[Todos CMAT] Invitación Seminario IngeMat 2012 - Giovanni Migliorati - martes 24 de julio - 10:00 hs.

[Marco Scavino - IMERL]

La Maestría en Ingeniería Matemática tiene el agrado de invitarles a
participar de la próxima ponencia 

del ciclo de Seminario IngeMat 2012, 

a cargo de Giovanni Migliorati (Dipartimento di Matematica "Francesco
Brioschi" - Politecnico di Milano) 

quien presentará el trabajo titulado 
"Uncertainty Quantification in computational models via the random discrete
L2 projection on polynomial spaces".
 
Horario y lugar: martes 24 de julio, 10:00 horas, Salón de Seminarios del
IMERL, Facultad de Ingeniería. 
 
 ** Resumen **


In many PDE models the parameters are not known with enough accuracy, or
they naturally feature randomness and can be treated therefore as random
variables. The challenge is then to efficiently compute the law of the
solution of the PDE or some quantities of interest (outputs), given the
probability distribution of the random input parameters.
We consider cases in which the parameter to solution map is smooth, and look
for a multivariate polynomial approximation of it (polynomial chaos
expansion).
An approach that has been advocated recently consists in evaluating the
solution on randomly chosen parameters and doing a discrete L2 projection on
the polynomial space. This problem can be analyzed in a regression framework
with random design. As usual, the regression function minimizes the L2 risk,
but here the observations are noise-free evaluations on random points.
We consider univariate or multivariate target functions and study the
approximation properties of the random L2 projection with respect to the
number of sampling points, the maximum polynomial degree, and the smoothness
of the function to approximate.
We prove optimality estimates (up to a logarithmic factor) when the random
points are sampled from bounded random variables with strictly positive
probability density functions. Our analysis of the random projection proves
that the optimal convergence rate is achieved when the number of sampling
points scales as the square of the dimension of the polynomial space.
Moreover, it gives an insight on the role of smoothness and the conditioning
of the random projection operator in the accuracy and stability of the L2
regression.
In this talk we will present the application of this methodology to compute
Quantities of Interest associated to the solution of stochastic PDEs. We
will deal with stochastic coefficients and with random domains, i.e. domains
whose shape is described by random variables. Numerical examples in low and
high dimensions will be shown as well.
 
La próxima ponencia, prevista para el viernes 3 de agosto, estará a cargo de
Raúl Tempone.


SCAPA de Ingeniería Matemática

 

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