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 1251 prodotti
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69/2022 - 19/10/2022
Franco, N.R; Manzoni, A.; Zunino, P.
Learning Operators with Mesh-Informed Neural Networks | Abstract | | Thanks to their universal approximation properties and new efficient training strategies, Deep Neural Networks are becoming a valuable tool for the approximation of mathematical operators. In the present work, we introduce Mesh-Informed Neural Networks (MINNs), a class of architectures specifically tailored to handle mesh based functional data, and thus of particular interest for reduced order modeling of parametrized Partial Differential Equations (PDEs). The driving idea behind MINNs is to embed hidden layers into discrete functional spaces of increasing complexity, obtained through a sequence of meshes defined over the underlying spatial domain. The approach leads to a natural pruning strategy which enables the design of sparse architectures that are able to learn general nonlinear operators. We assess this strategy through an extensive set of numerical experiments, ranging from nonlocal operators to nonlinear diffusion PDEs, where MINNs are compared to classical fully connected Deep Neural Networks. Our results show that MINNs can handle functional data defined on general domains of any shape, while ensuring reduced training times, lower computational costs, and better generalization capabilities, thus making MINNs very well-suited for demanding applications such as Reduced Order Modeling and Uncertainty Quantification for PDEs. |
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68/2022 - 19/10/2022
Orlando, G.; Benacchio, T.; Bonaventura, L.
An IMEX-DG solver for atmospheric dynamics simulations with adaptive mesh refinement | Abstract | | We present an accurate and efficient solver for atmospheric dynamics simulations that allows for non-conforming mesh refinement. The model equations are the conservative Euler equations for compressible flows. The numerical method is based on an h-adaptive Discontinuous Galerkin spatial discretization and on a second order Additive Runge Kutta IMEX method for time discretization, especially designed for low Mach regimes. The solver is implemented in the framework of the deal.II library, whose mesh refinement capabilities are employed to enhance efficiency. A number of numerical experiments based on classical benchmarks for atmosphere dynamics demonstrate the properties and advantages of the proposed method. |
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67/2022 - 19/10/2022
Alghamdi, M. M.; Boffi, D.; Bonizzoni, F.
A greedy MOR method for the tracking of eigensolutions to parametric elliptic PDEs | Abstract | | In this paper we introduce an algorithm based on a sparse grid adaptive refinement, for the approximation of the eigensolutions to parametric problems arising from elliptic partial differential equations. In particular, we are interested in detecting the crossing of the hypersurfaces describing the eigenvalues as a function of the parameters.
The a priori matching is followed by an a posteriori verification, driven by a suitably defined error indicator. At a given refinement level, a sparse grid approach is adopted for the construction of the grid of the next level, by using the marking given by the a
posteriori indicator.
Various numerical tests confirm the good performance of the scheme. |
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65/2022 - 05/10/2022
Dassi, F.; Fumagalli, A.; Mazzieri, I.; Vacca, G.
Mixed Virtual Element approximation of linear acoustic wave equation | Abstract | | We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In absence of external load, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical theta-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering application of the method. |
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64/2022 - 05/10/2022
Massi, M.C., Dominoni, L., Ieva, F., Fiorito, G.
A Deep Survival EWAS approach estimating risk profile based on pre-diagnostic DNA methylation: an application to Breast Cancer time to diagnosis | Abstract | | Previous studies for cancer biomarker discovery based on pre-diagnostic blood DNA methylation profiles, either ignore the explicit modeling of the time to diagnosis (TTD) as in a survival analysis setting, or provide inconsistent results. This lack of consistency is likely due to the limitations of standard EWAS approaches, that model the effect of
DNAm at CpG sites on TTD independently. In this work, we argue that a global approach to estimate CpG sites effect profile is needed, and we claim that such approach should capture the complex (potentially non-linear) relationships interplaying between sites. To prove our concept, we develop a new Deep Learning-based approach assessing the relevance of individual CpG Islands (i.e., assigning a weight to each site) in determining TTD while modeling their combined effect in a survival analysis scenario.
The algorithm combines a tailored sampling procedure with DNAm sites agglomeration, deep non-linear survival modeling and SHapley Additive exPlanations (SHAP) values estimation to aid robustness of the derived effects profile. The proposed approach deal with the common complexities arising from epidemiological studies, such as small sample size, noise, and low signal-to-noise ratio of blood-derived DNAm. We apply our approach to a prospective case-control study on breast cancer nested in the EPIC Italy cohort and we perform weighted gene-set enrichment analyses to demonstrate the biological meaningfulness of the obtained results. We compared the results of Deep Survival EWAS with those of a traditional EWAS approach, demonstrating that our method performs better than the standard approach in identifying biologically relevant pathways. |
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63/2022 - 05/10/2022
Corti, M.; Antonietti, P.F.; Dede', L.; Quarteroni, A.
Numerical Modelling of the Brain Poromechanics by High-Order Discontinuous Galerkin Methods | Abstract | | We introduce and analyze a discontinuous Galerkin method for the numerical modelling of the equations of Multiple-Network Poroelastic Theory (MPET) in the dynamic formulation. The MPET model can comprehensively describe functional changes in the brain considering multiple scales of fluids. Concerning the spatial discretization, we employ a high-order discontinuous Galerkin method on polygonal and polyhedral grids and we derive stability and a priori error estimates. The temporal discretization is based on a coupling between a Newmark $beta$-method for the momentum equation and a $theta$-method for the pressure equations. After the presentation of some verification numerical tests, we perform a convergence analysis using an agglomerated mesh of a geometry of a brain slice. Finally we present a simulation in a three dimensional patient-specific brain reconstructed from magnetic resonance images. The model presented in this paper can be regarded as a preliminary attempt to model the perfusion in the brain. |
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66/2022 - 05/10/2022
Antonietti, P.F.; Liverani, L.; Pata, V.
Lack of superstable trajectories in linear viscoelasticity: A numerical approach | Abstract | | Given a positive operator $A$ on some Hilbert space,
and a nonnegative decreasing summable function $mu$,
we consider the abstract equation with memory
$$
ddot u(t)+ A u(t)- int_0^t mu(s)Au(t-s) ds=0
$$
modeling the dynamics of linearly viscoelastic solids.
The purpose of this work is to provide numerical evidence
of the fact that the energy
$$E(t)=Big(1-int_0^tmu(s)dsBig)|u(t)|^2_1+|dot u(t)|^2
+int_0^tmu(s)|u(t)-u(t-s)|^2_1ds,$$
of any nontrivial solution cannot decay faster than exponential,
no matter how fast might be the decay of the memory kernel $mu$.
This will be accomplished by simulating the integro-differential
equation for different choices of the memory kernel $mu$
and of the initial data. |
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62/2022 - 05/09/2022
Ciaramella, G.; Halpern, L.; Mechelli, L.
Convergence analysis and optimization of a Robin Schwarz waveform relaxation method for periodic parabolic optimal control problems | Abstract | | This paper is concerned with a novel convergence analysis of the
optimized Schwarz waveform relaxation method (OSWRM) for the
solution of optimal control problems governed by periodic parabolic
partial differential equations (PDEs). The new analysis is based on
Fourier-type technique applied to a semidiscrete in time form of the
optimality condition. This leads to a precise characterization of the
convergence factor of the method at the semidiscrete level. Using
this characterization, the optimal transmission condition parameter
is obtained at the semidiscrete level and its asymptotic behavior
as the time discretization converges to zero is analyzed in detail. |
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