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 1322 prodotti
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17/2013 - 15/04/2013
Chen, P.; Quarteroni, A.
Accurate and efficient evaluation of failure probability for partial different equations with random input data | Abstract | | Several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis. When the dimension of the uncertainties becomes high, well established direct numerical methods can not be employed because of the “curse-of-dimensionality”. Many surrogate models have been proposed with the aim of reducing computational effort. However, most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness. In addition, for a stochastic system modeled by partial differential equations (PDEs) with random input, only a limited number of the underlying PDEs (order of a few tens) are affordable to solve in practice due to the considerable computational effort, therefore preventing the application of many numerical methods especially for high dimensional random inputs. In this work we develop hybrid and goal-oriented reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic PDE. The curse-of-dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal-oriented adaptation. Moreover, an accurate evaluation of the failure probability for PDE system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error. At the end of this paper we suitably extend our proposed method to deal with more general PDE models. Finally we perform several numerical experiments to illustrate its computational accuracy and efficiency. Keywords: failure probability evaluation, model order reduction, reduced basis method, goal-oriented adaptation, partial differential equations, random input data |
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16/2013 - 09/04/2013
Faggiano, E. ; Lorenzi, T. ; Quarteroni, A.
Metal Artifact Reduction in Computed Tomography Images by Variational Inpainting Methods | Abstract | | Permanent metallic implants such as dental fillings, hip prostheses and cardiac devices generate streaks-like artifacts in computed tomography images. In this paper, two methods based on partial differential equations (PDEs), the Cahn-Hilliard equation and the TV-H-1 inpainting equation, are proposed to reduce metal artifacts. Although already profitably employed in other branches of image processing, these two fourth-order variational methods have never been used to perform metal artifact reduction. A systematic evaluation of the performances of the two methods is carried out. Comparisons are made with the results obtained with classical linear interpolation and two other PDE-based approaches using, respectively, the Fourier heat equation and a nonlinear version of the heat equation relying on total variation flow. Visual inspection of both synthetic and real computed tomography images, as well as computation of similarity indexes, suggest that the Cahn-Hilliard method behaves comparably with more classical approaches, whereas the TV-H-1 method outperforms the others as it provides best image restoration, highest similarity indexes and for being the only one able to recover hidden structures, a task of primary importance in the medical field. |
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15/2013 - 08/04/2013
Antonietti, P.F.; Giani, S.; Houston, P.
Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains | Abstract | | In this article we consider the application of Schwarz-type domain decomposition preconditioners for discontinuous Galerkin finite element approximations of elliptic partial differential equations posed on complicated domains, which are characterized by small details in the computational domain or microstructures. In this setting, it is necessary to define a suitable coarse-level solver, in order to guarantee the scalability of the preconditioner under mesh refinement. To this end, we exploit recent ideas developed in the so-called composite finite element framework, which allows for the definition of finite element methods on general meshes consisting of agglomerated elements. Numerical experiments highlighting the practical performance of the proposed preconditioner are presented. |
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14/2013 - 23/03/2013
Gianni Arioli, Filippo Gazzola
A new mathematical explanation of the Tacoma Narrows Bridge collapse | Abstract | | The spectacular collapse of the Tacoma Narrows Bridge, which occurred in 1940, has attracted the attention of engineers, physicists, and mathematicians in the last 70 years. There have been many attempts to explain this amazing event. Nevertheless, none of these attempts gives a satisfactory and universally accepted explanation of the phenomena visible the day of the collapse. The purpose of the present paper is to suggest a new mathematical model for the study of the dynamical behavior of suspension bridges which provides a realistic explanation of the Tacoma collapse. |
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13/2013 - 14/03/2013
Pini, A.; Vantini, S.
