Quaderni MOX
Pubblicazioni
del Laboratorio di Modellistica e Calcolo Scientifico MOX. I lavori riguardano prevalentemente il campo dell'analisi numerica, della statistica e della modellistica matematica applicata a problemi di interesse ingegneristico. Il sito del Laboratorio MOX è raggiungibile
all'indirizzo mox.polimi.it
Trovati 1237 prodotti
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36/2012 - 13/09/2012
Canuto, C.; Verani, M.
On the Numerical Analysis of Adaptive Spectral/hp Methods for Elliptic Problems | Abstract | | We provide an overview of the state of the art of adaptive strategies for high-order $hp$ discretizations of partial differential
equations; at the same time, we draw attention on some recent results of ours concerning the convergence and complexity analysis of adaptive algorithm of spectral and spectral-element type. Complexity is studied under the assumption that the solution belongs to a sparsity class of exponential type, which means that its best $N$-term approximation error in the chosen piecewise polynomial basis decays at an exponential rate with respect to $N$. |
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35/2012 - 10/09/2012
Pigoli, D.; Aston, J.A.D.; Dryden, I.L.; Secchi, P.
Distances and Inference for Covariance Functions | Abstract | | A framework is developed for inference concerning the covariance operator of a functional random process, where the covariance operator itself is an object of interest for the statistical analysis. Distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable for functional data. In particular, an infinite dimensional analogue of
the Procrustes size and shape distance is developed. The convergence of the finite dimensional approximations to the infinite dimensional distance metrics is also shown. To perform inference, a Fréchet estimator for the average covariance function is introduced, and a
permutation procedure to test the equality of the covariance operator between two groups is then considered. The proposed techniques are applied to two problems where inference concerning the covariance is of interest. Firstly, in data arising from a study into cerebral aneurysms, it is of interest to determine whether two groups of data can be combined when comparing with a third group. For this to be done, it is necessary to assess whether the covariance structures of the two groups are the same or different. Secondly, in a philological study of cross-linguistic dependence, the use of covariance operators has been suggested as a way to incorporate quantitative phonetic information. It is shown
that distances between languages derived from phonetic covariance functions can provide insight into relationships between the Romance languages. |
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34/2012 - 04/09/2012
Menafoglio, A.; Dalla Rosa, M.; Secchi, P.
A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space | Abstract | | We address the problem of predicting spatially dependent functional data belonging to a Hilbert space, with a Functional Data Analysis approach. Having defined new global measures of spatial variability for functional random processes, we derive a Universal Kriging predictor for functional data.
Consistently with the new established theoretical results, we develop a two-step procedure for predicting georeferenced functional data: first model selection and estimation of the spatial mean (drift), then Universal Kriging prediction on the basis of the identified dichotomy model, sum of deterministic drift and stochastic residuals.
The proposed methodology is tested by means of a simulation study and finally applied to daily mean temperatures curves aiming at reconstructing the space-time field of temperatures of Canada s Maritimes Provinces. |
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33/2012 - 24/08/2012
Motamed, M.; Nobile, F.; Tempone, R.
Analysis and computation of the elastic wave equation with random coefficients | Abstract | | We analyze the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics of some given quantities of interest. We study the convergence rate of the error in the stochastic collocation method. In particular, we show that, the rate of convergence depends on the regularity of the solution or the quantity of interest in the stochastic space, which is in turn related to the regularity of the deterministic data in the physical space and the type of the quantity of interest. We demonstrate that a fast rate of convergence is possible in two cases: for the elastic wave solutions with high regular data; and for some high regular quantities of interest even in the presence of low regular data. We perform numerical examples, including a simplified earthquake, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo sampling method for problems with high stochastic regularity. |
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32/2012 - 22/08/2012
Formaggia, L.; Fumagalli, A.; Scotti A.; Ruffo, P
A reduced model for Darcy s problem in networks of fractures | Abstract | | Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In
literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and
effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across the intersection. This latter property permits to describe more accurately the flow when fractures are characterised by different properties, than other models that impose
pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analysed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the
extended finite element method (XFEM), to increase the flexibility of the method in the case of complex geometries characterized by a high number of fractures. |
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31/2012 - 20/08/2012
Bonizzoni, F.; Buffa, A; Nobile, F:
Moment equations for the mixed formulation of the Hodge Laplacian with stochastic data | Abstract | | We study the mixed formulation of the stochastic Hodge-Laplace problem defined on a n-dimensional domain D (n>=1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the m-th moment (m>=1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces. November 2012, ERRATA added at the end of the report |
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30/2012 - 26/07/2012
Beck, J.; Nobile, F.; Tamellini, L.; Tempone, R.;
Convergence of quasi-optimal Stochastic Galerkin Methods for a class of PDES with random coefficients | Abstract | | In this work we consider quasi-optimal versions of the Stochastic Galerkin Method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number N of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane C^N. We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates. |
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29/2012 - 13/07/2012
Chen, P.; Quarteroni, A.; Rozza, G.
Stochastic Optimal Robin Boundary Control Problems of Advection-Dominated Elliptic Equations | Abstract | | In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-dusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection led and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the rst order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized nite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided. |
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