24 Gennaio, 2005 14:30

Partial Differential Equations with Uncertainty: the Stochastic input case

Raul Tempone, ICES, University of Texas at Austin

Aula Seminari MOX - 6° piano dip di matematica

abstract

abstract

The main aim of applied numerical simulations is to derive predictions. Since these predictions are the basis for decision making it is natural to question their accuracy, specially because in most of the cases there is uncertainty in data of the problem to solve.

We consider numerical approximations of partial differential equations (PDE) with stochastic coefficients, which is one way to address uncertainty quantification for PDEs.

We discuss efficient discretization strategies, give convergence results and present numerical results.

19 Gennaio, 2005 12:30

Seminari di Cultura Matematica

Economia ed Ambiente: lo sviluppo sostenibile

G. Caiati, Univ. di Milano

aula T2.2

The main aim of applied numerical simulations is to derive predictions. Since these predictions are the basis for decision making it is natural to question their accuracy, specially because in most of the cases there is uncertainty in data of the problem to solve.

We consider numerical approximations of partial differential equations (PDE) with stochastic coefficients, which is one way to address uncertainty quantification for PDEs.

We discuss efficient discretization strategies, give convergence results and present numerical results.

18 Gennaio, 2005 11:00

Seminari di Cultura Matematica

Some well-balanced Finite Volume solvers for Shallow Water Equations

Tomás Chacón Rebollo, Universidad de Sevilla

Aula Seminari MOX-6° piano dip mat.

abstract

abstract

The accurate computation of stationary solutions of hyperbolic systems with source terms has been found in the past years as closely related to the accurate computation of transient solutions. Numerical solvers that do not solve steady solutions up to second order, at least, yield transient solutions that present large errors that grow as time increases, unacceptable from the physical point of view.

These solvers are called “well-balanced”, as the flow and source terms must balance each other, up to high precision, for steady solutions.This talk focus on the systematic derivation of numerical schemes satisfying this property.

Starting from standard Finite Volume solvers that can be written in viscous form, we shall derive associated well-balanced solvers in a specific way: These compute all steady solutions up to second order, in all the domain but on a subdomain whose measure vanishes as the grid size goes to zero.

We shall apply these solvers to Shallow Water equations with variable bottom and friction effects. In particular we shall present a numerical simulation of the toxic waste due to the breaking of Aznalcóllar mine pond in 1998.

17 Gennaio, 2005 14:00

Seminari di Cultura Matematica

Tecniche a basi ridotte per l'ottimizzazione in emodinamica

Gianluigi Rozza, EPFL

Aula Seminari MOX-6° piano dip di matematica

abstract

abstract

In questo seminario viene descritto il metodo a basi ridotte per le equazioni di
Stokes in domini parametrizzati, utilizzato per approssimare flussi sanguigni in
un bypass coronarico. Lo scopo e' fornire (a) un'analisi di sensitivita' per
grandezze geometriche rilevanti nelle configurazioni di bypass e (b) rapide e
affidabili previsioni del comportamento di certe grandezze di interesse (indici
di merito fluidodinamici legati per esempio a sforzi e vorticita').
Le linee guida della ricerca vogliono (i) fornire indicazioni di progettazione
per futuri bypass bioartificiali; (ii) sviluppare metodi nuemrici di
ottimizzazione in biomeccanica e (iii) fornire una relazione input-output
costituita da modelli con una complessita' e un costo computazionale piu' basso
rispetto alla soluzione completa delle equazioni della meccanica dei fluidi
tramite un metodo classico a elementi finiti.
Keywords:
PDE parametrizzate, problema di Stokes generalizzato, metodi a basi ridotte,
ottimizzazione di bypass, emodinamica.

12 Gennaio, 2005 12:30

Seminari di Cultura Matematica

Un viaggio attraverso la biologia computazionale

G. Lancia, Univ. di Udine

aula T2.2

In questo seminario viene descritto il metodo a basi ridotte per le equazioni di
Stokes in domini parametrizzati, utilizzato per approssimare flussi sanguigni in
un bypass coronarico. Lo scopo e' fornire (a) un'analisi di sensitivita' per
grandezze geometriche rilevanti nelle configurazioni di bypass e (b) rapide e
affidabili previsioni del comportamento di certe grandezze di interesse (indici
di merito fluidodinamici legati per esempio a sforzi e vorticita').
Le linee guida della ricerca vogliono (i) fornire indicazioni di progettazione
per futuri bypass bioartificiali; (ii) sviluppare metodi nuemrici di
ottimizzazione in biomeccanica e (iii) fornire una relazione input-output
costituita da modelli con una complessita' e un costo computazionale piu' basso
rispetto alla soluzione completa delle equazioni della meccanica dei fluidi
tramite un metodo classico a elementi finiti.
Keywords:
PDE parametrizzate, problema di Stokes generalizzato, metodi a basi ridotte,
ottimizzazione di bypass, emodinamica.

