29 Settembre, 2014 16:30

Lezioni Leonardesche

HEAT DIFFUSION IN GEOMETRY

Gerhard Huisken, Tubingen University and Mathematische Forshingsinstitut Oberwolfach

Università di Milano, Dipartimento di Matematica, Via Saldini

18 Settembre, 2014 16:30

Lezioni Leonardesche

CROSSROADS OF SYMPLECTIC RIGIDITY AND FLEXIBILITY

Yakov Eliashberg, Stanford University

Università di Milano, Dipartimento di Matematica, Via Saldini

18 Settembre, 2014 14:30 in punto

Sezione di Geometria, Algebra e loro applicazioni

Reproducing kernel Hilbert spaces, linear (deterministic and stochastic) systems and inverse scattering

Daniel Alpay, Ben Gurion University, Beer Sheva

Aula Seminari del 3 piano Dip. Matematica (Ed. 14)

abstract

abstract

Functions analytic and contractive in the open unit disk play a key role in a host of mathematical domains, for instance operator models, inverse scattering and linear systems. We first discuss the main points in this circle of ideas. The reproducing kernel Hilbert spaces of the kind introduced by de Branges and Rovnyak are an important tool in the arguments.

In the second part of the talk we study the stochastic analogs of the situation above. The complex numbers are then replaced by an algebra of stochastic distributions, with a very special structure. it is the dual of a nuclear Fréchet space, and a countable union of increasing Hilbert spaces with decreasing norms, the norms being related by a family of inequalities.

Finally we briefly outline the free analog of these spaces of stochastic distributions.

References:

D. Alpay. The Schur algorithm, reproducing kernel spaces and system theory, volume 5, SMF/AMS Texts and Monographs of the Société mathématique de France (2001).

D. Alpay and I. Gohberg. Discrete systems and their characteristic spectral functions. Mediterrean Journal of Mathematics, vol. 4 (2007), pp. 1-32.

D. Alpay, P. Jorgensen and D. Levanony. A class of Gaussian processes with fractional spectral measures. Journal of Functional Analysis, Volume 261 (2011), pp. 507-541.

D. Alpay and G. Salomon. Non-commutative stochastic distributions and applications to linear systems theory. Stochastic Processes and their Applications, vol. 123 (2013), pp. 2303-2322.

D. Alpay, P. Jorgensen and G. Salomon: On free stochastic processes and their derivatives. Stochastic Processes and their Applications, vol. 214 (2014), 3392-3411.

16 Settembre, 2014 14:00 in punto

MOX Seminar

L'uso di artefatti per l' insegnamento-apprendimento della matematica.

Mirko Maracci, Universita` di Pavia

AULA SALERI 6° PIANO DIPARTIMENTO DI MATEMATICA - POLITECNICO DI MILANO

abstract

abstract

Il seminario si propone di introdurre e discutere alcuni aspetti della problematica propria dell'uso di 'artefatti' nell'insegnamento
apprendimento della matematica. Il focus sarà principalmente sull'uso di artefatti informatici per la matematica, ma l'analisi che si vuole sviluppare si inserisce in un quadro più ampio che riguarda problematiche connesse, da un lato, alle relazioni tra uso di artefatti, sviluppo cognitivo e sviluppo culturale, e,dall'altro, a come queste si articolano in contesto educativo.

Referente: chiara.andra@polimi.it
Progetto: BetOnMath

16 Settembre, 2014 14:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

Invariant measures, conjugacies and renormalizations of piecewise-smooth circle maps

Akhtam Dzhalilov, Turin Polytechnic University in Tashkent, Uzbekistan

Aula seminari del 3 piano

abstract

abstract

The class of orientation preserving circle homeomorphisms is one of important part of one-dimensional dynamical systems. We will consider piecewise-smooth circle homeomorphisms f with finite many break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative Df has a jump. It is well known that the invariant measures of sufficiently smooth circle diffeomorphisms are absolutely continuous w.r.t. Lebesque measure. But in the case maps with break points the results are quite different. We formulate the main results on invariant measures, conjugacies and renormalizations of piecewise-smooth diffeomorphisms with finite many break points.

