20 Giugno, 2006 15:00

Weak approximation of stochastic differential equations with jumps

Raul Tempone, Florida State University, Tallahassee

Aula Seminari MOX, VI piano Dipartimento

abstract

abstract

We develop adaptive time stepping methods, based on the Monte Carlo Euler method, for weak approximation of jump diffusion driven stochastic differential equations. The main result is new expansions of the computational error, with computable leading order term in a posteriori form, based on stochastic flows and discrete dual back-ward problems. The expansions lead to efficient and accurate computation of error estimates. We describe adaptive algorithms for either stochastic time steps or quasi-deterministic time steps and present numerical examples illustrating their behavior.

16 Giugno, 2006 15:00

Seminari di Cultura Matematica

I poligoni stellati

E. Reyes, Univ. di Vallodolid (Spagna)

Laboratorio didattico

We develop adaptive time stepping methods, based on the Monte Carlo Euler method, for weak approximation of jump diffusion driven stochastic differential equations. The main result is new expansions of the computational error, with computable leading order term in a posteriori form, based on stochastic flows and discrete dual back-ward problems. The expansions lead to efficient and accurate computation of error estimates. We describe adaptive algorithms for either stochastic time steps or quasi-deterministic time steps and present numerical examples illustrating their behavior.

14 Giugno, 2006 12:30

Seminari di Cultura Matematica

La turbolenza dal bicchiere d’acqua alle stelle

L. Valdettaro, Dip. di Matematica

Aula B.21

We develop adaptive time stepping methods, based on the Monte Carlo Euler method, for weak approximation of jump diffusion driven stochastic differential equations. The main result is new expansions of the computational error, with computable leading order term in a posteriori form, based on stochastic flows and discrete dual back-ward problems. The expansions lead to efficient and accurate computation of error estimates. We describe adaptive algorithms for either stochastic time steps or quasi-deterministic time steps and present numerical examples illustrating their behavior.

13 Giugno, 2006 17:00

Seminario Matematico e Fisico di Milano

Information and Entropy Flow in Filtering and Control

Sanjoy Mitter, Massachusetts Institute of Technology (Cambridge, Stati Uniti)

Dipartimento di Matematica - Politecnico di Milano - Via Bonardi 9 - Milano - Aula Seminari VI piano

abstract

abstract

Recent applications such as control over networks, system state estimation using networks of sensors and real-time embedded control systems have blurred the boundaries of the disciplines of communications, control and computation. Increasingly, communication, control and computation take place through interconnection of systems leading to desirable interactions. From a methodological point of view, simple models where the nature of these desirable interactions can be studied in some depth are needed. We examine the structure of interaction between sensing, communicating and controlling in the context of statistical filtering for signals described as Hidden Markov processes and in the context of stabilization of an unstable control system where the sensor and controller are linked via a noisy communication channel. We argue amongst other things, that a fundamental reexamination of information theory is needed to address these questions. These ideas appear to have nontrivial connections to recent work in nonequilibrium statistical mechanics.

13 Giugno, 2006 17:00

Seminario Matematico e Fisico di Milano

Schwarz Reflection Geometry

Joel Langer, Case Western Reserve University (Cleveland, Stati Uniti)

Dipartimento di Matematica e Applicazioni - Università degli Studi di Milano Bicocca - Via Cozzi, 53 - Aula 3014

abstract

abstract

"The Schwarz function of an analytic arc in the complex plane was introduced by Davis and Pollak in the 1950’s and extensively developed and generalized as a basic analytical tool in monographs by Davis (1974) and Shapiro (1992). Applications of the Schwarz function, since its initial introduction, have appeared occasionally in the literature—e.g., in a series of papers by Gustafsson on quadrature domains and in recent work of Wiegmann, Zabrodin and others on completely integrable planar models. My talk will begin with a brief introduction to the notion of Schwarz function and a review of some of its basic properties and applications. Then I will describe the role of Schwarz functions in Schwarz reflection geometry. Namely, Schwarzian reflection of one analytic plane curve in another may be used to define a formal symmetric space structure on the infinite dimensional space of such curves (joint work with A. Calini). We introduce a computational formalism which takes advantage of the remarkable fit between the Schwarz function and Ottmar Loos’s abstract approach to symmetric spaces. We discuss the resulting geodesic equation, a second order partial differential equation for a time-dependent Schwarz function S(t,z), and describe solutions in terms of meromorphic differentials. Data associated with poles and periods of such differentials will be seen to govern the conformal geometric features of the corresponding singular foliations. The above ideas and constructions will be explained with the help of concrete graphical examples, starting with familiar circle patterns of potential theory—interpreted as geodesics in a three-dimensional Lorentzian manifold of circles. Further, we describe a local normal form for geodesics in M obtained by examining the local action of the group G of analytic circle diffeomorphisms