16 Dicembre, 2004 11:30

Seminari di Cultura Matematica

High Order Reconstruction Methods for Piecewise Smooth Functions

Anne Gelb, Arizona state university

Aula Seminari MOX - 6° piano dip. di Matematica

abstract

abstract

It is well known that while spectral methods yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Specifically, unlike standard filtering, the convergence does not deteriorate as the discontinuities area approached. The most familar ad easily analyzed spectral reprojection method is the Gegenbauer reconstruction method. However it is apparent that the Gegenbauer reconstruction method is not robust and in particular it suffer from round off error. Methods to alleviate these difficulties have been recently developed. All high order recontruction methods require a-priori knowledge of the jump discontinuity locations, since these edges determine the intervarls of smoothness in which the spectral reprojection reconstruction method can be applied. The local edge detection method has been recently been developed to determine the location of edges on scattered grid point data. It can also be applied to detect the discontinuities in the derivates of functions. In this talk I discuss recent advances in edge detection and high order reconstruction methods. Examples include applications from medical imaging, where noise is an additional impediment to the reconstruction. Other spectral reprojection methods are briefly discussed.

15 Dicembre, 2004 12:30

Seminari di Cultura Matematica

Medicina, Sport, Ecologia: la Modellistica Matematico-Numerica nella vita di tutti i giorni

A. Veneziani, Dip. di Matematica

Aula B.21

It is well known that while spectral methods yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Specifically, unlike standard filtering, the convergence does not deteriorate as the discontinuities area approached. The most familar ad easily analyzed spectral reprojection method is the Gegenbauer reconstruction method. However it is apparent that the Gegenbauer reconstruction method is not robust and in particular it suffer from round off error. Methods to alleviate these difficulties have been recently developed. All high order recontruction methods require a-priori knowledge of the jump discontinuity locations, since these edges determine the intervarls of smoothness in which the spectral reprojection reconstruction method can be applied. The local edge detection method has been recently been developed to determine the location of edges on scattered grid point data. It can also be applied to detect the discontinuities in the derivates of functions. In this talk I discuss recent advances in edge detection and high order reconstruction methods. Examples include applications from medical imaging, where noise is an additional impediment to the reconstruction. Other spectral reprojection methods are briefly discussed.

10 Dicembre, 2004 10:30

Sezione di Geometria, Algebra e loro applicazioni

Intersecting longest cycles and paths

Tudor Zamfirescu, Universität Dortmund, Germany

Dipartimento di Matematica - Politecnico di Milano

It is well known that while spectral methods yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Specifically, unlike standard filtering, the convergence does not deteriorate as the discontinuities area approached. The most familar ad easily analyzed spectral reprojection method is the Gegenbauer reconstruction method. However it is apparent that the Gegenbauer reconstruction method is not robust and in particular it suffer from round off error. Methods to alleviate these difficulties have been recently developed. All high order recontruction methods require a-priori knowledge of the jump discontinuity locations, since these edges determine the intervarls of smoothness in which the spectral reprojection reconstruction method can be applied. The local edge detection method has been recently been developed to determine the location of edges on scattered grid point data. It can also be applied to detect the discontinuities in the derivates of functions. In this talk I discuss recent advances in edge detection and high order reconstruction methods. Examples include applications from medical imaging, where noise is an additional impediment to the reconstruction. Other spectral reprojection methods are briefly discussed.

10 Dicembre, 2004 15:00

Sezione di Analisi

Problemi di controllo ottimo con ritardo sul controllo

Carlo Marinelli, Universita di Bonn

Dipartimento di Matematica

It is well known that while spectral methods yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Specifically, unlike standard filtering, the convergence does not deteriorate as the discontinuities area approached. The most familar ad easily analyzed spectral reprojection method is the Gegenbauer reconstruction method. However it is apparent that the Gegenbauer reconstruction method is not robust and in particular it suffer from round off error. Methods to alleviate these difficulties have been recently developed. All high order recontruction methods require a-priori knowledge of the jump discontinuity locations, since these edges determine the intervarls of smoothness in which the spectral reprojection reconstruction method can be applied. The local edge detection method has been recently been developed to determine the location of edges on scattered grid point data. It can also be applied to detect the discontinuities in the derivates of functions. In this talk I discuss recent advances in edge detection and high order reconstruction methods. Examples include applications from medical imaging, where noise is an additional impediment to the reconstruction. Other spectral reprojection methods are briefly discussed.

1 Dicembre, 2004 12:30

Seminari di Cultura Matematica

Architettura e matematica: 1000 anni di motivi geometrici

K. Williams, professionista

Aula B.21

It is well known that while spectral methods yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Specifically, unlike standard filtering, the convergence does not deteriorate as the discontinuities area approached. The most familar ad easily analyzed spectral reprojection method is the Gegenbauer reconstruction method. However it is apparent that the Gegenbauer reconstruction method is not robust and in particular it suffer from round off error. Methods to alleviate these difficulties have been recently developed. All high order recontruction methods require a-priori knowledge of the jump discontinuity locations, since these edges determine the intervarls of smoothness in which the spectral reprojection reconstruction method can be applied. The local edge detection method has been recently been developed to determine the location of edges on scattered grid point data. It can also be applied to detect the discontinuities in the derivates of functions. In this talk I discuss recent advances in edge detection and high order reconstruction methods. Examples include applications from medical imaging, where noise is an additional impediment to the reconstruction. Other spectral reprojection methods are briefly discussed.