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 1238 prodotti
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05/2012 - 21/01/2012
Ambrosi, D; Arioli, G; Koch, H.
A homoclinic solution for excitation waves on a contractile substratum | Abstract | | We analyze a model of electric signalling in biological tissues
and prove that this model admits a travelling wave solution.
Our result is based on a new technique for computing rigorous bounds on the stable and unstable manifolds at an equilibrium point of a dynamical system depending on a parameter. |
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04/2012 - 18/01/2012
Tumolo, G.; Bonaventura, L.; Restelli, M.
A semi-implicit, semi-Lagrangian, p-adaptive Discontinuous Galerkin method for the shallow water equations | Abstract | | A semi-implicit and semi-Lagrangian Discontinuous Galerkin (SISLDG) method for the shallow water equations is proposed, for applications to geophysical scale flows. A non conservative formulation of the advection equation is employed, in order to achieve a more treat- able form of the linear system to be solved at each time step. The method is equipped with a simple p−adaptivity criterion, that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution. Numerical results show that the method captures well the main features of gravity and inertial gravity waves, as well as reproducing correct solutions in nonlinear test cases with analytic solutions. The accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high Courant numbers and with automatic choice of the local approximation degree. |
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03/2012 - 17/01/2012
Fumagalli, A.; Scotti, A.
A reduced model for flow and transport in fractured porous media with non-matching grids | Abstract | | In this work we focus on a model reduction approach for the treatment of fractures in a porous medium, represented as interfaces embedded in a n-dimensional domain, in the form of a (n- 1)-dimensional manifold, to describe fluid flow and transport in both domains. We employ a method that allows for non-matching grids, thus very advantageous if the position of the fractures is uncertain and multiple simulations are required. To this purpose we adopt an XFEM approach to
represent discontinuities of the variables at the interfaces, which can arbitrarily cut the elements of the grid. The method is applied to the numerical solution of the Darcy problem, and advection-diffusion problems in porous media. |
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02/2012 - 15/01/2012
Arioli, G.
Optimization of the forcing term for the solution of two point boundary value problems | Abstract | | We present a new numerical method for the computation of the forcing term of minimal norm such that a two point boundary value problem admits a solution. The method relies on the following steps. The forcing term is written as a (truncated) Chebyshev series, whose coefficients are free parameters. A technique derived from automatic differentiation is used to solve the initial value problem, so that the final value is obtained as a series of polynomials whose coefficients depend explicitly
on (the coefficients of) the forcing term. Then the minimization problem becomes purely algebraic, and can be solved by standard methods of constrained optimization, e.g. with Lagrange multipliers. We provide an application of this algorithm to the restricted three body problem in order to study the planning of low thrust transfer orbits. |
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01/2012 - 12/01/2012
Lassila, T.; Manzoni, A.; Quarteroni, A.; Rozza, G.
A reduced computational and geometrical framework for inverse problems in haemodynamics | Abstract | | The solution of inverse problems in cardiovascular mathematics is computationally
expensive. In this paper we apply a domain parametrization technique to reduce both the geometrical and computational complexity of the forward problem, and replace the finite element solution of the incompressible Navier-Stokes equations by a computationally less expensive
reduced basis approximation. This greatly reduces the cost of simulating the forward problem. We then consider the solution of inverse problems in
both the deterministic sense, by solving a least-squares problem, and in the statistical sense, by using a Bayesian framework for quantifying uncertainty.
Two inverse problems arising in haemodynamics modelling are considered: (i) a simplified fluid-structure interaction model problem in a portion of a stenosed artery for quantifying the risk of atherosclerosis by identifying the material parameters of the arterial wall based on pressure measurements;
(ii) a simplied femoral bypass graft model for robust shape design under uncertain residual flow in the main arterial branch identified from pressure measurements.
Keywords: Inverse problems; model reduction; shape optimization; fluid-structure interaction; reduced basis methods; haemodynamics; parametrized Navier-Stokes equations |
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47/2011 - 30/12/2011
Antonietti, P.F.; Borzì, A.; Verani, M.
Multigrid shape optimization governed by elliptic PDEs | Abstract | | This paper presents and analyzes a new multigrid framework to solve shape op- timization problems governed by elliptic PDEs. The boundary of the domain, i.e., the control variable, is represented as the graph of a continuous function that is approximated at various levels of discretization. The proposed multigrid shape opti- mization scheme acts directly on the function describing the geometry of the domain and it combines a single-grid shape gradient optimizer with a coarse-grid correction (minimization) step, recursively within a hierarchy of levels. The convergence of the proposed multigrid shape optimization method is proved and several numerical experiments assess its effectiveness. |
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46/2011 - 21/12/2011
Migliorati, G.; Nobile, F.; von Schwerin, E.; Tempone, R.
Analysis of the discrete L2 projection on polynomial spaces with random evaluations | Abstract | | We analyse the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is Uncertainty Quantification (UQ) for computational models.
We prove an optimal convergence estimate, up to a logarithmic factor, in the monovariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero, provided the number of samples scales quadratically with the dimension
of the polynomial space.
Several numerical tests are presented both in the monovariate and multivariate case, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target
function. |
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45/2011 - 20/12/2011
Canuto, C.; Nochetto, R. H.; Verani, M.
Adaptive Fourier-Galerkin Methods | Abstract | | We study the performance of adaptive Fourier-Galerkin methods in a periodic box in $ mathbb{R}^d$ with dimension $d ge 1$. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the $hp$-FEM. We examine two nonlinear approximation classes, one classical corresponding to algebraic decay of Fourier coefficients and another associated with exponential decay. We study the sparsity classes of the residual and
show that they are the same as the solution for the algebraic class but not for the exponential one. This possible sparsity degradation for the exponential class can be compensated with coarsening, which we discuss in detail. We present several adaptive Fourier algorithms, and prove their contraction and optimal cardinality properties.
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