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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01/2019 - 01/17/2019
Regazzoni, F.; Dedè, L.; Quarteroni, A.
Machine learning for fast and reliable solution of time-dependent differential equations | Abstract | | We propose a data-driven Model Order Reduction (MOR) technique, based on Artificial Neural Networks (ANNs), applicable to dynamical systems arising from Ordinary Differential Equations (ODEs) or time-dependent Partial Differential Equations (PDEs). Unlike model-based approaches, the proposed approach is non-intrusive since it just requires a collection of input-output pairs generated through the high-fidelity (HF) ODE or PDE model. We formulate our model reduction problem as a maximum-likelihood problem, in which we look for the model that minimizes, in a class of candidate models, the error on the available input-output pairs. Specifically, we represent candidate models by means of ANNs, which we train to learn the dynamics of the HF model from the training input-output data. We prove that ANN models are able to approximate every time-dependent model described by ODEs with any desired level of accuracy. We test the proposed technique on different problems, including the model reduction of two large-scale models. One of the HF systems of ODEs here considered stems from the spatial discretization of a parabolic PDE, which sheds light on a promising field of application of the proposed technique. |
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66/2018 - 12/24/2018
Riccobelli, D.; Agosti, A.; Ciarletta, P.
On the existence of elastic minimizers for initially stressed materials | Abstract | | A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration.
Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of the divergence-free symmetric tensors satisfying the boundary condition in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed accordingly to the symmetry group of the material at fixed initial stresses. These results allow to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. |
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65/2018 - 12/24/2018
Boschi, T.; Chiaromonte, F.; Secchi, P.; Li, B.
Covariance based low-dimensional registration for function-on-function regression | Abstract | | We propose a new low-dimensional registration procedure that exploits the relationship between response and predictor in a function-on-function regression. In this context, Functional Covariance Components (FCC) provide a flexible and powerful tool to represent the data in a low-dimensional space, capturing the most meaningful modes of dependency between the two set of curves. Based on this reduced representation, our procedure aligns simultaneously the two sets of curves, in a way that optimizes the subsequent regression analysis. To implement our procedure, we use both the Continuous Registration algorithm (CR) and a novel parallel algorithm coded in R. We then compare it to other common registration approaches via simulations and an application to the AneuRisk data. |
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64/2018 - 12/24/2018
Menafoglio, A.; Pigoli, D.; Secchi, P.
Kriging Riemannian Data via Random Domain Decompositions | Abstract | | Data taking value on a Riemannian manifold and observed over a complex
spatial domain are becoming more frequent in applications, e.g. in
environmental sciences and in geoscience. The analysis of these data needs to rely on local models to account for the non stationarity of the generating random process, the non linearity of the manifold and the complex topology of the domain. In this paper, we propose to use a random domain decomposition approach to estimate an ensemble of local models and then to aggregate the predictions of the local models through Frechet averaging. The algorithm is introduced in complete generality and is valid for data belonging to any smooth Riemannian manifold but it is then described in details for the case of the manifold of positive definite matrices, the hypersphere and the Cholesky manifold. The predictive performance of the method are explored via simulation studies for covariance matrices and correlation matrices, where the Cholesky manifold geometry is used. Finally, the method is illustrated on an environmental dataset observed over the Chesapeake Bay (USA). |
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63/2018 - 12/24/2018
Brugiapaglia, S.; Micheletti, S.; Nobile, F.; Perotto, S.
Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations | Abstract | | We present and analyze a wavelet-Fourier technique for the numerical treatment of multi-dimensional advection-diffusion-reaction equations with periodic boundary condi- tions. Combining the Petrov-Galerkin technique with the compressed sensing approach, the proposed method is able to approximate the largest coefficients of the solution with respect to a biorthogonal wavelet basis. Namely, we assemble a compressed discretization based on randomized subsampling of the Fourier test space and we employ sparse recovery tech- niques. The proposed theoretical analysis is based on the local a-coherence and provides effective recipes for a practical implementation. The stability and robustness of the pro- posed scheme is shown by numerical illustrations in the one-, two-, and three-dimensional case. |
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62/2018 - 12/13/2018
Perotto, S.; Carlino, M.G.; Ballarin, F.
Model reduction by separation of variables: a comparison between Hierarchical Model reduction and Proper Generalized Decomposition | Abstract | | Hierarchical Model reduction and Proper Generalized
Decomposition both exploit separation of variables
to perform a model reduction. After setting the basics, we exemplify
these techniques on some standard elliptic problems
to highlight pros and cons of the two procedures, both from a
methodological and a numerical viewpoint. |
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61/2018 - 12/13/2018
Zakerzadeh, R.; Zunino P.
A Computational Framework for Fluid-Porous Structure Interaction with Large Structural Deformation | Abstract | | We study the effect of poroelasticity on fluid-structure interaction. More precisely, we analyze the role of fluid flow through a deformable porous matrix in the energy dissipation behavior of a poroelastic structure. For this purpose, we develop and use a nonlinear poroelastic computational model and apply it to the fluid-structure interaction simulations. We discretize the problem by means of the finite element method for the spatial approximation and using finite differences in time. The numerical discretization leads to a system of non-linear equations that are solved by Newton’s method. We adopt a moving mesh algorithm, based on the Arbitrary Lagrangian Eulerian (ALE) method to handle large deformations of the structure. To reduce the computational cost, the coupled problem of free fluid, porous media flow and solid mechanics is split among its components and solved using a partitioned approach. Numerical results show that the flow through the porous matrix is responsible for generating a hysteresis loop in the stress versus displacement diagrams of the poroelastic structure. The sensitivity of this effect with respect to the parameters of the problem is also analyzed. |
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59/2018 - 11/20/2018
Martino, A.; Guatteri, G.; Paganoni, A. M.
Multivariate Hidden Markov Models for disease progression | Abstract | | Disease progression models are a powerful tool for understanding the development of a disease, given some clinical measurements obtained from longitudinal events related to a sample of patients. These models are able to give some insights about the disease progression through the analysis of patients histories and can be also used to predict the future course of the disease for an individual. In particular, Hidden Markov Models are suitable for disease progression since they model the latent unobservable states of the disease. In this work we introduce a novel HMM where the outcome is multivariate and its components are not independent; to accomplish our aim, since we do not make any usual normality assumptions, we model the outcome using copulas. We first test the performance of our model in a simulation setting and show the validity of the method. Then, we study the course of Heart Failure, applying our model to an administrative dataset from Lombardia Region in Italy, showing how episodes of hospitalization can give information about the disease status of a patient. |
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