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 1251 products
-
02/2012 - 01/15/2012
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. |
-
01/2012 - 01/12/2012
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 |
-
47/2011 - 12/30/2011
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. |
-
46/2011 - 12/21/2011
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. |
-
45/2011 - 12/20/2011
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.
|
-
44/2011 - 12/16/2011
Fumagalli,A.; Scotti,A.
Numerical modelling of multiphase subsurface flow in the presence of fractures | Abstract | | Subsurface flow is influenced by the heterogeneity of the porous medium
and in particular by the presence of faults and large fractures which act as
preferential paths for the flow. In this work we present a robust numerical
method for the simulation of two-phase Darcy flows in heterogeneous media
and propose a possible treatment of fractures by means of the extended
finite element method, XFEM, and the coupling with a reduced model for
the flow inside the fracture. The use of extended finite elements allows to
handle fractures that are non conforming with the underlying mesh, thus
increasing the applicability of the proposed scheme to the simulation of
realistic problems such as oil migration in fractured basins, CO2 storage or
pollutant dispersion in groundwater flows.
|
-
43/2011 - 12/09/2011
L. Formaggia, A. Quarteroni, C. Vergara
On the physical consistency of the coupling between three-dimensional compliant and one-dimensional problems in haemodynamics | Abstract | | In this work we discuss the reliability of the coupling among three-dimensional (3D)
and one-dimensional (1D) models, that describe blood flowing into the circulatory tree.
In particular, we study the physical consistency of the 1D model with respect to the
3D one. To this aim, we introduce a general criterion based on energy balance for the
proper choice of coupling conditions between models. We also propose a way to include
in the 1D model the effect of the external tissue surrounding the vessel and we discuss
its importance whenever this effect is considered in the 3D model. Finally, we propose
several numerical results in real human carotids, studying different configurations for
the 1D model and highlighting the best one in view of the physical consistency. |
-
42/2011 - 11/27/2011
Antonietti, P.F.; Quarteroni, A.
Numerical performance of discontinuous and stabilized continuous Galerkin methods or convection-diffusion problems | Abstract | | We compare the performance of two classes of numerical methods for the approximation of linear steady-state convection-diffusion equations, namely, the discontinuous Galerkin (DG) method and the continuous streamline upwind Petrov-Galerkin (SUPG) method. We present a fair comparison of such schemes considering both diffusion--dominated and convection-dominated regimes, and present numerical results obtained on a series of test problems including smooth solutions, and test cases with sharp internal and boundary layers. |
|
