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 1237 products
-
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.
|
-
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. |
-
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. |
-
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. |
-
49/2013 - 10/24/2013
Micheletti, S.
Fast simulations in Matlab for Scientific Computing | Abstract | | We show how the numerical simulation of typical problems found in Scientific Computing can be run efficiently even under the serial Matlab environment. This is made possible by a strong employment of vectorization
and sparse matrix manipulation. Numerical examples based on FEMs on
2D unstructured triangular grids assess the flexibility and efficiency of the
simulation tool, both on simple elliptic problems as well as on the steady
and unsteady incompressible Navier-Stokes equations. Any type of finite
elements, and 1D and 2D quadrature rules can be easily accommodated
within our framework. Emphasis is focused on vectorization programming
and sparse matrix storage and operations, which allow one to obtain very
efficient programs which run in a few minutes on a common notebook.
|
-
48/2013 - 10/23/2013
Simone Palamara, Christian Vergara, Elena Faggiano, Fabio Nobile
An effective algorithm for the generation of patient-specific Purkinje networks in computational electrocardiology | Abstract | | The Purkinje network is responsible for the fast and coordinated distribution
of the electrical impulse in the ventricle that triggers its contraction.
Therefore, it is necessary to model its presence to obtain an accurate
patient-specific model of the ventricular electrical activation. In this paper,
we present an efficient algorithm for the generation of a patient-specific
Purkinje network, driven by measures of the electrical activation acquired
on the endocardium. The proposed method provides a correction of an
initial network, generated by means of a fractal law, and it is based on
the solution of Eikonal problems both in the muscle and in the Purkinje
network. We present several numerical results both in an ideal geometry
with synthetic data and in a real geometry with patient-specific clinical
measures. These results highlight an improvement of the accuracy of the patient-specific Purkinje network with respect to the initial one, also in the
cases of a cross-validation test and of noisy data. |
-
46/2013 - 10/08/2013
Marron, J.S.; Ramsay, J.O.; Sangalli, L.M.; Srivastava, A.
Statistics of Time Warpings and Phase Variations | Abstract | | Many methods exist for one dimensional curve registration, and how methods compare has not been made clear in the literature. This special section is a summary of a detailed comparison of a number of major methods, done during a recent workshop. The basis of the comparison was simultaneous analysis of a set of four real data sets, which engendered a high level of informative discussion. Most research groups in this area were represented, and many insights were gained, which are discussed here. The format of this special section is four papers introducing the data, each accompanied by a number of analyses by di erent groups, plus a discussion summary of the lessons learned. |
-
45/2013 - 10/08/2013
Sangalli, L.M.; Secchi, P.; Vantini, S.
Analysis of AneuRisk65 data: K-mean Alignment | Abstract | | We describe the k-mean alignment procedure, for the joint alignment and clustering of functional data and we apply it to the analysis of
AneuRisk65 data. Thanks to the efficient separation of the variability in phase variability and within/between clusters amplitude variability, we are able to discriminate subjects having aneurysms in di fferent cerebral districts and identifying diff erent morphological shapes of Inner Carotid Arteries, unveiling a strong association between vessel morphologies and the aneurysmal pathology. |
|
