arXiv subjects

Tommaso Taddei

Publications and source records attributed to Tommaso Taddei.

28 records · Page 2Linked to original sources

arXiv2021

A projection-based model reduction method for nonlinear mechanics with internal variables: application to thermo-hydro-mechanical systems

We propose a projection-based monolithic model order reduction (MOR) procedure for a class of problems in nonlinear mechanics with internal variables. The work is is motivated by applications to thermo-hydro-mechanical (THM) systems for radioactive waste disposal. THM equations model the behaviour of temperature, pore water pressure and solid displacement in the neighborhood of geological repositories, which contain radioactive waste and are responsible for a significant thermal flux towards the Earth's surface. We develop an adaptive sampling strategy based on the POD-Greedy method, and we develop an element-wise empirical quadrature hyper-reduction procedure to reduce assembling costs. We present numerical results for a two-dimensional THM system to illustrate and validate the proposed methodology.

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arXiv2021

Registration-based model reduction of parameterized two-dimensional conservation laws

We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping $Φ$ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.

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arXiv2021

Registration-based model reduction in complex two-dimensional geometries

We present a general -- i.e., independent of the underlying equation -- registration procedure for parameterized model order reduction. Given the spatial domain $Ω\subset \mathbb{R}^2$ and the manifold $\mathcal{M}= \{ u_μ : μ\in \mathcal{P} \}$ associated with the parameter domain $\mathcal{P} \subset \mathbb{R}^P$ and the parametric field $μ\mapsto u_μ \in L^2(Ω)$, our approach takes as input a set of snapshots $\{ u^k \}_{k=1}^{n_{\rm train}} \subset \mathcal{M}$ and returns a parameter-dependent bijective mapping $Φ: Ω\times \mathcal{P} \to \mathbb{R}^2$: the mapping is designed to make the mapped manifold $\{ u_μ \circ Φ_μ: \, μ\in \mathcal{P} \}$ more amenable for linear compression methods. In this work, we extend and further analyze the registration approach proposed in [Taddei, SISC, 2020]. The contributions of the present work are twofold. First, we extend the approach to deal with annular domains by introducing a suitable transformation of the coordinate system. Second, we discuss the extension to general two-dimensional geometries: towards this end, we introduce a spectral element approximation, which relies on a partition $\{ Ω_{q} \}_{q=1} ^{N_{\rm dd}}$ of the domain $Ω$ such that $Ω_1,\ldots,Ω_{N_{\rm dd}}$ are isomorphic to the unit square. We further show that our spectral element approximation can cope with parameterized geometries. We present rigorous mathematical analysis to justify our proposal; furthermore, we present numerical results for a heat-transfer problem in an annular domain, a potential flow past a rotating symmetric airfoil, and an inviscid transonic compressible flow past a non-symmetric airfoil, to demonstrate the effectiveness of our method.

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arXiv2021

A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

We present a general approach for the treatment of parameterized geometries in projection-based model order reduction. During the offline stage, given (i) a family of parameterized domains $\{ Ω_μ: μ\in \mathcal{P} \} \subset \mathbb{R}^D$ where $μ\in \mathcal{P} \subset \mathbb{R}^P$ denotes a vector of parameters, (ii) a parameterized mapping $Φ_μ$ between a reference domain $Ω$ and the parameter-dependent domain $Ω_μ$, and (iii) a finite element triangulation of $Ω$, we resort to an empirical quadrature procedure to select a subset of the elements of the grid. During the online stage, we first use the mapping to "move" the nodes of the selected elements and then we use standard element-wise residual evaluation routines to evaluate the residual and possibly its Jacobian. We discuss how to devise an online-efficient reduced-order model and we discuss the differences with the more standard "map-then-discretize" approach (e.g., Rozza, Huynh, Patera, ACME, 2007); in particular, we show how the discretize-then-map framework greatly simplifies the implementation of the reduced-order model. We apply our approach to a two-dimensional potential flow problem past a parameterized airfoil, and to the two-dimensional RANS simulations of the flow past the Ahmed body.

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arXiv2020

Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs

We propose a model reduction procedure for rapid and reliable solution of parameterized hyperbolic partial differential equations. Due to the presence of parameter-dependent shock waves and contact discontinuities, these problems are extremely challenging for traditional model reduction approaches based on linear approximation spaces. The main ingredients of the proposed approach are (i) an adaptive space-time registration-based data compression procedure to align local features in a fixed reference domain, (ii) a space-time Petrov-Galerkin (minimum residual) formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure to speed up online computations. We present numerical results for a Burgers model problem and a shallow water model problem, to empirically demonstrate the potential of the method.

