Martedì 21 Aprile 2015, ore 15 **Dr. Lorenzo Tamellini** (EPFL, Lausanne) **Dr. Diego Irisarri** (Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza)

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Martedì 21 Aprile 2015, ore 15 precise, presso la sala conferenze dell'IMATI-CNR di Pavia, il **Dr. Lorenzo Tamellini** (EPFL, Lausanne) terrà un seminario dal titolo:

"*Polynomial approximation of PDEs with stochastic coefficients*",

e alle ore 16 il

**Dr. Diego Irisarri** (Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza) terrà un seminario dal titolo:

"*A posteriori error estimation in Finite element method based on **Variational Multiscale method. Application to linear elasticity and **transport equations*"

nell'ambito del Seminario di Matematica Applicata (IMATI-CNR e Dipartimento di Matematica, Pavia).

Nell'intervallo tra le due conferenze sarà organizzato un piccolo rinfresco.

----------------------**Abstract seminario Tamellini**

Partial differential equations with stochastic coefficients conveniently model problems in which the data of a given PDE (coefficients, forcing terms, boundary conditions) are affected by uncertainty, due e.g. to measurement errors, limited data availability or intrinsic variability of the described system.

Assuming that the uncertainty in the problem can be described by a set of N random variables y_1,...,y_N, the solution u of the PDE at hand can be seen as an N-variate random function, u = u(y_1 , . . . , y_N), for which one may wish to compute mean and variance, or the probability that it exceeds a given threshold; such analysis is usually referred to as "Uncertainty Quantification". This could be achieved with a straightforward Monte Carlo method, that may however be very demanding in terms of computational costs. Methods based on polynomial approximations of u have thus been introduced, aiming at exploiting the possible degree of regularity of u with respect to y_1, . . . , y_N to alleviate the computational burden. Such polynomial approximations can be obtained e.g. with Galerkin projections or collocation methods (e.g. the sparse grids collocation method) over the support of the random variables. Although effective for problems with a moderate number of random dimensions, these methods suffer from a degradation of their performance as N increases ("curse of dimensionality"). Minimizing the impact of the "curse of

dimensionality" is therefore a key point for the application of polynomial methods to high-dimensional problems.

In this talk we will discuss these methods and explore possible strategies to determine efficient polynomial approximations of u (the so-called "best M-terms" approximation of u). In particular, we will consider a "knapsack approach", in which we estimate the cost and error contribution of each possible component of the polynomial approximation, and then we choose the components with the highest error/cost ratio. We will present theoretical convergence results obtained for some specific problems as well as numerical results showing the efficiency of the proposed approach.

------------------------**Abstract seminario Irisarri**

The need of attaining proper and reliable solutions of differential equations and the computational advancements have caused the widespread development of numerical methods such as Finite element Methods (FEM). However, numerical methods have an inherent error which must be quantified in order to assess the quality of the numerical solution. In this talk, we present an a posteriori error estimator based on the variational multiscale method (VMS) [2]. Basically, the VMS consists in splitting the solution and the weighting functions into coarse (or resolved) scales and fine (or unresolved) scales. This framework enables the analysis of the fine scales and their

interaction with the coarse scales. A rigorous study of the fine scales was made by Hughes and Sangalli [3] in which a explicit expression for the fine-scale Green's functions is proposed. Taking into account these concepts, we present a residual-based error estimation in which the error is carried out post-processing the information provided by the FEM solution. The analysis of the fine scales and the computation of the fine-scale Green's functions are tackled in order to estimate both point-wise and elemental error. Numerical examples related to linear elasticity and transport equations will be shown [1,4].

References

[1] Hauke, G., Irisarri, D.: Variational multiscale a posteriori error estimation for systems. application to linear elasticity. Computer

Methods in Applied Mechanics and Engineering (2014)

[2] Hughes, T.: Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Meth. Appl. Mech. Engrng. 127, 387–401 (1995)

[3] Hughes, T., Sangalli, G.: Variational multiscale analysis: the fine-scale green's function, projection, optimization, localization and stabilized methods. SIAM J. Numer. Anal. 45(2), 539–557 (2007)

[4] Irisarri, D., Hauke, G.: Variational multiscale a posteriori error estimation for 2nd and 4th-order ODES. Int. J. Numer. Anal.

Model accepted (2014)

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Pagina web del Seminario di Matematica Applicata:

http://www-dimat.unipv.it/~seminari/matematica-applicata.html