15 Settembre, 2014 14:30 in punto

Sezione di Geometria, Algebra e loro applicazioni

Multi-Scale Modeling of Healing Myocardial Infarcts:Interactions Between Mechanical Environment, Scar Formation, and LV Function

JEFFREY HOLMES, Faculty Member, Robert M. Berne Cardiovascular Research Center University of Virginia

AULA SALERI 6° PIANO DIPARTIMENTO DI MATEMATICA -Politecnico di Milano

abstract

abstract

The mechanics of healing myocardial infarcts are a critical determinant of left ventricular function. We recently showed that infarcts healing in different mechanical environments develop different collagen fiber structures and mechanical properties. We developed an agent-based model to evaluate the mechanisms by which mechanical environment directs collagen deposition and remodeling by cardiac fibroblasts and to better understand the effects of therapeutic interventions on the evolving scar structure. Parameters for this model were derived from a combination of published literature and new experiments in engineered tissue-equivalents, focusing particularly on features of cell-matrix interaction such as contact guidance and collagen remodeling. Our results suggest that different environmental cues regulate scar formation in different tissues. In contrast to previous models of skin wound healing showing that chemokine gradients are a dominant regulator of scar formation, we find that mechanical environment is the dominant regulator of evolving scar structure following myocardial infarction. Our results suggest that a number of regenerative and device therapies that alter infarct mechanics, including stem cell injection, polymer injection, surgical reinforcement, and peri-infarct pacing, will also alter scar structure. Our computational model of infarct healing should enable in silico screening of novel therapies for potential adverse effects on scar formation.

contact: davide.ambrosi@polimi.it

12 Settembre, 2014 12:00 in punto

Sezione di Calcolo delle Variazioni ed Equazioni Differenziali

Hénon type equations and concentration on spheres

Ederson Moreira dos Santos, Università di Sao Paulo, Brasile

Aula seminari III piano

abstract

abstract

In this talk, I will present the concentration profile of various kind of symmetric solutions of some semilinear elliptic problems arising in astrophysics and in diffusion phenomena. Motivated by these elliptic equations and exploiting their symmetry, I will discuss about solutions that concentrate and blow up at points and around spheres as the concentration parameter tends to infinity.



This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni - Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola

9 Settembre, 2014 11:00

MOX Seminar

A Fokker-Planck Strategy to Control Stochastic Processes

Alfio Borzi', University of Würzburg

Aula Saleri 6° Piano - Dipartimento di Matematica del Politecnico di Milano

abstract

abstract

An efficient framework for the optimal control of probability density functions (PDF) of multidimensional stochastic processes and piecewise deterministic processes is presented.
This framework is based on Fokker-Planck-type equations that govern the time evolution of the PDF of stochastic processes.
In this approach, the control objectives may require to follow a given PDF trajectory or to minimize an expectation functional.
Theoretical results concerning the forward and the optimal control problems are provided. In the case of stochastic (Ito) processes, the Fokker-Planck equation is of parabolic type and it is shown that under appropriate assumptions the open-loop bilinear control function is unique. In the case of piecewise deterministic processes (PDP), the Fokker-Planck equation consists of a first-order hyperbolic system. Discretization schemes are discussed that guarantee positivity and conservativeness of the forward solution. The proposed control framework is validated with multidimensional biological, quantum mechanical, and financial models.

