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. 2018-2019

Martedì 26 marzo 2019, ore 14.15, Aula B

Luigi Preziosi
Dipartimento di Scienze Matematiche, Politecnico di Torino
Degenerate Parabolic Models for Sand Slides

Abstract: Four phenomena contribute to wind-induced sand movement and eventually to the formation and evolution of dunes: erosion from the sand bed, transport by the wind, sedimentation due to gravity, and sand grain slides occurring when the slope of the accumulated sand exceeds a critical angle of repose. In particular, erosion, sedimentation, and the formation of such small avalanches determine the evolution of the free-boundary over which wind blows and transports the sand. The need to couple the multiphase turbulent fluid-dynamics with the dynamics occurring at the sand surface requires to deduce mathematical models for such phenomena that are able to describe the evolution of the surface in an accurate way, but that is at the same time computationally fast.
Starting from this need, the aim of this talk is to propose a new mathematical model based on using classical continuum mechanics tools under the assumptions that the thickness of the creep layer is small and that the grains in it move in the direction of the steepest descent with a speed that is determined by several constitutive closures (Coulomb-like or pseudo-plastic fluids). The mathematical models deduced as degenerate parabolic equations for the height of the sand pile. In spite of their simplicity, all the models reply many well known behaviours characterizing the evolution of sand piles, such as the non-uniqueness of static configurations in subcritical conditions and the link between critical stationary configurations and the eikonal equation, and behave well in all tested set-ups. The only slight difference among them lies in the temporal evolution of the interface.

Martedì 26 marzo 2019, ore 15.15, Aula B

Athena Picarelli
Dipartimento di Scienze Economiche, Università di Verona
Some high order filtered schemes for parabolic Hamilton-Jacobi-Bellman equations

Abstract: We consider high order numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations. For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. We present a class of "filtered" schemes: a suitable local modification of the high order scheme is introduced by "filtering" it with a monotone one. The resulting scheme can be proven to converge and it still shows an overall high order behavior for smooth enough solutions. We give theoretical proofs of these claims and validate the results with numerical tests.

Martedì 12 marzo 2019, ore 15.00, Aula di Consiglio

Lehel Banjai
Heriot-Watt University, Edinburgh (UK), Maxwell Institute for Mathematical Sciences
Tensor Finite Element Methods for the Spectral Fractional Laplacian

Abstract: In this talk we discuss the analysis and fast implementation of numerical algorithms for the solution of fractional linear elliptic problems on a bounded domain with a homogeneous Dirichlet boundary condition. We focus on the spectral fractional Laplacian and the corresponding Caffarelli-Silvestre extension. We make use of tensor product finite element spaces with hp finite elements in the extended direction and piecewise linear or hp finite elements in the bounded domain and prove convergence estimates. We describe fast methods for solving the resulting linear system resulting in algorithms of almost optimal complexity. Numerical results will be shown that confirm the theoretical predictions.

Martedì 5 marzo 2019, ore 15.00, Aula di Consiglio

Corrado Mascia
SAPIENZA Università di Roma
Modelling reaction-diffusion beyond the Fourier law

Abstract: The classical modelling of diffusion processes posits that the flux is proportional to the gradient of the density, with the final output that the resulting equation is parabolic. Such description has a number of flaws, the most famous being the infinite speed of propagation. The main problem lies in the fact that the system's response is assumed to be instantaneous. Differently, the presence of a time-lag, corresponding to the presence of inertial terms, has to be taken into account. In fact, philosophically speaking, parabolicity is an exceptional event, and the generic case (which preserves well-posedness) is the hyperbolic one. In this presentation, attention is given to the Maxwell-Cattaneo law -a constitutive equation for the evolution of the flux- which is the simplest mechanism to incorporate the presence of a relaxation time-scale and furnishing a hyperbolic model. Specifically, the talk concentrates on the main differences when the standard balance equation is coupled with the Maxwell-Cattaneo law in place of the Fourier law. Specific attention will be given to the presence of propagating fronts and their (analytical and numerical) stability.

