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 1242 prodotti
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75/2021 - 28/11/2021
Cicci, L.; Fresca, S.; Pagani, S.; Manzoni, A.; Quarteroni, A.
Projection-based reduced order models for parameterized nonlinear time-dependent problems arising in cardiac mechanics | Abstract | | The numerical simulation of several virtual scenarios arising in cardiac mechanics poses a computational challenge that can be alleviated if traditional full-order models (FOMs) are replaced by reduced order models (ROMs). For example, in the case of problems involving a vector of input parameters related, e.g., to material coefficients, projection-based ROMs provide mathematically rigorous physics-driven surrogate ROMs. In this work we demonstrate how, once trained, ROMs yield extremely accurate predictions (according to a prescribed tolerance) - yet cheaper than the ones provided by FOMs - of the structural deformation of the left ventricular tissue over an entire heartbeat, and of related output quantities of interest, such as the pressure-volume loop, for any desired input parameter values within a prescribed parameter range. However, the construction of ROM approximations for time-dependent cardiac mechanics is not straightforward, because of the highly nonlinear and multiscale nature of the problem, and almost never addressed. Our approach relies on the reduced basis method for parameterized partial differential equations. This technique performs a Galerkin projection onto a low-dimensional space for the displacement variable; the reduced space is built from a set of solution snapshots - obtained for different input parameter values and time instances - of the high-fidelity FOM, through the proper orthogonal decomposition technique. Then, suitable hyper-reduction techniques, such as the Discrete Empirical Interpolation Method, are exploited to efficiently handle nonlinear and parameter-dependent terms. In this work we show how a fast and reliable approximation of the time-dependent cardiac mechanical model can be achieved by a projection-based ROM, taking into account both passive and active mechanics for the left ventricle providing all the building blocks of the methodology, and highlighting those challenging aspects that are still open. |
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74/2021 - 28/11/2021
Orlando,G.; Barbante, P. F.; Bonaventura, L.
An efficient IMEX-DG solver for the compressible Navier-Stokes equations with a general equation of state | Abstract | | We propose an efficient, accurate and robust IMEX solver for the compressible Navier-Stokes equation with general equation of state. The method, which is based on an h-adaptive Discontinuos Galerkin spatial discretization and on an Additive Runge Kutta IMEX method for time discretization, is tailored for low Mach number applications and allows to simulate low Mach regimes at a significantly reduced computational cost, while maintaining full second order accuracy also for higher Mach number regimes. The method has been implemented in the framework of the deal.II numerical library, whose adaptive mesh refinement capabilities are employed to enhance efficiency. Refinement indicators appropriate for real gas phenomena have been introduced. A number of numerical experiments on classical benchmarks for compressible flows and their extension to real gases demonstrate the properties of the proposed method. |
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73/2021 - 28/11/2021
Marcinno, F.; Zingaro, A.; Fumagalli, I.; Dede', L.; Vergara, C.
A computational study of blood flow dynamics in the pulmonary arteries | Abstract | | In this work we study for the first time the blood dynamics in the pulmonary arteries
by means of a 3D-0D geometric multiscale approach, where a detailed 3D model for the pulmonary arteries is coupled with
a lumped parameters (0D) model of the cardiocirculatory system.
We propose to investigate two strategies for the numerical solution of the 3D-0D coupled problem: a Splitting Algorithm,
where information are exchanged between 3D and 0D models at each time step at the interfaces, and a One-Way Algorithm,
where the 0D is solved first off-line.
In our numerical experiments performed in a realistic patient-specific 3D domain with a physiologically calibrated 0D model,
we discuss first the issue on instabilities that may arise when not suitable connections are considered between 3D and 0D models;
second we compare the performance and accuracy of the two proposed numerical strategies. Finally, we report a
comparison between an healthy and an hypertensive case, providing a preliminary result highlighting how our method could
be used in future for clinical purposes. |
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72/2021 - 16/11/2021
Fresca, S.; Manzoni, A.
POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition | Abstract | | Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional reduced order models (ROMs) - built, e.g., through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized partial differential equations (PDEs). These might be related to (i) the need to deal with projections onto high dimensional linear approximating trial manifolds, (ii) expensive hyper-reduction strategies, or (iii) the intrinsic difficulty to handle physical complexity with linear superimpositions of modes. All these aspects are avoided when employing DL-ROMs, which learn in a non-intrusive way both the nonlinear trial manifold and the reduced dynamics, by relying on deep (e.g., feedforward, convolutional, autoencoder) neural networks. Although extremely efficient at testing time, when evaluating the PDE solution for any new testing-parameter instance, DL-ROMs require an expensive training stage, because of the extremely large number of network parameters to be estimated. In this paper we propose a possible way to avoid an expensive training stage of DL-ROMs, by (i) performing a prior dimensionality reduction through POD, and (ii) relying on a multi-fidelity pretraining stage, where different physical models can be efficiently combined. The proposed POD-DL-ROM is tested on several (both scalar and vector, linear and nonlinear) time-dependent parametrized PDEs (such as, e.g., linear advection-diffusion-reaction, nonlinear diffusion-reaction, nonlinear elastodynamics, and Navier-Stokes equations) to show the generality of this approach and its remarkable computational savings. |
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71/2021 - 16/11/2021
Franco, N.; Manzoni, A.; Zunino, P.