6 Giugno, 2006 12:30

Seminari di Cultura Matematica

La statistica in medicina: esempi, risultati, prospettive

A. Paganoni, Dip. di Matematica

Aula B.21

"The Schwarz function of an analytic arc in the complex plane was introduced by Davis and Pollak in the 1950’s and extensively developed and generalized as a basic analytical tool in monographs by Davis (1974) and Shapiro (1992). Applications of the Schwarz function, since its initial introduction, have appeared occasionally in the literature—e.g., in a series of papers by Gustafsson on quadrature domains and in recent work of Wiegmann, Zabrodin and others on completely integrable planar models. My talk will begin with a brief introduction to the notion of Schwarz function and a review of some of its basic properties and applications. Then I will describe the role of Schwarz functions in Schwarz reflection geometry. Namely, Schwarzian reflection of one analytic plane curve in another may be used to define a formal symmetric space structure on the infinite dimensional space of such curves (joint work with A. Calini). We introduce a computational formalism which takes advantage of the remarkable fit between the Schwarz function and Ottmar Loos’s abstract approach to symmetric spaces. We discuss the resulting geodesic equation, a second order partial differential equation for a time-dependent Schwarz function S(t,z), and describe solutions in terms of meromorphic differentials. Data associated with poles and periods of such differentials will be seen to govern the conformal geometric features of the corresponding singular foliations. The above ideas and constructions will be explained with the help of concrete graphical examples, starting with familiar circle patterns of potential theory—interpreted as geodesics in a three-dimensional Lorentzian manifold of circles. Further, we describe a local normal form for geodesics in M obtained by examining the local action of the group G of analytic circle diffeomorphisms

31 Maggio, 2006 12:30

Seminari di Cultura Matematica

Bellezza e complessità dei modelli matematici in ecologia

M. Gatto, Dip. di Elettronica e Informazione

Aula B.21

"The Schwarz function of an analytic arc in the complex plane was introduced by Davis and Pollak in the 1950’s and extensively developed and generalized as a basic analytical tool in monographs by Davis (1974) and Shapiro (1992). Applications of the Schwarz function, since its initial introduction, have appeared occasionally in the literature—e.g., in a series of papers by Gustafsson on quadrature domains and in recent work of Wiegmann, Zabrodin and others on completely integrable planar models. My talk will begin with a brief introduction to the notion of Schwarz function and a review of some of its basic properties and applications. Then I will describe the role of Schwarz functions in Schwarz reflection geometry. Namely, Schwarzian reflection of one analytic plane curve in another may be used to define a formal symmetric space structure on the infinite dimensional space of such curves (joint work with A. Calini). We introduce a computational formalism which takes advantage of the remarkable fit between the Schwarz function and Ottmar Loos’s abstract approach to symmetric spaces. We discuss the resulting geodesic equation, a second order partial differential equation for a time-dependent Schwarz function S(t,z), and describe solutions in terms of meromorphic differentials. Data associated with poles and periods of such differentials will be seen to govern the conformal geometric features of the corresponding singular foliations. The above ideas and constructions will be explained with the help of concrete graphical examples, starting with familiar circle patterns of potential theory—interpreted as geodesics in a three-dimensional Lorentzian manifold of circles. Further, we describe a local normal form for geodesics in M obtained by examining the local action of the group G of analytic circle diffeomorphisms

29 Maggio, 2006 16:30

Seminari di Cultura Matematica

Stati coerenti di grande ampiezza in strati limite

prof. Alessandro Bottaro, Dipartimento DIAM,Università degli Studi di Genova

Aula Seminari VI piano Dipartimento

abstract

abstract

Un nuovo paradigma e' emerso recentemente per la descrizione della transizione alla turbolenza di moti fluidi vicino a pareti, incentrato sull'esistenza di soluzioni nonlineari instabili delle equazioni del moto. Tali soluzioni sono generalmente delle onde viaggianti; si specula che il moto turbolento si organizzi attorno ad esse. Si discuteranno recenti studi volti all'identificazione di tali stati coerenti non lineari per il caso dello strato limite che si sviluppa su piastra piana.