Martedì 26 febbraio 2019, ore 15.00, Aula F

Ourania Giannopoulou
Dottorato, Matematica SAPIENZA
Chorin’s approaches revisited: Particle Vortex Method vs Finite Volume Method

Abstract: In this work a Vortex Particle Method is combined with a Boundary Element Method for the study of viscous incompressible planar flow around solid bodies. The method is based on Chorin’s operator splitting approach, consisting of an advection step followed by a diffusion step. The evaluation of the advection velocity exploits the Helmholtz-Hodge Decomposition, while the no–slip condition is enforced by an indirect boundary integral equation. No mesh is used for the solution of the Poisson equation for the velocity (advection step) and the diffusion step is performed on a Regular Point Distribution with no topological connection; therefore, the resulting algorithm is completely meshless. We also revise the use of the same decomposition for the solution of the Navier–Stokes equations in primitive variables and its role in maintaining the divergence–free constraint. The results are compared with those obtained by a mesh-based Finite Volume Method, where the pseudo-compressible iteration is exploited to enforce the solenoidal constraint on the velocity field. Several benchmark tests were performed for verification and validation purposes; in particular, the unsteady flow past a circular cylinder, an ellipse with incidence and an equilateral triangle was simulated for several values of the Reynolds number.

Martedì 12 febbraio 2019, ore 15.00, Aula di Consiglio

Sergio Pirozzoli
Department of Mechanical and Aerospace Engineering, SAPIENZA
Numerical methods for compressible Navier-Stokes equations

Abstract: We review numerical methods for direct numerical simulation of turbulent compressible flow in the presence of shock waves. Ideal numerical methods should be accurate and free from numerical dissipation in smooth parts of the flow, and at the same time they must robustly capture shock waves without significant Gibbs ringing, which may lead to nonlinear instability. Adapting to these conflicting goals leads to the design of strongly nonlinear numerical schemes that depend on the geometrical properties of the solution. For low-dissipation methods for smooth flows, numerical stability can be based on physical conservation principles for kinetic energy and/or entropy. Shock-capturing requires the addition of artificial dissipation, in more or less explicit form, as a surrogate for physical viscosity, to obtain non-oscillatory transitions. Methods suitable for both smooth and shocked flows will be discussed, and the potential for hybridization is highlighted. Examples of the application of advanced algorithms to DNS/LES of turbulent, compressible flows will be presented.

Martedì 5 febbraio 2019, ore 15.00, Aula di Consiglio

Simona Perotto
MOX, Dipartimento di Matematica, Politecnico di Milano
Minimizzazione di funzionali energia attraverso un’adattazione anisotropa della mesh

Abstract: In questa presentazione ci focalizziamo su due diversi tipi di fenomeni che possono essere modellati attraverso la minimizzazione di un opportuno funzionale energia, ovvero la propagazione di fratture in mezzi fragili e la segmentazione di immagini. In entrambe le applicazioni, l’obiettivo è quello di identificare un contorno di interesse all’interno di un certo dominio. Il modello di riferimento, per entrambi i casi, è rappresentato dal funzionale di Mumford-Shah. Tale modello risulta, in generale, poco utile da un punto di vista pratico, soprattutto se si è interessati ad una sua controparte discreta che risulti computazionalmente efficiente. Questo limite giustifica l’introduzione sul panorama scientifico di diverse approssimazioni del funzionale di Mumford-Shah. Nello specifico, noi ci riferiremo all’approssimazione proposta da Ambrosio e Tortorelli. Per migliorarne l’efficienza computazionale, abbiamo arricchito l’algoritmo standard basato sul funzionale di Ambrosio-Tortorelli con una procedura di adattazione anisotropa della griglia computazionale. I vantaggi dovuti ad una scelta ad-hoc della mesh si concretizzano in una considerevole riduzione dell’onere computazionale richiesto per garantire una certa accuratezza nell’identificazione del contorno di interesse. Dopo aver introdotto la parte più teorica, verrà presentato l’algoritmo adattattivo, successivamente validato su casi test di interesse per gli ambiti applicativi oggetto della presentazione.

