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 1288 prodotti
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53/2024 - 12/08/2024
Caldana, M.; Hesthaven, J. S.
Neural ordinary differential equations for model order reduction of stiff systems | Abstract | | Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying neural ODEs in practical applications often encounters the challenge of stiffness, a condition where rapid variations in some components of the solution demand prohibitively small time steps for explicit solvers. This work addresses the stiffness issue when employing neural ODEs for model order reduction by introducing a suitable reparametrization in time. The considered map is data-driven and it is induced by the adaptive time-stepping of an implicit solver on a reference solution. We show the map produces a nonstiff system that can be cheaply solved with an explicit time integration scheme. The original, stiff, time dynamic is recovered by means of a map learnt by a neural network that connects the state space to the time reparametrization. We validate our method through extensive experiments, demonstrating improvements in efficiency for the neural ODE inference while maintaining robustness and accuracy. The neural network model also showcases good generalization properties for times beyond the training data. |
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52/2024 - 02/08/2024
Botti, M.; Mascotto, L.
A Necas-Lions inequality with symmetric gradients on star-shaped domains based on a first order Babuska-Aziz inequality | Abstract | | We prove a Necas-Lions inequality with symmetric gradients on two and three dimensional domains that are star-shaped with respect to a ball B; the constants in the inequality are explicit with respect to the diameter and the radius of B. Crucial tools in deriving this inequality are a first order Babuska-Aziz inequality based on Bogovskii's construction of a right-inverse of the divergence and Fourier transform techniques proposed by Duran. As a byproduct, we derive arbitrary order estimates in arbitrary dimension for that operator. |
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51/2024 - 23/07/2024
Antonietti, P.F.; Corti, M.; Lorenzon, G.
A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins’ dynamics | Abstract | | Neurocognitive disorders, such as Alzheimer's and Parkinson's, have a wide social impact. These proteinopathies involve misfolded proteins accumulating into neurotoxic aggregates. Mathematical and computational models describing the prion-like dynamics offer an analytical basis to study the diseases' evolution and a computational framework for exploring potential therapies. This work focuses on the heterodimer model in a three-dimensional setting, a reactive-diffusive system of nonlinear partial differential equations describing the evolution of both healthy and misfolded proteins. We investigate traveling wave solutions and diffusion-driven instabilities as a mechanism of neurotoxic pattern formation. For the considered mathematical model, we propose a space discretization, relying on the Discontinuous Galerkin method on polytopal/polyhedral grids, allowing high-order accuracy and flexible handling of the complicated brain’s geometry. Further, we present a-priori error estimates for the semi-discrete formulation and we perform convergence tests to verify the theoretical results. Finally, we conduct simulations using realistic data on a three-dimensional brain mesh reconstructed from medical images. |
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50/2024 - 13/07/2024
Fumagalli, I.; Parolini, N.; Verani, M.
A posteriori error analysis for a coupled Stokes-poroelastic system with multiple compartments | Abstract | | The computational effort entailed in the discretization of fluid-poromechanics systems is typically highly demanding. This is particularly true for models of multiphysics flows in the brain, due to the geometrical complexity of the cerebral anatomy - requiring a very fine computational mesh for finite element discretization - and to the high number of variables involved. Indeed, this kind of problems can be modeled by a coupled system encompassing the Stokes equations for the cerebrospinal fluid in the brain ventricles and Multiple-network Poro-Elasticity (MPE) equations describing the brain tissue, the interstitial fluid, and the blood vascular networks at different space scales. The present work aims to rigorously derive a-posteriori error estimates for the coupled Stokes-MPE problem, as a first step towards the design of adaptive refinement strategies or reduced order models to decrease the computational demand of the problem. Through numerical experiments, we verify the reliability and optimal efficiency of the proposed a-posteriori estimator and identify the role of the different solution variables in its composition. |
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49/2024 - 06/07/2024
Ballini, E.; Formaggia, L.; Fumagalli, A.; Keilegavlen, E.; Scotti, A.
