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 1268 products
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77/2021 - 11/28/2021
Guo, M.; Manzoni, A.; Amendt, M.; Conti, P.; Hesthaven, J.S.
Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities | Abstract | | Highly accurate numerical or physical experiments are often very time-consuming or expensive to obtain. When time or budget restrictions prohibit the generation of additional data, the amount of available samples may be too limited to provide satisfactory model results. Multi-fidelity methods deal with such problems by incorporating information from other sources, which are ideally well-correlated with the high-fidelity data, but can be obtained at a lower cost. By leveraging correlations between different data sets, multi-fidelity methods often yield superior generalization when compared to models based solely on a small amount of high fidelity data. In the current work, we present the use of artificial neural networks applied to multi- fidelity regression problems. By elaborating a few existing approaches, we propose new neural network architectures for multi-fidelity regression. The introduced models are compared against a traditional multi- fidelity regression scheme - co-kriging. A collection of artificial benchmarks are presented to measure the performance of the analyzed models. The results show that cross-validation in combination with Bayesian optimization leads to neural network models that outperform the co-kriging scheme. Additionally, we show an application of multi-fidelity regression to an engineering problem. The propagation of a pressure wave into an acoustic horn with parametrized shape and frequency is considered, and the index of reflection intensity is approximated using the proposed multi-fidelity models. A finite element, full-order model and a reduced-order model built through the reduced basis method are adopted as the high- and low-fidelity, respectively. It is shown that the multi-fidelity neural networks return outputs that achieve a comparable accuracy to those from the expensive, full-order model, using only very few full-order evaluations combined with a larger amount of inaccurate but cheap evaluations of the reduced order model. |
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76/2021 - 11/28/2021
Ponti, L.; Perotto, S.; Sangalli, L.M.
A PDE-regularized smoothing method for space-time data over manifolds with application to medical data | Abstract | | We propose an innovative statistical-numerical method to model spatio-temporal data, observed over a generic two-dimensional Riemanian manifold.
The proposed approach consists of a regression model completed with a regularizing term based on the heat equation.
The model is discretized through a finite element scheme set on the manifold, and solved by resorting to a fixed point-based iterative algorithm.
This choice leads to a procedure which is highly efficient when compared with a monolithic approach, and which allows us to deal with
massive datasets. After a preliminary assessment on simulation study cases, we investigate the performance of the new estimation tool
in practical contexts by dealing with neuroimaging and hemodynamic data. |
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75/2021 - 11/28/2021
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 - 11/28/2021
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 - 11/28/2021
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 - 11/16/2021
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 - 11/16/2021
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 - 11/16/2021
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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