Martedì 29 gennaio 2019, ore 15.00, Aula di Consiglio

P. Moschetta
Dottorato, Matematica SAPIENZA
Analisi ed approssimazione del modello di Gatenby-Gawlinski per la crescita tumorale

Abstract: Studieremo il modello di Gatenby-Gawlinski, ideato per la descrizione matematica di fenomeni biomedici legati all'invasione tumorale, mediante un approccio di tipo analitico-numerico. Si propone un'opportuna strategia numerica per testare una stima di approssimazione per la velocità dei fronti quale strumento efficace per lo studio del fenomeno delle onde viaggianti. Illustreremo i risultati relativi alle simulazioni condotte con particolare enfasi sul riconoscimento di evidenze sperimentali dal profondo ed utile significato biologico. Infine, porremo le basi per giustificare alcune semplificazioni del modello matematico cercando però di salvaguardare gli aspetti tecnici qualitativamente più rilevanti.

Martedì 22 gennaio 2019, ore 15.00, Aula E

J. Melou
IRIT e Mikros Image
A splitting-based algorithm for multi-view shape-from-shading

Abstract: Shape-from-shading is a classic ill-posed technique for 3D-reconstruction, which suffers from the well-known concave/convex ambiguity. However, if may recover thin details of the 3D-shape. On the other hand, multi-view stereo is not satisfactory either: if the overall 3D-shape is reasonable, thin details are missing and artefacts appears in the textureless areas. We propose a simple yet effective strategy for combining the advantages of both techniques.

Martedì 15 gennaio 2019, ore 15.00, Aula di Consiglio

Adriano Festa
Università de L'Aquila
Uno schema semi-Lagrangiano per equazioni di Hamilton-Jacobi su networks e la sua applicazione a modelli di traffico veicolare

Abstract: In questo seminario presentiamo uno schema numerico per l'approssimazione di soluzioni di viscosità di una classe di equazioni di Hamilton-Jacobi su domini ramificati. Lo schema è esplicito e stabile sotto appropriate condizioni tecniche. Mostriamo un teorema di convergenza e delle stime a priori dell'errore numerico. I risultati teorici sono confermati da specifici test. Per concludere, applichiamo lo schema alla simulazione di problemi di traffico veicolare.
Lavoro sviluppato in collaborazione con E. Carlini (Sapienza, Roma) e N. Forcadel (INSA, Rouen).

Martedì 18 dicembre 2018, ore 15.00, Aula di Consiglio

Alessandro Alla
PUC, Rio de Janeiro
Discovery mathematical models from experimental data

Abstract: In this talk, we will present two recent techniques to discover mathematical models from data using machine learning techniques. In the first part of the talk, we advocate the use of Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the data with a linear model. DMD is a spatio-temporal matrix decomposition of a data matrix that correlates spatial features while simultaneously associating the activity with periodic temporal behavior. With this decomposition, one can obtain a reduced dimensional surrogate model and use for future state predictions or extrapolate missing information from the data. In the second part, we address the problem of discovering nonlinear ODEs and PDEs from data. We will show that we can recover the mathematical problem by means of sparse optimization methods such as LASSO and RIDGE regression. Examples and applications of the methods will be showed during the talk.

