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 1238 products
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45/2012 - 10/24/2012
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 - 10/19/2012
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 - 10/19/2012
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 - 10/15/2012
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 - 10/14/2012
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 - 10/13/2012
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 - 09/16/2012
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 - 09/15/2012
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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