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 1287 products
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22/2013 - 05/03/2013
Falcone, M.; Verani, M.
Recent Results in Shape Optimization and Optimal Control for PDEs | Abstract | | In this paper we will present some recent advances in the numerical approximation of two classical problems: shape optimization and optimal control for evolutive partial differential equations. For shape optimization we present two novel techniques which have shown to be rather efficient on some applications. The first technique is based on multigrid methods whereas the second relies on an adaptive sequential quadratic programming. With respect to the optimal control of evolutive problems, the approximation is based on the coupling between a POD representation of the dynamical system and the classical Dynamic Programming approach. We look for an approximation of the value function characterized as the weak solution (in the viscosity sense) of the corresponding Hamilton-Jacobi equation.
Several tests illustrate the main features of the above methods. |
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21/2013 - 05/02/2013
Perotto, S.; Veneziani, A.
Coupled model and grid adaptivity in hierarchical reduction of elliptic problems | Abstract | | In this paper we propose a surrogate model for advection-diffusion-reaction problems characterized by a dominant direction in their dynamics. We resort to a hierarchical-model reduction where we couple a modal representation of the transverse dynamics with a finite element approximation along the mainstream. This different treatment of the dynamics entails a surrogate model enhancing a purely 1D description related to the leading direction. The coefficients of the finite element expansion along this direction introduce a generally non-constant description of the transversal dynamics. Aim of this paper is to provide an automatic adaptive approach to locally determine the dimension of the modal expansion as well as the finite element step in order to satisfy a prescribed tolerance on a goal functional of interest. |
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20/2013 - 05/01/2013
Azzimonti, L.; Nobile, F.; Sangalli, L.M.; Secchi, P.
Mixed Finite Elements for spatial regression with PDE penalization | Abstract | | We study a class of models at the interface between statistics and numerical analysis. Specifically, we consider non-parametric regression models for the estimation of spatial fields from pointwise and noisy observations, that account for problem specific prior information, described in terms of a PDE governing the phenomenon under study. The prior information is incorporated in the model via a roughness term using a penalized regression framework. We prove the well-posedness of the estimation problem and we resort to a mixed equal order Finite Element method for its discretization. We prove the well posedness and the optimal convergence rate of the proposed discretization method. Finally the smoothing technique is extended to the case of areal data, particularly interesting in many applications. Keywords: mixed Finite Element method, fourth order problems, non-parametric regression, smoothing. |
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19/2013 - 04/30/2013
Azzimonti, L.; Sangalli, L.M.; Secchi, P.; Domanin, M.; Nobile, F.
Blood flow velocity field estimation via spatial regression with PDE penalization | Abstract | | We propose an innovative method for the accurate estimation of surfaces and spatial fields when a prior knowledge on the phenomenon under study is available. The prior knowledge included in the model derives from physics, physiology or mechanics of the problem at hand, and is formalized in terms of a partial differential equation governing the phenomenon behavior, as well as conditions that the phenomenon has to satisfy at the boundary of the problem domain. The proposed models exploit advanced scientific computing techniques and specifically make use of the Finite Element method. The estimators have a typical penalized regression form and the usual inferential tools are derived. Both the pointwise and the areal data frameworks are considered. The driving application concerns the estimation of the blood flow velocity field in a section of a carotid artery, using data provided by echo-color doppler; this applied problem arises within a research project that aims at studying atherosclerosis pathogenesis. Keywords: functional data analysis, spatial data analysis, object-oriented data analysis, penalized regression, Finite Elements. |
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18/2013 - 04/29/2013
Discacciati, M.; Gervasio, P.; Quarteroni, A.
Interface Control Domain Decomposition (ICDD) Methods for Coupled Diffusion and Advection-Diffusion Problems | Abstract | | This paper is concerned with ICDD (Interface Control Domain Decomposition) method, a strategy introduced for the solution of partial differential equations (PDEs) in computational domains partitioned into subdomains that overlap. After reformulating the original boundary value problem with the introduction of new additional control variables, the unknown traces of the solution at internal subdomain interfaces, the determination of the latter is made possible by the requirement that the (a-priori) independent solutions in each subdomain undergo a minimization of a suitable cost functional. We illustrate the method on two kinds of boundary value problems, one homogeneous (an elliptic PDE), the other heterogeneous (a coupling between a second order advection-diffusion equation and a first order advection equation). We derive the associated optimality system, analyze its well posedness, and illustrate efficient algorithms based on the solution of the Schur-complement system restricted solely to the interface control variables. Finally, we validate numerically our method through a family of numerical tests and investigate the excellent convergence properties of our iterative solution algorithm. |
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17/2013 - 04/15/2013
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 - 04/09/2013
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 - 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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