A Deep Learning approach to Reduced Order Modelling of parameter dependent Partial Differential Equations | Abstract | | Within the framework of parameter dependent PDEs, we develop
a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n-width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. In order to fully exploit the approximation capabilities of neural networks, we consider a nonlinear version of the Kolmogorov n-width over which we base the concept of a minimal latent dimension. We show that this minimal dimension is intimately related to the topological properties of the solution manifold, and we provide some theoretical results with
particular emphasis on second order elliptic PDEs. Finally, we report numerical experiments where we compare the proposed approach with classical POD-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients. |
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70/2021 - 16/11/2021
Beirao da Veiga, L.; Canuto, C.; Nochetto, R.H.; Vacca, G.; Verani, M.
Adaptive VEM: Stabilization-Free A Posteriori Error Analysis | Abstract | | In the present paper we initiate the challenging task of building a mathematically sound theory for Adaptive Virtual Element Methods (AVEMs). Among the realm of polygonal meshes, we restrict our analysis to triangular meshes with hanging nodes in 2d -- the simplest meshes with a systematic refinement procedure that preserves shape regularity and optimal complexity. A major challenge in the a posteriori error analysis of AVEMs is the presence of the stabilization term, which is of the same order as the residual-type error estimator but prevents the equivalence of the latter with the energy error.
Under the assumption that any chain of recursively created hanging nodes has uniformly bounded length, we show that the stabilization term can be made arbitrarily small relative to the error estimator provided the stabilization parameter of the scheme is sufficiently large. This quantitative estimate leads to stabilization-free upper and lower a posteriori bounds for the energy error. This novel and crucial property of VEMs hinges on the largest subspace of continuous piecewise linear functions and the delicate interplay between its coarser scales and the finer ones of the VEM space. Our results apply to H^1-conforming (lowest order) VEMs of any kind, including the classical and enhanced VEMs. |
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69/2021 - 06/11/2021
Antonietti, P.F.; Caldana, M.; Dede', L.
Accelerating Algebraic Multigrid Methods via Artificial Neural Networks | Abstract | | We present a novel Deep Learning-based algorithm to accelerate - through the use of Artificial Neural Networks (ANNs) - the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations stemming from Finite Element discretizations of Partial Differential Equations. We show that ANNs can be be successfully used to predict the strong connection parameter that enters in the construction of the sequence of increasingly smaller matrix problems standing at the basis of the AMG algorithm, so as to maximize the corresponding convergence factor of the AMG scheme. To demonstrate the practical capabilities of the proposed algorithm, which we call AMG-ANN,
we consider the iterative solution via the AMG method of the algebraic system of equations stemming from Finite Element discretizations of a two-dimensional elliptic equation with a highly heterogeneous diffusion coefficient. We train (off-line) our ANN with a rich data-set and present an in-depth analysis of the effects of tuning the strong threshold parameter on the convergence factor of the resulting AMG iterative scheme. |
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68/2021 - 03/11/2021
Regazzoni, F.; Salvador, M.; Dede', L.; Quarteroni, A.
A machine learning method for real-time numerical simulations of cardiac electromechanics | Abstract | | We propose a machine learning-based method to build a system of differential equations that approximates the dynamics of 3D electromechanical models for the human heart, accounting for the dependence on a set of parameters. Specifically, our method permits to create a reduced-order model (ROM), written as a system of Ordinary Differential Equations (ODEs) wherein the forcing term, given by the right-hand side, consists of an Artificial Neural Network (ANN), that possibly depends on a set of parameters associated with the electromechanical model to be surrogated. This method is non-intrusive, as it only requires a collection of pressure and volume transients obtained from the full-order model (FOM) of cardiac electromechanics. Once trained, the ANN-based ROM can be coupled with hemodynamic models for the blood circulation external to the heart, in the same manner as the original electromechanical model, but at a dramatically lower computational cost. Indeed, our method allows for real-time numerical simulations of the cardiac function. Our results show that the ANN-based ROM is accurate with respect to the FOM (relative error between $10^{-3}$ and $10^{-2}$ for biomarkers of clinical interest), while requiring very small training datasets (30-40 samples). We demonstrate the effectiveness of the proposed method on two relevant contexts in cardiac modeling. First, we employ the ANN-based ROM to perform a global sensitivity analysis on both the electromechanical and hemodynamic models. Second, we perform a Bayesian estimation of two parameters starting from noisy measurements of two scalar outputs. In both these cases, replacing the FOM of cardiac electromechanics with the ANN-based ROM makes it possible to perform in a few hours of computational time all the numerical simulations that would be otherwise unaffordable, because of their overwhelming computational cost, if carried out with the FOM. As a matter of fact, our ANN-based ROM is able to speedup the numerical simulations by more than three orders of magnitude. |
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