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 1237 prodotti
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25/2014 - 02/07/2014
Hron, K.; Menafoglio, A.; Templ, M.; Hruzova K.; Filzmoser, P.
Simplicial principal component analysis for density functions in Bayes spaces | Abstract | | Probability density functions are frequently used to characterize the distributional properties of large-scale database systems. As functional compositions, densities carry primarily relative information. As such, standard methods of functional data analysis (FDA) are not appropriate for their statistical processing. The specific features of density functions are accounted for in Bayes spaces, which result from the generalization to the infinite dimensional setting of the Aitchison geometry for compositional data. The aim of the paper is to build up a concise methodology for functional principal component analysis of densities. We propose the simplicial functional principal component analysis (SFPCA), which is based on the geometry of the Bayes space B^2 of functional compositions. We perform SFPCA by exploiting the centred log-ratio transform, an isometric isomorphism between B^2 and L^2 which enables one to resort to standard FDA tools. Advances of the proposed approach are demonstrated using a real-world example of population pyramids in Upper Austria. |
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24/2014 - 01/07/2014
Ieva, F., Jackson, C.H., Sharples, L.D.
Multi-State modelling of repeated hospitalisation and death in patients with Heart Failure: the use of large administrative databases in clinical epidemiology | Abstract | | In chronic diseases like Heart Failure (HF), the disease course and associated clinical event histories for the patient population vary widely. To improve understanding of the prognosis of patients and enable health-care providers to assess and manage resources, we wish to jointly model disease progression, mortality and their relation with patient characteristics. We show how episodes of hospitalisation for disease-related events, obtained from administrative data, can be used as a surrogate for disease status. We propose flexible multi-state models for serial hospital admissions and death in HF patients, that are able to accommodate important features of disease progression, such as multiple ordered events and competing risks. Markov and semi-Markov models are implemented using freely available software in R. The models were applied to a dataset from the administrative data bank of the Lombardia region in Northern Italy, which included 15,298 patients who had a first hospitalisation ending in 2006 and 4 years of follow up thereafter. This provided estimates of the associations of of age and gender with rates of hospital admission and length of stay in hospital, and estimates of the expected total time spent in hospital. For example, older patients and men were readmitted more frequently, though the total time in hospital was roughly constant with age. We also discuss the relative merits of parametric and semi-parametric multi-state models, and assessment of the Markov assumption. |
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23/2014 - 13/06/2014
Ieva, F., Paganoni, A.M., Tarabelloni, N.
Covariance Based Unsupervised Classification in Functional Data Analysis | Abstract | | In this paper we propose a new algorithm to perform unsupervised classification of multivariate and functional data when the difference between the two populations lies in their covariances, rather than in their means. The algorithm relies on a proper quantification of distance between the estimated covariance operators of the populations, and identifies as clusters those groups maximising the distance between their induced covariances. The naive implementation of such an algorithm is computationally forbidding, so we propose an heuristic formulation with a much lighter complexity and we study its convergence properties, along with its computational cost. We also propose to use an enhanced estimator for the estimation of discrete covariances of functional data, namely a linear shrinkage estimator, in order to improve the precision of the classification. We establish the effectiveness of our algorithm through applications to both synthetic data and a real dataset coming from a biomedical context, showing also how the use of shrinkage estimation may lead to substantially better results. |
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22/2014 - 10/06/2014
Arioli, G.
Insegnare Matematica con Mathematica | Abstract | | n questo articolo vogliamo discutere le possibilità offerte da soft- ware di calcolo simbolico, in particolare parliamo di Mathematica, nell’insegnamento della Matematica nelle scuole primarie e secondarie. Dopo un breve excursus sulla ricerca recente in didattica della matematica e sull’importanza del problem solving nell’insegnamento della matematica, illustriamo come Mathematica possa essere un ottimo strumento didattico, e introduciamo tre applet create con Mathematica per illustrare le possibilità di questo strumento. |
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21/2014 - 09/06/2014
Artina, M.; Fornasier, M.; Micheletti, S.; Perotto, S
The benefits of anisotropic mesh adaptation for brittle fractures under plane-strain conditions | Abstract | | We develop a reliable a posteriori anisotropic first order estimator for the numerical simulation of the Francfort and Marigo model of brittle fracture, after its approximation by means of the Ambrosio-Tortorelli variational model. We show that an adaptive algorithm based on this estimator reproduces all the previously obtained well-known benchmarks on fracture development with particular attention to the fracture directionality. Additionally, we explain why our method, based on an extremely careful tuning of the anisotropic adaptation, has the potential of outperforming significantly in terms of numerical complexity the ones used to achieve similar degrees of accuracy in previous studies. |
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20/2014 - 26/05/2014
Artina, M.; Fornasier, M.; Micheletti, S.; Perotto, S.
Anisotropic mesh adaptation for crack detection in brittle materials | Abstract | | The quasistatic brittle fracture model proposed by G. Francfort and J.-J. Marigo can be Gamma-approximated at each time evolution step by the Ambrosio-Tortorelli functional. In this paper, we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field. Secondly, we build on this variational model an adapted discretization to numerically compute the time-evolving minimizing solution. We present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure. The main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack, but only far away from the crack tip, while they show a highly isotropic behavior in a neighborhood of the crack tip instead. As a consequence of these properties, the resulting discretizations follow very closely the propagation of the fracture, which is not significantly influenced by the discretization itself, delivering a physically sound prediction of the crack path, with a reasonable computational effort. In fact, we provide numerical tests which assess the balance between accuracy and complexity of the algorithm. We compare our results with isotropic mesh adaptation and we highlight the remarkable improvements both in terms of accuracy and computational cost with respect to simulations in the pertinent most recent literature. |
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19/2014 - 22/05/2014
Bonaventura, L.; Ferretti, R.
Semi-Lagrangian methods for parabolic problems in divergence form | Abstract | | Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection-diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analysed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low order time discretization approaches. |
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18/2014 - 21/05/2014
Tumolo, G.; Bonaventura, L.
An accurate and efficient numerical framework for adaptive numerical weather prediction | Abstract | | We present an accurate and efficient discretization approach for the adaptive discretization of typical model equations employed in numerical weather prediction. A semi-Lagrangian approach is combined with the TR-BDF2 semi-implicit time discretization method and with a spatial discretization based on adaptive discontinuous finite elements. The resulting method has full second order accuracy in time and can employ polynomial bases of arbitrarily high degree in space, is unconditionally stable and can effectively adapt the number of degrees of freedom employed in each element, in order to balance accuracy and computational cost. The p-adaptivity approach employed does not require remeshing, therefore it is especially suitable for applications, such as numerical weather prediction, in which a large number of physical quantities are associated with a given mesh. Furthermore, although the proposed method can be implemented on arbitrary unstructured and nonconforming meshes, even its application on simple Cartesian meshes in spherical coordinates can cure effectively the pole problem by reducing the polynomial degree used in the polar elements. Numerical simulations of classical benchmarks for the shallow water and for the fully compressible Euler equations validate the method and demonstrate its capability to achieve accurate results also at large Courant numbers, with time steps up to 100 times larger than those of typical explicit discretizations of the same problems, while reducing the computational cost thanks to the adaptivity algorithm. |
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