28 Luglio, 2014 10:30 in punto

MOX Seminar

Higher order methods for simulating fracturing with applications in multi-physics problems

MAURIZIO M. CHIARAMONTE, Dept. of Mechanical Engineering - Stanford University

Aula Seminari - VI piano

abstract

abstract

We develop higher order nite element methods for fracture mechanics. The framework is cast within the context of conforming nite elements [3]. The method [1] exploits the a priori knowledge of the singular behavior of the elds to construct an alternate regular solution. Solving for the alternate problem yields optimal rates of convergence and high order of accuracy. The salient feature of the method is the lack of additional degrees of freedom in comparison with its standard Galerking nite element formulation. E ectively for the same computational cost we obtain a higher order of accuracy.
Along with the above we employ interaction integrals for curvilinear fractures as presented in [2] and generalize their denition for the proposed higher order method. Along with the optimality of the convergence of the solution we showcase the accuracy and the convergent behavior of the computed stress intensity factors. The method is verified with respect several analytical solutions. The application of the framework are showcased for complex fracturing problems. In particular, simulations of fracture instabilities in thermoelastic materials subjected to large temperature gradients, where oscillatory fracture behavior is expected, will be used to demonstrate the robustness and capabilities of the presented tools.

References
[1] Chiaramonte M.M., Shen Y., and Lew A.J. Higher order methods for fracture mechanics. preprint, 2014.
[2] Chiaramonte M.M., Shen Y., Keer L.M., and Lew A.J. Computing stress intensity factors for curvilinear fractures. preprint, 2014.
[3] Rangarajan R., Chiaramonte M.M., Shen Y. Hunsweck M.J., and Lew A.J. Simulating curvilinear crack propagation with universal meshes. Int Journal For Numerical Methods in Engineering, Submitted.


CONTACT: piercesare.secchi@polimi.it

18 Luglio, 2014 10:00

MOX Seminar

A family of methods for DFN flow simulations with non-conforming meshes

STEFANO BERRONE, Politecnico di Torino

Aula seminari MOX

abstract

abstract

A novel approach for simulating subsurface flow in a discrete fracture network (DFN) is presented. The method relies on the reformulation of the problem as a PDE-constrained optimization problem. The minimization of a properly defined functional is adopted to enforce fluid flow continuity and conservation at fracture intersections. The minimization is subject to PDE-constraints, given by the flow equations defined on each fracture. The discretization of the underlying flow equations is performed with finite element based methods.
Since matching conditions are not explicitly imposed, non-conforming meshes at fracture intersections can be used: this crucial point allows for a meshing process which can be independently performed on each fracture, thus being extremely reliable and computationally inexpensive. The figure displays a detail of the computational mesh used for the simulation, highlighting mesh non-conformities at fracture intersections.
As far as space discretization on the fractures is concerned, several different discretization strategies can be used and mixed in order to improve approximation properties and provide high levels of accuracy, especially at fracture intersections, where a non-smooth behavior of the solution is expected.
In particular, Extended Finite Elements and Virtual Elements have been effectively used when dealing with complex DFN configurations.
By using a gradient based method for the minimization of the functional, the solution of the flow equations on each fracture of the network is carried on independently of the solution on the other fractures. This in turn, together with the meshing process independently performed on each fracture, allows in a natural way for parallel implementation of the overall method, thus providing an efficient handling of problem size and complexity. This is of paramount importance both for addressing computations on huge networks, and for performing massive simulations for uncertainty quantification in stochastically generated networks.
References
[1] S. Berrone, S. Pieraccini, S. Scialo' A PDE-constrained optimization formulation for discrete fracture network flows, SIAM J. Sci. Comput., 2013, 35, B487–B510.
[2] S. Berrone, S. Pieraccini, S. Scialo' On simulations of discrete fracture network flows with an optimization-based extended finite element method, SIAM J. Sci. Comput., 2013, 35, A908–A935.
[3] S. Berrone, S. Pieraccini, S. Scialo' An optimization approach for large scale simulations of discrete fracture network flows, J. Comput. Phys., 2014, 256, 838–853.
[4] M. F. Benedetto, S. Berrone, S. Pieraccini, S. Scialo' The Virtual Element Method for Discrete Fracture Network simulations, submitted.