MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1251 products
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34/2012 - 09/04/2012
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 - 08/24/2012
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 - 08/22/2012
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 - 08/20/2012
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 - 07/26/2012
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 - 07/13/2012
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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28/2012 - 06/19/2012
Canuto, C.; Nochetto, R.H.; Verani, M.
Contraction and optimality properties of adaptive Legendre-Galerkin methods: the 1-dimensional case | Abstract | | As a first step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value
problems, we study the performance of adaptive Legendre-Galerkin methods in one space dimension. These methods offer unlimited approximation power
only restricted by solution and data regularity. Our investigation is inspired by a similar study that we recently carried out for Fourier-Galerkin methods in a periodic box. We first consider an ideal algorithm, which we prove to be convergent at a fixed rate. Next we enhance its performance, consistently with the expected fast error decay of high-order methods, by activating a larger set of degrees of freedom at
each iteration. We guarantee optimality (in the non-linear approximation sense) by incorporating a coarsening step. Optimality is measured in terms
of certain sparsity classes of the Gevrey type, which describe a (sub-)exponential decay of the best approximation error. |
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27/2012 - 06/11/2012
Pigoli, D.; Secchi,P.
Estimation of the mean for spatially dependent data belonging to a Riemannian manifold | Abstract | | The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications. The aim of this work is to introduce models for spatial dependence among Riemannian data, with a special focus on the case of positive definite symmetric matrices. First, the Riemannian semivariogram of
a field of positive definite symmetric matrices is defined. Then, we propose an estimator for the mean which considers both the non Euclidean nature of the data and their spatial correlation. Simulated data are used to evaluate the performance of the proposed estimator: taking into account spatial dependence leads to better estimates when observations are irregularly spaced in the region of interest. Finally, we address a meteorological problem, namely, the estimation of the covariance matrix between temperature and
precipitation for the province of Quebec in Canada. |
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