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 1249 products
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56/2013 - 11/15/2013
Antonietti, P.F.; Ayuso de Dios, B.; Mazzieri, I.; Quarteroni, A.
Stability analysis for Discontinuous Galerkin approximations of the elastodynamics problem | Abstract | | We consider semi-discrete discontinuous Galerkin approximations of a general elastodynamics problem, in both displacement and displacement-stress formulations. We present the stability analysis of all the methods in the natural energy norm and derive optimal a-priori error estimates. For the displacement-stress formulation, schemes preserving the total energy of the system are introduced and discussed. We include some numerical experiments in three dimensions to verify the theory. |
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54/2013 - 11/13/2013
Biasi, R.; Ieva, F.; Paganoni, A.M.; Tarabelloni, N.
Use of depth measure for multivariate functional data in disease prediction: an application to electrocardiographic signals | Abstract | | In this paper we develop statistical methods to compare two independent samples of multivariate functional data that differ in terms of covariance operators. In particular we generalize the concept of depth measure to this kind of data, exploiting the role of the covariance operators in weighting the components that define the depth.
Two simulation studies are carried out to validate the robustness of the proposed methods.
We present an application to Electrocardiographic (ECG) signals aimed at comparing physiological subjects and patients affected by Left Bundle Branch Block. The proposed depth measures computed on data are then used to perform a nonparametric comparison test among these two populations. They are also introduced into a generalized regression model aimed at classifying the ECG signals.
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55/2013 - 11/13/2013
Laadhari, A.; Ruiz-Baier, R.; Quarteroni, A.
Fully Eulerian finite element approximation of a fluid-structure interaction problem in cardiac cells | Abstract | | We propose in this paper an Eulerian finite element approximation of a coupled chemical fluid-structure interaction problem arising in the study of mesoscopic cardiac biomechanics. We simulate the active response of a myocardial cell (here considered as an anisotropic, hyperelastic, and incompressible material), the propagation of calcium concentrations inside it, and the presence of a surrounding Newtonian fluid. An active strain approach is employed to account for the mechanical activation, and the deformation of the cell membrane is captured using a level set strategy. We address in detail the main features of the proposed method, and we report several numerical experiments aimed at model validation. Copyright © 2013 John Wiley & Sons, Ltd.
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53/2013 - 11/12/2013
Micheletti, S.
A continuum variational approach based on optimal control to adaptive moving mesh methods | Abstract | | We cast mesh adaptation based on point relocation in a continuum mechanics analogy. The movement of the mesh points is thus interpreted as a displacement of points of the continuum. We describe our approach on the
Dirichlet problem for the Poisson equation in 2D. It is well known that, for a fixed mesh, the best approximation in the energy norm, |||·|||, to the exact solution, u, is the Galerkin approximation, uh , in a finite element space, and that uh minimizes also a suitable energy functional. The best error, however, still depends on the mesh. The energy functional is then rewritten in terms of the displacement through its displacement-gradient tensor.
Thus finding the optimal mesh, where |||u − uh ||| is a minimum, among a family of possible meshes, amounts to computing the displacement field
which minimizes the energy functional. This is carried out via the optimal control approach, after enforcing the constraint that the displacement satisfies a diffusion equation with the control functions in the role of a variable
diffusivity. This in turn yields the optimal movement of the mesh nodes. An algorithm based on a gradient flow delivers the actual adapted mesh.
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51/2013 - 10/30/2013
Chen, P.; Quarteroni, A.
Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraint | Abstract | | In this paper we develop and analyze an efficient computational method for solving stochastic optimal control problems constrained by elliptic partial differential equation (PDE) with random input data. We first prove both existence and uniqueness of the optimal solution. Regularity of the optimal solution in the stochastic space is studied in view of the analysis of stochastic approximation error. For numerical approximation, we employ finite element method for the discretization of physical variables and stochastic collocation method for the discretization of random variables. In order to alleviate the computational effort, we develop a model order reduction strategy based on a weighted reduced basis method. A global error analysis of the numerical approximation is carried out and several numerical tests are performed to verify our analysis.
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52/2013 - 10/30/2013
Chen, P.; Quarteroni, A.; Rozza, G.
Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations | Abstract | | In this paper we develop and analyze a multilevel weighted reduced basis method
for solving stochastic optimal control problems constrained by Stokes equations. Existence and uniqueness of the stochastic optimal solution is proved by establishing the equivalence between the constrained optimization problem and a stochastic saddle point problem. Analytic regularity of the optimal solution in the probability space is obtained under certain assumptions on the random input data. Finite element method and stochastic collocation method are employed for numerical approx imation of the problem in deterministic space and probability space, respectively. A reduced basis method using a multilevel greedy algorithm based on isotropic and anisotropic sparse-grid techniques and weighted a posteriori error estimate is proposed in order to reduce the computational effort. A global error is obtained based on estimate results of error contribution from each method. Numerical experiments are performed with stochastic dimension ranging from 10 to 100, demonstrating that the proposed method is very efficient, especially for high dimensional and large-scale optimization problems. |
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47/2013 - 10/26/2013
Chkifa, A.; Cohen, A.; Migliorati, G.; Nobile, F.; Tempone, R.
Discrete least squares polynomial approximation with random evaluations - application to parametric and stochastic elliptic PDEs | Abstract | | Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case and for the uniform distribution, the least-squares method is optimal in expectation in [1] and in probability in [7], under the condition that the number of samples scales quadratically with respect to the dimension of the polynomial space. Here optimal means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary
multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss inclusion type elliptic PDE models, and derive an exponential convergence estimate for the least-squares method.
Numerical results con rm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate. |
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50/2013 - 10/25/2013
Antonietti, P.F.; Verani, M.; Zikatanov, L.
A two-level method for Mimetic Finite Difference discretizations of elliptic problems | Abstract | | We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to achieve convergence is uniformly bounded independently of the characteristic size of the underling partition. We also show that the resulting scheme provides a uniform preconditioner with respect to the number of degrees of freedom. Numerical results that validate the theory are also presented. |
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