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 1249 prodotti
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48/2012 - 26/11/2012
Ghiglietti, A.; Paganoni, A.M.
Statistical properties of two-color randomly reinforced urn design targeting fixed allocations | Abstract | | This paper deals with the statistical properties of a response adaptive design, described in terms of a two colors urn model, targeting prespecified asymptotic allocations. Results on the rate of convergence of number of patients assigned to each treatment are proved as well as on the asymptotic behavior of the urn composition. Suitable statistics are introduced and studied to test the hypothesis on treatment s differences. |
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47/2012 - 30/10/2012
Astorino, M.; Chouly, F.; Quarteroni, A.
Multiscale coupling of finite element and lattice Boltzmann methods for time dependent problems | Abstract | | In this work we propose a new numerical procedure for the simulation of time-dependent problems based on the coupling between the finite element method and the lattice Boltzmann method. The two methods are regarded as macroscale and mesoscale solvers, respectively. The procedure is based on the Parareal paradigm and allows for a truly multiscale coupling between two numerical methods having optimal efficiency at different space and time scales. The motivations behind this approach are manifold. Among others, we have that one technique may be more efficient, or physically more appropriate or less memory consuming than the other depending on the target of the simulation and/or on the sub-region of the computational
domain. The theoretical and numerical framework is presented for parabolic equations even though its potential applicability is much wider (e.g. Navier-Stokes equations). Various numerical examples on the heat equation will validate the proposed procedure and illustrate its multiple advantages.
Keywords: finite element method, lattice Boltzmann method, multiscale coupling, Parareal, parallel-in-time domain decomposition. |
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46/2012 - 26/10/2012
Dassi, F.; Perotto, S.; Formaggia, L.; Ruffo, P.
Efficient geometric reconstruction of complex geological structures | Abstract | | Complex geological structures pose a challenge to domain discretization. Indeed data are normally given as a set of intersecting surfaces, sometimes with incomplete data, from which one has to identify the computational domain to build a mesh suited for numerical simulations. In this paper we describe a set of tools which have been developed for this purpose. Specialized data structures have been developed to efficiently identify intersections of triangulated surfaces and to conformally include these intersections in the starting meshes, while improving the mesh quality. Then, an effective algorithm has been implemented to detect the different sub-regions forming the computational domain; this algorithm has been properly enhanced to take into account the specific characteristics involved in the simulation of geological basins. |
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45/2012 - 24/10/2012
Negri, F.; Rozza, G.; Manzoni, A.; Quarteroni, A.
Reduced basis method for parametrized elliptic optimal control problems | Abstract | | We propose a suitable model reduction paradigm -the certied reduced basis method (RB) - for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations (PDEs). In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint. Firstly, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of
the RB methodology are provided: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling to perform competitive Offine-Online splitting in the computational procedure; an efficient and rigorous a posteriori error estimate on the state, control and adjoint variables as well as on the cost functional. Finally, the reduction scheme is applied to some numerical tests conrming the theoretical results and showing the efficiency of the proposed technique.
Keywords: reduced basis methods, parametrized optimal control problems, saddle-point problems, model order reduction, PDE-constrained optimization, a posteriori error estimate |
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43/2012 - 19/10/2012
Secchi, P.; Vantini, S.; Vitelli, V.
A Case Study on Spatially Dependent Functional Data: the Analysis of Mobile Network Data for the Metropolitan Area of Milan | Abstract | | We analyze geo-referenced high-dimensional data describing the use over time of the mobile-phone network in the urban area of Milan, Italy. Aim of the analysis is segmenting the metropolitan area of Milan into subregions sharing a similar pattern along time, possibly related to activities taking place in specific locations and/or times within the city. To tackle this problem, we develop a non-parametric method for the analysis of spatially dependent functional data, named Bagging Voronoi Treelet Analysis. Indeed, this novel approach integrates the treelet decomposition with a proper treatment of spatial dependence, obtained through a Bagging Voronoi strategy. The latter relies on the aggregation of different replicates of the analysis, each involving a set of functional local representatives associated to random Voronoi-based neighborhoods covering the investigated area. In the presence of spatial dependence the method appears to be both computationally and statistically efficient. Indeed results clearly point out some interesting temporal patterns interpretable both in terms of population density and mobility (e.g., daily work activities in the tertiary district, leisure activities in residential areas in the evenings and in the weekend, commuters movements along the highways during rush hours, and localized mob concentrations related to occasional events). Moreover we perform two simulation studies, aimed at investigating the properties and performances of the method.
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44/2012 - 19/10/2012
Fumagalli, A.; Scotti, A.
A numerical method for two-phase flow in fractured porous media with non-matching grids | Abstract | | We propose a novel computational method for the efficient simulation of two-phase flow in fractured porous media. Instead of refining the grid to capture the flow along the faults or fractures, we represent them as immersed interfaces with a reduced model for the flow and suitable coupling conditions. We allow for non matching grids between the porous matrix and the fractures to increase the flexibility of the method in realistic cases. We employ the extended finite element method for the Darcy problem and a finite volume method that is able to handle cut cells and matrix-fracture interactions for the saturation equation. The choice of a suitable flux function in the case of discontinuous flux function at the interface between the fracture and the porous matrix is also addressed through numerical experiments.
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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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