Martedì 11 dicembre 2018, ore 15.00, Aula di Consiglio

Elisa Iacomini
Un nuovo modello multiscala per il traffico veicolare

Abstract: In questo seminario presenterò un nuovo modello per lo studio del traffico veicolare che riassume in un unico scenario il punto di vista macroscopico e microscopico, evitando qualsiasi tipo di interfaccia tra le due scale. Per l'approccio macroscopico considereremo il classico modello del primo ordine di Lighthill-Witham-Richards, mentre useremo un modello di tipo Follow-the-Leader del secondo ordine per quanto riguarda la scala microscopica. Saremo così in grado di descrivere alcune dinamiche reali, evitando implementazioni complesse e risparmiando in termini di memoria e di tempo.
Lo schema numerico associato risulta essere conservativo ed in grado di descrivere gli effetti tipici del secondo ordine come le onde di stop&go.
Lavoro in collaborazione con Emiliano Cristiani (IAC-CNR).

Martedì 4 dicembre 2018, ore 15.00, Aula di Consiglio

Gabriella Puppo
Dipartimento di Matematica, SAPIENZA
Ricostruzioni di alto ordine per problemi di tipo iperbolico

Abstract: Nell’integrazione numerica di problemi di tipo iperbolico, gli algoritmi di ricostruzione che permettono di passare dai valori delle medie di cella ai valori puntuali della soluzione sono uno degli aspetti più critici. In questo seminario parleremo di una tecnica nuova che permette di ricostruire un polinomio non-oscillante con proprietà di approssimazione uniforme su tutta la cella computazionale. Discuterò le proprietà spettrali di questa approssimazione, analizzando la diffusione e la dispersione numerica degli schemi ottenuti. Inoltre, in questo lavoro verrà introdotto anche il concetto di distorsione numerica, caratteristico delle tecniche di approssimazione non lineari. Infine, presenterò l’estensione che abbiamo ottenuto recentemente per l’integrazione delle equazioni di Eulero con termine di sorgente, che ci permette di ottenere schemi ben bilanciati di alto ordine per la gas dinamica in un campo gravitazionale.

Martedì 27 novembre 2018, ore 15.00, Aula di Consiglio

Thierry Goudon
INRIA, Sophia-Antipolis
A staggered discretization for solving the Euler equations

Abstract: The mathematical modeling of particulate flows naturally lead to systems of conservation laws involving constraints on velocity fields. The numerical treatment of the constrained systems of PDEs might lead to difficulties: it is not clear that different formulations of the equations remain equivalent at the discrete level, and a careless approach might give rise to spurious instabilities, or to unsatisfactory mass and energy balances. This is reminiscent to the difficulties that appear in the simulation of Euler equations in low Mach regimes, when using standard Riemann solvers. We introduce a new class of schemes for the Euler equations that work on staggered grids, numerical densities and velocities being stored in different locations. Moreover, the design of the numerical fluxes is inspired from the principles of the kinetic schemes. Stability conditions ensuring the positivity of the discrete density and energy can be identified, for both first and second order version of the scheme. The method can be incorporated into a suitable splitting strategy to handle low Mach simulations.

Martedì 20 novembre 2018, ore 14.30, Aula di Consiglio

Gerardo Toraldo
Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università di Napoli Federico II
Proportionality based two-phase gradient methods for large scale quadratic programming problems

