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 1256 products
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15/2013 - 04/08/2013
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 - 03/23/2013
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 - 03/14/2013
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 - 03/09/2013
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 - 03/01/2013
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 - 02/28/2013
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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09/2013 - 02/23/2013
Vergara, C.; Palamara, S.; Catanzariti, D.; Pangrazzi, C.; Nobile, F.; Centonze, M.; Faggiano, E.; Maines, M.; Quarteroni, A.; Vergara, G.
Patient-specific computational generation of the Purkinje network driven by clinical measuraments | Abstract | | Rationale: The propagation of the electrical signal in the Purkinje network is the starting point of the activation of the muscular cells in the ventricle and of the contraction of the heart. Anomalous propagation in such a network can cause disorders such as ventricular fibrillation. In the computational models describing the electrical activity of the ventricle is therefore important to account for the Purkinje fibers.
Objective: Aim of this work is to apply to real cases a method for the generation of the Purkinje network driven by patient-specific measures of the activation on the endocardium, to compute the activation maps in the ventricle and to compare the accuracy with that of other strategies proposed so far in the literature.
Methods and Results: We consider MRI images of two patients and data of the activation times on the endocardium acquired by means of the EnSite NavX system. To generate the Purkinje network we use a fractal law driven by the measures. Our results show that for a normal activation our algorithm is able to reduce considerably the errors (19.9±5.3% with our algorithm vs 33.9±6.8% with the best of the other strategies for patient 1, and 28.6±6.0% vs 63.9±8.6% for patient 2).
Conclusions: In this work we showed the reliability of the proposed method to generate a patient-specific Purkinje network. This allowed to improve the accuracy of computational models for the description of the electrical activation in the ventricle. |
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08/2013 - 02/22/2013
Chen, P.; Quarteroni, A.; Rozza, G.
A Weighted Reduced Basis Method for Elliptic Partial Differential Equations with Random Input Data | Abstract | | In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equation (PDE) with random input data. The PDE is first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance at different values of the parameters are taken into account by assigning different weight to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and stochastic collocation method in both univariate and multivariate stochastic problems. Keywords: weighted reduced basis method, stochastic partial differential equation, uncertainty quantication, stochastic collocation method, Kolmogorov N-width, exponential convergence |
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