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 1251 prodotti
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42/2012 - 15/10/2012
Lassila, T.; Manzoni, A.; Quarteroni, A.; Rozza, G.
Generalized reduced basis methods and n width estimates for the approximation of the solution manifold of parametric PDEs | Abstract | | The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in
a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affine parametric dependence, expressed as a linear combination of parameter-independent
operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in
its affine expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These
spaces can be constructed without any assumptions on the parametric regularity of the manifold - only spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to
parametrized elliptic equations conrming the predicted convergence rates.
Keywords: Reduced basis method, parametric PDEs, n-width estimates.
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41/2012 - 14/10/2012
Chen, P.; Quarteroni, A.; Rozza, G.
Comparison between reduced basis and stochastic collocation methods for elliptic problems | Abstract | | The stochastic collocation method has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method, primarily developed for solving parametric systems, has been recently used to deal with stochastic problems. In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: 1) convergence rate of each method referred to both a priori and a posteriori error estimate; 2) computational costs for oine construction and online evaluation of the two methods. Numerical experiments are performed in univariate problems as well as multivariate problems from low dimensions O(1) to moderate dimensions O(10) and to high dimensions O(100). The main result stemming from our comparison is that the reduced basis method converges no worse in theory and faster in practice than the stochastic collocation method, and is more suitable for large scale and high
dimensional stochastic problems when considering computational costs.
keywords: stochastic elliptic problem, reduced basis method, stochastic collocation method,
sparse grid, greedy algorithm, offline-online computational decomposition, convergence analysis |
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40/2012 - 13/10/2012
Lombardi, M.; Parolini, N.; Quarteroni, A.
Radial basis functions for inter-grid interpolation and mesh motion in FSI problems | Abstract | | When addressing multi-domain/multi-physics problems, the correct exchange of mathematical information at subdomain interfaces is crucial. In this paper, such transfer is analyzed in the particular framework of a fluid-structure interaction (FSI) problem. A genuine FEM-FEM formulation is considered firstly, followed by a mixed FVM-FEM formulation. In both cases, we focus on two critical issues: how to interpolate numerical quantities at the interface, and how to achieve the property of conservation of energy transfer. In the second part of this work, we analyze the use of Radial Basis Functions (RBF) to handle both mesh motion and interpolation of numerical variables over non-matching interface grids. Different kinds of radial basis functions are considered and numerical tests comparing their performances in terms of accuracy and stability are presented and discussed. |
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39/2012 - 16/09/2012
Ieva, F.; Paganoni, A.M.; Ziller, S.
Operational risk management: a statistical perspective | Abstract | | This work presents a statistical model for operational risk management. We distinguish different types of operational Event, we model the probability of event occurrence (the frequency distribution) and the economic impact of the single event (the severity distribution), and then the aggregated distribution is obtained through convolution of frequency and severity, for each event type. The main problem is the parameters estimation of the severity distribution above a suitable threshold, that we consider as an unknown parameter to be estimated as well. An application to a case study is also presented.
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38/2012 - 15/09/2012
Antonietti, P.F.; Bigoni, N.; Verani, M.
Mimetic finite difference approximation of quasilinear elliptic problems | Abstract | | In this work we approximate the solution of a quasilinear elliptic problem of monotone type by using the Mimetic Finite Difference (MFD) method. Under a suitable approximation assumption, we prove that the MFD approximate solution converges, with optimal rate, to the exact solution in a mesh-dependent energy norm. The resulting nonlinear discrete problem is then solved iteratively via linearization by applying the Kacanov method. The convergence of the Kacanov algorithm in the discrete mimetic framework is also proved. Several numerical experiments confirm the theoretical analysis. |
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37/2012 - 14/09/2012
Nobile, F.; Pozzoli, M.; Vergara, C.
Exact and inexact partitioned algorithms for fluid-structure interaction problems with finite elasticity in haemodynamics | Abstract | | In this paper we consider the numerical solution of the three-dimensional
(3D) fluid-structure interaction problem in haemodynamics, in the case of
physiological geometries and data, and finite elasticity vessel deformations.
We introduce new partitioned algorithms and compare their efficiency with
that of existing ones. We also study some new inexact variants, obtained
from semi-implicit approximations, and show that they allow to improve the
efficiency while preserving the accuracy of the related exact (implicit) scheme. |
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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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