Abstract: We propose a two-phase gradient-based method for general Quadratic Programming (QP) problems. Such kind of problems arise in many real-world applications, such as Support Vector Machines, multicommodity network flow and logistics or in variational approaches to image deblurring. Moreover, an effective QP solver is the basic building block in many algorithms for the solution of nonlinear constrained problems. The proposed approach alternates between two phases: an identification phase, which performs Gradient Projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or any spectral gradient method. A critical issue about a two-phase method stands in the design of an effective way to switch from phase 1 to phase 2. In our method, this is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the active variables, defined by extending the concept of proportional iterate, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in the case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.
This talk is based on joint work with Daniela di Serafino (Dipartimento di Matematica e Fisica dell’Università della Campania “Luigi Vanvitelli”) and Marco Viola (Dipartimento di Ingegneria Informatica Automatica e Gestionale “Antonio Ruberti” della Sapienza Università di Roma).
D. di Serafino, G. Toraldo, M. Viola, J. Barlow, A two-phase gradient method for quadratic programming problems with a single linear constraint and bounds on the variables, SIAM Journal on Optimization 28 (4), 2809-2838 (2018)
D. di Serafino, V. Ruggiero, G. Toraldo, and L. Zanni, On the steplength selection in gradient methods for unconstrained optimization, Appl. Math. Comput., 318, pp. 176-195 (2018).
Z. Dostál, G. Toraldo, M. Viola, O. Vlach, Proportionality-Based Gradient Methods with Applications in Contact Mechanics, In: Kozubek T. et al. (eds) High Performance Computing in Science and Engineering. HPCSE 2017. Lecture Notes in Computer Science, vol. 11087 (2018).
Z. Dostál, Box constrained quadratic programming with proportioning and projections, SIAM J. Optim., 7, pp. 871-887 (1997).
J. J. Moré and G. Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM J. Optim., 1, pp. 93-113 (1991).

Martedì 13 novembre 2018, ore 15.00, Aula di Consiglio

Luca Saluzzi
GSSI, L'Aquila
Schemi numerici a tempo discreto per problemi di controllo ottimo ad orizzonte finito

Abstract: Il principio della Programmazione Dinamica per problemi di controllo ottimo si basa sulla caratterizzazione della funzione valore come l'unica soluzione di viscosità dell'equazione di Hamilton-Jacobi-Bellman. Gli schemi per l'approssimazione numerica di queste equazioni sono tipicamente basati su una discretizzazione temporale che è successivamente proiettata su una griglia spaziale tramite interpolazione polinomiale. Presenterò un nuovo approccio per problemi di controllo ottimo ad orizzonte finito dove il calcolo della funzione valore viene eseguita su una struttura ad albero costruito direttamente dalla dinamica discreta, questo permette di eliminare il costo dell'interpolazione spaziale e di affrontare problemi in dimensione molto alta. Per ridurre l'occupazione di memoria, la complessità dell'albero viene ridotta da una tecnica di "potatura". Considererò lo schema di Eulero per discretizzare la dinamica e dimostrerò una stima dell'errore (ordine 1) per la funzione valore mostrando come questo approccio si possa estendere a schemi di ordine più alto. Verranno presentati anche alcuni risultati numerici.
Lavoro in collaborazione con Maurizio Falcone e Alessandro Alla (PUC, Rio de Janeiro).

Martedì 6 novembre 2018, ore 15.00, Aula di Consiglio

Claudio Estatico
Università degli Studi di Genova, Dipartimento di Matematica
Regularization methods in variable exponent Lebesgue spaces

Abstract: Let us consider a functional equation Ax=y characterized by an ill-posed linear operator A acting between two Banach spaces X and Y. In this talk, we propose an extension of the Tikhonov regularization approach to the (unconventional) setting where X and Y are both two variable exponent Lebesgue spaces. Basically, a variable exponent Lebesgue space is a (non-Hilbertian) Banach space where the exponent used in the definition of the norm is not constant, but rather is a function of the domain. This way, we can automatically assign different ``amount’’ of regularization, related to different values of the exponent function, on different regions of the domain.
In the case of image deblurring problems, different pointwise regularization is useful because background, low intensity, and high intensity values of the image to restore require different filtering (i.e., regularization) levels, depending on the local signal to noise ratios in all the different portions of the image domain. A numerical evidence will be also discussed.
Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M., Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics. vol. 2017, Springer, 2011.
Schuster, T., Kaltenbacher, B., Hofmann, B., and Kazimierski, K. S., Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10, De Gruyter, 2012
C. Estatico, S. Gratton, F. Lenti, D. Titley-Peloquin, “A conjugate gradient like method for p-norm minimization in functional spaces”, Numerische Mathematik,