Expositor: Giovanni Migliorati (Dipartimento di Matematica "Francesco Brioschi" - Politecnico di Milano)
Título: "Uncertainty Quantification in Computational Models via the Random Discrete L2 Projection on Polynomial Spaces".
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.