. Hairer (University of Geneva), B. Paternoster (University of Salerno).

Martedì 28 novembre 2017, ore 15.00, Aula di Consiglio

Michele Giuliano Carlino
Istituto Universitario di Studi Superiori, IUSS, Pavia
Riduzione di Modello PGD: Applicazione all'Elettrocardiologia

Abstract: Tra le varie tecniche di riduzione di modello note in letteratura, la Proper Generalized Decomposition (PGD) si propone come un metodo particolarmente adatto per l’approssimazione di problemi differenziali parametrici. A differenza di tecniche di riduzione di modello più note come le basi ridotte, la PGD costruisce un’approssimazione per la soluzione di tali problemi senza alcuna necessità di avere una conoscenza a priori, seppur parziale, della soluzione.
L’idea alla base della PGD è quella di trattare eventuali parametri di interesse per il problema in esame come variabili indipendenti addizionali. Il conseguente aumento della dimensionalità del problema viene gestita da un punto di vista formale con una classica separazione delle variabili, da un punto di vista computazionale tramite un algoritmo di tipo alternating direction, che consente di gestire separatamente la dipendenza della soluzione da ciascuna variabile indipendente. Questo artificio porta la complessità computazionale di tipo esponenziale rispetto alla dimensione del problema, tipica, ad esempio, di un approccio standard agli elementi finiti, a scalare linearmente, con un conseguente guadagno non trascurabile sui tempi di calcolo.
In questo talk, dopo aver introdotto la PGD nella sua versione più generica ed averla applicata a casi test benchmark in cui i parametri possono avere una diversa natura, utilizziamo la PGD per l’approssimazione di un problema inverso noto in letteratura come Inverse Conductivity (ICT) Problem nell’ambito della modellazione dell’elettrofisiologia cardiaca. In particolare, dopo aver descritto il modello monodomain di riferimento per la polarizzazione alla macro-scala delle cellule del tessuto cardiaco, vengono stimati i parametri di diffusione del potenziale elettrico tramite la soluzione parametrica PGD, minimizzando un certo funzionale di costo.
Questo lavoro è stato svolto in collaborazione con Simona Perotto, MOX-Politecnico di Milano, e Alessandro Veneziani, Emory University, Atlanta, GA, USA.

Martedì 21 novembre 2017, ore 15.00, Aula di Consiglio

M. Semplice
Università di Torino
Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws driven by numerical entropy production

I will present a third order accurate finite volume scheme under Adaptive Mesh Refinement (AMR) on quad-tree type grids.
In the scheme, AMR is driven by the, so called, numerical entropy production. This is a residual of the entropy inequality that is computable in each space-time finite volume during the simulation and that is of the same size of the truncation error, which has been successfully exploited to control adaptive behavior of schemes in various ways in one and two space dimensions. Of course, the reconstruction of point values from cell averages requires a procedure that is third order accurate, non-oscillatory, but also versatile enough to handle data on unstructured and non-conforming grids and efficient in computing point data at very many points in each cell: the Central WENO (CWENO) technique is employed here for this task.

Martedì 14 novembre 2017, ore 15.00, Aula di Consiglio

E. Iacomini
Dottorato, SBAI Sapienza
Sensitivity analysis of the LWR model for traffic forecast on large networks using Wasserstein distance

Abstract: In this talk we present a sensitivity analysis of a PDE model for traffic forecast on networks. The analysis is made with respect to the parameters and to the network. In order to compare different numerical solutions coming from different inputs, we will use the Wasserstein distance, computed by a suitable numerical method. Traffic uncertainty is evaluated for different initial data, different fundamental diagrams, different vehicle distributions at junctions and different network sizes.
Joint work with M. Briani and E. Cristiani.

Martedì 7 novembre 2017, ore 15.00, Aula di Consiglio

Roberto Ferretti
Roma Tre
Un solutore diffusione-trasporto esplicito, a grandi passi in tempo, per l'equazione di Navier-Stokes

Abstract: Si discuterà l'introduzione di un solutore diffusione-trasporto totalmente semi-Lagrangiano nella soluzione della equazione di Navier-Stokes, sia nella formulazione vorticita'-funzione di corrente che in quella pressione-velocità. Nonostante il basso ordine di consistenza, questo schema si dimostra efficace e di basso costo computazionale, permettendo numeri di Courant relativamente grandi ed evitando l'introduzione di viscosità numerica indesiderata. Si presenterà lo schema, in particolare le strategie di upwinding e l'implementazione delle condizioni al bordo, e si mostreranno test numerici su benchmark classici, sia in regime laminare che turbolento.