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 1238 products
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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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26/2012 - 05/28/2012
Tamellini, L.; Le Maitre, O.; Nouy, A.
Model reduction based on Proper Generalized Decomposition for the Stochastic steady incompressible Navier-Stokes equations | Abstract | | In this paper we consider a Proper Generalized Decomposition method to solve the steady incompressible Navier-Stokes equations with random Reynolds number and forcing term. The aim of such technique is to compute a low-cost reduced basis approximation of the full Stochastic Galerkin solution of the problem at hand. A particular algorithm, inspired by the Arnoldi method for solving eigenproblems, is proposed for an efficient greedy construction of a deterministic reduced basis approximation. This algorithm decouples the computation of the deterministic and stochastic components of the solution, thus allowing reuse of pre-existing deterministic Navier-Stokes solvers. It has the remarkable property of only requiring the solution of m deterministic problems for the construction of a m-dimensional reduced basis. |
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25/2012 - 05/17/2012
Manfredini, F.; Pucci, P.; Secchi, P.; Tagliolato, P.; Vantini, S.; Vitelli, V.
Treelet decomposition of mobile phone data for deriving city usage and mobility pattern in the Milan urban region | Abstract | | The paper presents a novel geo-statistical unsupervised learning technique aimed at identifying useful information on hidden patterns of mobile phone use. These hidden patterns regard dierent usages of the city in time and in space which are related to individual mobility, outlining the potential of this technology for the urban planning community. The methodology allows to obtain a reference basis that reports the speci c eect of some activities on the Erlang data recorded and a set of maps showing the contribution of each activity to the local Erlang signal. We selected some results as signi cant for explaining speci c mobility and city usages patterns (commuting, nightly activities, distribution of residences, non systematic mobility) and tested their signi cance and their interpretation from an urban analysis and planning perspective at the Milan urban region scale. |
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24/2012 - 05/16/2012
Antonietti, P.F.; Giani, S.; Houston, P.
hp–Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains | Abstract | | In this paper we introduce the hp-version discontinuous Galerkin composite finite element method for the discretization of second–order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or micro-structures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain $ Omega$ is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in $ Omega$. In this article, we extend these ideas to the discontinuous Galerkin setting, based on employing the hp-version of the finite element method. Numerical experiments highlighting the practical application of the proposed numerical scheme will be presented. |
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23/2012 - 05/15/2012
Fabio Nobile, Christian Vergara
Partitioned algorithms for fluid-structure interaction problems in haemodynamics | Abstract | | We consider the fluid-structure interaction problem arising in
haemodynamic applications. The finite elasticity equations
for the vessel are written in Lagrangian form,
while the Navier-Stokes equations for the blood in Arbitrary
Lagrangian Eulerian form. The resulting three fields problem
(fluid/ structure/ fluid domain) is formalized via the introduction of
three Lagrange multipliers and consistently discretized by p-th order backward differentiation
formulae (BDFp).
We focus on partitioned algorithms for its numerical solution, which
consist in the successive solution of the three subproblems. We review
several strategies that all rely on the exchange of Robin interface
conditions and review their performances reported recently in the literature.
We also analyze the stability of explicit partitioned procedures and
convergence of iterative implicit partitioned procedures on a simple
linear FSI problem for a general BDFp temporal discretizations. |
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22/2012 - 05/11/2012
Ettinger, B.; Passerini, T.;Perotto, S.; Sangalli, L.M.
Regression models for data distributed over non-planar domains | Abstract | | We consider the problem of surface estimation and spatial smoothing over non-planar domains. In particular, we deal with the case where the data or signals occur on a domain that is a surface in a three-dimensional space. The application driving our research is the modeling of hemodynamic data, such as the shear stress and the pressure exerted by blood flow on the wall of a carotid artery. The regression model we propose consists of two key phases. First, we conformally map the surface domain to a region in the plane. Then, we apply existing regression methods for planar domains, suitably modified to respect the geometry of the original surface
domain. |
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