A hybrid upwind scheme for two-phase flow in fractured porous media | Abstract | | Simulating the flow of two fluid phases in porous media is a challenging task, especially when fractures are included in the simulation. Fractures may have highly heterogeneous properties compared to the surrounding rock matrix, significantly affecting fluid flow, and at the same time hydraulic aperture that are much smaller than any other characteristic sizes in the domain. Generally, flow simulators face difficulties with counter-current flow, generated by gravity and pressure gradients, which hinders the convergence of non-linear solvers (Newton).
In this work, we model the fracture geometry with a mixed-dimensional discrete fracture network, thus lightening the computational burden associated to an equi-dimensional representation. We address the issue of counter-current flows with appropriate spatial discretization of the advective fluid fluxes, with the aim of improving the convergence speed of the non-linear solver. In particular, the extension of the hybrid upwinding to the mixed-dimensional framework, with the use of a phase potential upstreaming at the interfaces of subdomains.
We test the method across several cases with different flow regimes and fracture network geometry. Results show robustness of the chosen discretization and a consistent improvements, in terms of Newton iterations, compared to use the phase potential upstreaming everywhere.
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48/2024 - 04/07/2024
Cicalese, G.; Ciaramella, G.; Mazzieri, I.
Addressing Atmospheric Absorption in Adaptive Rectangular Decomposition | Abstract | | This paper focuses on the Adaptive Rectangular Decomposition (ARD) scheme, a wave-based method utilized for acoustic simulations. ARD holds promise for diverse applications. In architectural design, it can forecast acoustical parameters, facilitating the creation of spaces with superior sound quality. Moreover, in the domain of Acoustic Virtual Reality, ARD can offer users a more immersive and lifelike acoustic environment. Our enhancement proves advantageous for all these applications, enabling the simulation of larger environments with heightened precision. Despite its notable efficiency, ARD faces a significant drawback: the absence of atmospheric absorption modeling. The principal aim of this study is to rectify this limitation, thereby augmenting the capabilities of the ARD algorithm. |
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47/2024 - 24/06/2024
Franco, N.R.; Brugiapaglia, S.
A practical existence theorem for reduced order models based on convolutional autoencoders | Abstract | | In recent years, deep learning has gained increasing popularity in
the fields of Partial Differential Equations (PDEs) and Reduced Order Modeling (ROM), providing domain practitioners with new powerful data-driven techniques such as Physics-Informed Neural Networks (PINNs), Neural Operators, Deep Operator Networks (DeepONets) and Deep-Learning based ROMs (DL-ROMs). In this context, deep autoencoders based on Convolutional Neural Networks (CNNs) have proven extremely effective, outperforming established techniques, such as the reduced basis method, when dealing with complex nonlinear problems. However, despite the empirical success of CNN-based autoencoders, there are only a few theoretical results supporting these architectures, usually stated in the form of universal approximation theorems. In particular, although the existing literature provides users with guidelines for designing convolutional autoencoders, the subsequent challenge of learning the latent features has been barely investigated. Furthermore, many practical questions remain unanswered, e.g., the number of snapshots needed for convergence or the neural network training strategy. In this work, using recent techniques from sparse high-dimensional function approximation, we fill some of these gaps by providing a new practical existence theorem for CNN-based autoencoders when the parameter-to-solution map is holomorphic. This regularity assumption arises in many relevant classes of parametric PDEs, such as the parametric diffusion equation, for which we discuss an explicit application of our general theory. |
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45/2024 - 17/06/2024
Fumagalli, A.; Patacchini, F. S.
Numerical validation of an adaptive model for the determination of nonlinear-flow regions in highly heterogeneos porous media | Abstract | | An adaptive model for the description of flows in highly heterogeneous porous media is developed in [13,14]. There, depending on the magnitude of the fluid's velocity, the constitutive law linking velocity and pressure gradient is selected between two possible options, one better adapted to slow motion and the other to fast motion. We propose here to validate further this adaptive approach by means of more extensive numerical experiments, including a three-dimensional case, as well as to use such approach to determine a partition of the domain into slow- and fast-flow regions. |
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