24 Maggio, 2006 15:00

Seminari di Cultura Matematica

Large Eddy Simulation of Engineering and Geophysical Turbulent Flows

Traian Iliescu, Virginia Polytechnic Institute State University

Aula Seminari, VI piano Dipartimento

abstract

abstract

Large Eddy Simulation (LES) is one of the most successful approaches in the numerical simulation of turbulent flows. LES aims at approximating the large structures in the flow, while modeling the effect of the small structures which are not captured on the numerical mesh.

This introductory talk will present some of the achievements and challenges in LES of engineering and geophysical applications.
In particular, we will focus on the closure problem and the boundary conditions in LES.

First, we will present a rigorous numerical analysis for a bounded artificial viscosity model for the numerical simulation of turbulent flows. In practice, the commonly used Smagorinsky model is overly dissipative, and yields unphysical results. To date, several methods for ``clipping'' the Smagorinsky viscosity have proved useful in improving the physical characteristics of the simulated flow. However, such heuristic strategies rely strongly upon a priori knowledge of the flow regime. The bounded artificial viscosity model relies on a highly nonlinear, but monotone and smooth, semilinear elliptic form for the artificial viscosity.
For such a bounded model, we have introduced a variational computational strategy, provided finite element error convergence estimates, and included several computational examples indicating its improvement over the overly diffusive Smagorinsky model.

Second, we will introduce new boundary conditions for LES based on approximate deconvolution. An advantage of these Approximate Deconvolution Boundary Conditions (ADBC) is that they are suited for turbulent flows with time-dependent boundary conditions. One such application is flow control, where for example, blowing and suction on the surface of an airfoil can be used to reduce the skin-friction drag. Note that current LES boundary conditions are not up to this task: they either lead to a prohibitively high computational cost or need boundary layer theory which might not be available. Our new boundary conditions avoid these two roadblocks - they are efficient and general.

24 Maggio, 2006 12:30

Seminari di Cultura Matematica

Numeri sacri e numeri quotidiani, calcolo sacro e calcolo quotidiano presso gli Inca

L. Laurencich Minelli, Univ. di Bologna

Aula B.21

Large Eddy Simulation (LES) is one of the most successful approaches in the numerical simulation of turbulent flows. LES aims at approximating the large structures in the flow, while modeling the effect of the small structures which are not captured on the numerical mesh.

This introductory talk will present some of the achievements and challenges in LES of engineering and geophysical applications.
In particular, we will focus on the closure problem and the boundary conditions in LES.

First, we will present a rigorous numerical analysis for a bounded artificial viscosity model for the numerical simulation of turbulent flows. In practice, the commonly used Smagorinsky model is overly dissipative, and yields unphysical results. To date, several methods for ``clipping'' the Smagorinsky viscosity have proved useful in improving the physical characteristics of the simulated flow. However, such heuristic strategies rely strongly upon a priori knowledge of the flow regime. The bounded artificial viscosity model relies on a highly nonlinear, but monotone and smooth, semilinear elliptic form for the artificial viscosity.
For such a bounded model, we have introduced a variational computational strategy, provided finite element error convergence estimates, and included several computational examples indicating its improvement over the overly diffusive Smagorinsky model.

Second, we will introduce new boundary conditions for LES based on approximate deconvolution. An advantage of these Approximate Deconvolution Boundary Conditions (ADBC) is that they are suited for turbulent flows with time-dependent boundary conditions. One such application is flow control, where for example, blowing and suction on the surface of an airfoil can be used to reduce the skin-friction drag. Note that current LES boundary conditions are not up to this task: they either lead to a prohibitively high computational cost or need boundary layer theory which might not be available. Our new boundary conditions avoid these two roadblocks - they are efficient and general.