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arXiv2019

A registration method for model order reduction: data compression and geometry reduction

We propose a general --- i.e., independent of the underlying equation --- registration method for parameterized Model Order Reduction. Given the spatial domain $Ω\subset \mathbb{R}^d$ and a set of snapshots $\{ u^k \}_{k=1}^{n_{\rm train}}$ over $Ω$ associated with $n_{\rm train}$ values of the model parameters $μ^1,\ldots, μ^{n_{\rm train}} \in \mathcal{P}$, the algorithm returns a parameter-dependent bijective mapping $\boldsymbolΦ: Ω\times \mathcal{P} \to \mathbb{R}^d$: the mapping is designed to make the mapped manifold $\{ u_μ \circ \boldsymbolΦ_μ: \, μ\in \mathcal{P} \}$ more suited for linear compression methods. We apply the registration procedure, in combination with a linear compression method, to devise low-dimensional representations of solution manifolds with slowly-decaying Kolmogorov $N$-widths; we also consider the application to problems in parameterized geometries. We present a theoretical result to show the mathematical rigor of the registration procedure. We further present numerical results for several two-dimensional problems, to empirically demonstrate the effectivity of our proposal.

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arXiv2019

PBDW method for state estimation: error analysis for noisy data and nonlinear formulation

We present an error analysis and further numerical investigations of the Parameterized-Background Data-Weak (PBDW) formulation to variational Data Assimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a state estimation method involving reduced models. It aims at approximating an unknown function $u^{\rm true}$ living in a high-dimensional Hilbert space from $M$ measurement observations given in the form $y_m = \ell_m(u^{\rm true}),\, m=1,\dots,M$, where $\ell_m$ are linear functionals. The method approximates $u^{\rm true}$ with $\hat{u} = \hat{z} + \hatη$. The \emph{background} $\hat{z}$ belongs to an $N$-dimensional linear space $\mathcal{Z}_N$ built from reduced modelling of a parameterized mathematical model, and the \emph{update} $\hatη$ belongs to the space $\mathcal{U}_M$ spanned by the Riesz representers of $(\ell_1,\dots, \ell_M)$. When the measurements are noisy {--- i.e., $y_m = \ell_m(u^{\rm true})+ε_m$ with $ε_m$ being a noise term --- } the classical PBDW formulation is not robust in the sense that, if $N$ increases, the reconstruction accuracy degrades. In this paper, we propose to address this issue with an extension of the classical formulation, {which consists in} searching for the background $\hat{z}$ either on the whole $\mathcal{Z}_N$ in the noise-free case, or on a well-chosen subset $\mathcal{K}_N \subset \mathcal{Z}_N$ in presence of noise. The restriction to $\mathcal{K}_N$ makes the reconstruction be nonlinear and is the key to make the algorithm significantly more robust against noise. We {further} present an \emph{a priori} error and stability analysis, and we illustrate the efficiency of the approach on several numerical examples.

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arXiv2019

An offline/online procedure for dual norm calculations of parameterized functionals: empirical quadrature and empirical test spaces

We present an offline/online computational procedure for computing the dual norm of parameterized linear functionals. The key elements of the approach are (i) an empirical test space for the manifold of Riesz elements associated with the parameterized functional, and (ii) an empirical quadrature procedure to efficiently deal with parametrically non-affine terms. We present a number of theoretical results to identify the different sources of error and to motivate the technique. Finally, we show the effectiveness of our approach to reduce both offline and online costs associated with the computation of the time-averaged residual indicator proposed in [Fick, Maday, Patera, Taddei, Journal of Computational Physics, 2018 (accepted)].

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arXiv2017

Adaptive PBDW approach to state estimation: noisy observations; user-defined update spaces

We provide a number of extensions and further interpretations of the Parameterized-Background Data-Weak (PBDW) formulation, a real-time and in-situ Data Assimilation (DA) framework for physical systems modeled by parametrized Partial Differential Equations (PDEs), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. Given $M$ noisy measurements of the state, PBDW seeks an approximation of the form $u^{\star} = z^{\star} + η^{\star}$, where the \emph{background} $z^{\star}$ belongs to a $N$-dimensional \emph{background space} informed by a parameterized mathematical model, and the \emph{update} $η^{\star}$ belongs to a $M$-dimensional \emph{update space} informed by the experimental observations. The contributions of the present work are threefold: first, we extend the adaptive formulation proposed in [T Taddei, M2AN, 51(5), 1827-1858] to general linear observation functionals, to effectively deal with noisy observations; second, we consider an user-defined choice of the update space, to improve convergence with respect to the number of measurements; third, we propose an \emph{a priori} error analysis for general linear functionals in the presence of noise, to identify the different sources of state estimation error and ultimately motivate the adaptive procedure. We present results for two synthetic model problems in Acoustics, to illustrate the elements of the methodology and to prove its effectiveness. We further present results for a synthetic problem in Fluid Mechanics to demonstrate the applicability of the approach to vector-valued fields.

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arXiv2017

A Reduced Basis Technique for Long-Time Unsteady Turbulent Flows

We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. The new contributions are threefold. First, we propose a constrained Galerkin formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the $N$-term expansion. Second, we propose an \emph{a posteriori} error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the \emph{a posteriori} error indicator developed in this paper. We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.

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