The Interval Testing Procedure: Inference for Functional Data Controlling the Family Wise Error Rate on Intervals. | Abstract | | We propose a novel inferential technique based on permutation tests that enables the statistical comparison between two functional populations. The procedure (i.e., Interval Testing Procedure) involves three steps: (i) representing functional data on a suitable high-dimensional ordered functional basis;
(ii) jointly performing univariate permutation tests on the coefficients of the expansion;
(iii) combining the results obtaining a suitable family of multivariate tests and a p-value heat-map to be used to correct the univariate p-values. The procedure is provided with an interval-wise control of the Family Wise Error Rate. For instance this control, which lies in between the weak and the strong control of the Family Wise Error Rate, can imply that, given any interval of the domain in which there is no difference between the two functional populations, the probability that at least a part of the domain is wrongly detected as significant is always controlled. Moreover, we prove that the statistical power of the Interval Testing Procedure is always higher than the one provided by the Closed Testing Procedure (which provides a strong control of the Family Wise Error Rate but it is computationally unfeasible in the functional framework). On the contrary, we prove that the power of the Interval Testing Procedure is always lower than the Global Testing Procedure one (which however provides only a weak control of the Family Wise Error Rate and does not provide any guide to the interpretation of the test result). The Interval Testing Procedure is also extended to the comparison of several functional populations and to the estimation of the central function of a symmetric functional population. Finally, we apply the Interval Testing Procedure to two case studies: Fourier-based inference for the mean function of yearly recorded daily temperature profiles in Milan, Italy; and B-spline-based inference for the difference between curvature, radius and wall shear stress profiles along the Internal Carotid Artery of two pathologically-different groups of subjects. In the supplementary materials we report the results of a simulation study aiming at comparing the novel procedure with other possible approaches. An R-package implementing the Interval Testing Procedure is available as supplementary material. |
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12/2013 - 09/03/2013
Antonietti, P.F.; Beirao da Veiga, L.; Bigoni, N.; Verani, M.
Mimetic finite differences for nonlinear and control problems | Abstract | | In this paper we review some recent applications of the mimetic finite difference method to nonlinear problems (variational inequalities and quasilinear elliptic equations) and optimal control problems governed by linear elliptic partial differential equations. Several numerical examples show the effectiveness of mimetic finite differences in building accurate and robust numerical approximations. Finally, we draw some conclusions highlighting possible further applications of the mimetic finite difference method to nonlinear Stokes equations and shape optimization problems. |
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11/2013 - 01/03/2013
Discacciati, M.; Gervasio, P.; Quarteroni, A.
The Interface Control Domain Decomposition (ICDD) Method for Elliptic Problems | Abstract | | Interface controls are unknown functions used as Dirichlet or Robin boundary data on the interfaces of an overlapping decomposition designed for solving second order elliptic boundary value problems. The controls are computed through an optimal control problem with either distributed or interface observation. Numerical results show that, when interface observation is considered, the resulting Interface Control Domain Decomposition (ICDD) method is robust with respect to coefficients variations; it can exploit non-conforming meshes and provides optimal convergence with respect to the discretization parameters; finally it can be easily used to face heterogeneous advection - advection/diffusion couplings. |
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10/2013 - 28/02/2013
Antonietti, P.F.; Beirao da Veiga, L.; Mora, D.; Verani, M.
A stream virtual element formulation of the Stokes problem on polygonal meshes | Abstract | | In this paper we propose and analyze a novel stream formulation
of the Virtual Element Method (VEM) for the solution of the
Stokes problem. The new formulation hinges upon the introduction
of a suitable stream function space (characterizing the divergence
free subspace of discrete velocities) and it is equivalent to the
velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beirao da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes, J. Comput. Phys. (2009)]
(up to a suitable reformulation into the VEM framework). Both schemes are
thus stable and linearly convergent but the new method results to be more
desirable as it employs much less degrees of freedom and it is based on a
positive definite algebraic problem. Several numerical experiments assess
the convergence properties of the new method and show its computational
advantages with respect to the mimetic one.
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