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Journal articles

A technique for non-intrusive greedy piecewise-rational model reduction of frequency response problems over wide frequency bands

Davide Pradovera and Fabio Nobile

Published in J. Math. Ind. 12, 2022

DOI: 10.1186/s13362-021-00117-4


In the field of model order reduction for frequency response problems, the minimal rational interpolation (MRI) method has been shown to be quite effective. However, in some cases, numerical instabilities may arise when applying MRI to build a surrogate model over a large frequency range, spanning several orders of magnitude. We propose a strategy to overcome these instabilities, replacing an unstable global MRI surrogate with a union of stable local rational models. The partitioning of the frequency range into local frequency sub-ranges is performed automatically and adaptively, and is complemented by a (greedy) adaptive selection of the sampled frequencies over each sub-range. We verify the effectiveness of our proposed method with two numerical examples.

Non-intrusive double-greedy parametric model reduction by interpolation of frequency-domain rational surrogates

Fabio Nobile and Davide Pradovera

Published in ESAIM:M2AN 55, 2021

DOI: 10.1051/m2an/2021040


We propose a model order reduction approach for non-intrusive surrogate modeling of parametric dynamical systems. The reduced model over the whole parameter space is built by combining surrogates in frequency only, built at few selected values of the parameters. This, in particular, requires matching the respective poles by solving an optimization problem. If the frequency surrogates are constructed by a suitable rational interpolation strategy, frequency and parameters can both be sampled in an adaptive fashion. This, in general, yields frequency surrogates with different numbers of poles, a situation addressed by our proposed algorithm. Moreover, we explain how our method can be applied even in high-dimensional settings, by employing locally-refined sparse grids in parameter space to weaken the curse of dimensionality. Numerical examples are used to showcase the effectiveness of the method, and to highlight some of its limitations in dealing with unbalanced pole matching, as well as with a large number of parameters.

Least-Squares Padé approximation of parametric and stochastic Helmholtz maps

Francesca Bonizzoni, Fabio Nobile, Ilaria Perugia, and Davide Pradovera

Published in Adv. Comput. Math. 46, 2020

DOI: 10.1007/s10444-020-09749-3


The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Padé approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., 52(4), 1261-1284 2018, Math. Comput. 89, 1229-1257 2020). Algorithmical aspects concerning the construction of rational LS-Padé approximants are described. In particular, we show that the computation of the Padé denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Padé techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both cases, we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Padé approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods.

Interpolatory minimal rational model order reduction of parametric problems lacking uniform inf-sup stability

Davide Pradovera

Published in SIAM J. Numer. Anal. 58, 2020

DOI: 10.1137/19M1269695


We present a technique for the approximation of a class of Hilbert space-valued maps which arise within the framework of Model Order Reduction for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR strategy consists in constructing an explicit rational approximation based on few snapshots of the solution, in an interpolatory fashion. Under some restrictions on the structure of the original problem, we describe a priori convergence results for our technique, hereafter called minimal rational interpolation, which show its ability to identify the main features (e.g. resonance locations) of the target solution map. We also investigate some procedures to obtain a posteriori error indicators, which may be employed to adapt the degree and the sampling points of the minimal rational interpolant. Finally, some numerical experiments are carried out to confirm the theoretical results and the effectiveness of our technique.

Fast Least-Squares Padé approximation of problems with normal operators and meromorphic structure

Francesca Bonizzoni, Fabio Nobile, Ilaria Perugia, and Davide Pradovera

Published in Math. Comput. 89, 2020

DOI: 10.1090/mcom/3511


In this work, we consider the approximation of Hilbert space-valued meromorphic functions that arise as solution maps of parametric PDEs whose operator is the shift of an operator with normal and compact resolvent, e.g., the Helmholtz equation. In this restrictive setting, we propose a simplified version of the Least-Squares Padé approximation technique studied in (ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284) following (J. Approx. Theory 95 (1998), pp. 203-2124). In particular, the estimation of the poles of the target function reduces to a low-dimensional eigenproblem for a Gramian matrix, allowing for a robust and efficient numerical implementation (hence the "fast" in the name). Moreover, we prove several theoretical results that improve and extend those in (ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284), including the exponential decay of the error in the approximation of the poles, and the convergence in measure of the approximant to the target function. The latter result extends the classical one for scalar Padé approximation to our functional framework. We provide numerical results that confirm the improved accuracy of the proposed method with respect to the one introduced in (ESAIM Math. Model. Numer. Anal. 52 (2018), pp. 1261-1284) for differential operators with normal and compact resolvent.


Conference proceedings

Shape optimization for a noise reduction problem by non-intrusive parametric reduced modeling

Francesca Bonizzoni and Davide Pradovera

Published in Proc. WCCM-ECCOMAS2020, 2021

DOI: 10.23967/wccm-eccomas.2020.300


We study a PDE-constrained optimization problem, where the shape and liner material of the nacelle of an aircraft engine are optimized in order to minimize the noise radiated by the engine. More precisely, the acoustic problem is modeled by the Helmholtz equation with varying wavenumber $k$ on an exterior domain. A model reduction strategy is employed to alleviate the cost of the design optimization: the minimal rational interpolation technique is used to construct a surrogate (w.r.t. $k$) for the quantity of interest at fixed shape/material parameter values, and a parametric model order reduction approach is employed to combine surrogates at different shape/material designs, resulting in a non-intrusive methodology. Numerical experiments for shape and shape/material optimization are provided, to showcase the effectiveness of the presented methodology.

Frequency-domain non-intrusive greedy Model Order Reduction based on minimal rational approximation

Davide Pradovera and Fabio Nobile

Published in Sci. Comput. Electr. Eng. 36, 2021

DOI: 10.1007/978-3-030-84238-3_16


We present a technique for Model Order Reduction (MOR) of frequency-domain problems relying on rational interpolation of vector-valued functions. The selection of the sample points is carried out adaptively according to a greedy procedure. We describe several options for the choice of a posteriori error indicators, which are used to drive the greedy algorithm and define its termination condition. Namely, we illustrate a tradeoff between each estimator's accuracy and its "intrusiveness", i.e. how much information on the underlying high-fidelity model needs to be available. We test numerically the effectiveness of this technique in solving a non-Hermitian eigenproblem and a microwave frequency response analysis.

Distributed sampling for rational approximation of the acoustic scattering of an airfoil

Francesca Bonizzoni and Davide Pradovera

Published in PAMM 19, 2019

DOI: 10.1002/pamm.201900422


In this paper we compute a reduced order model for a time-harmonic external acoustic scattering problem with parametric frequency. The employed technique is minimal rational interpolation, an explicit moment-matching method for Hilbert space-valued meromorphic maps. We study the approximation and stability properties of this technique for different choices of the sample point set, namely fully distributed in the parameter range, and partially and fully confluent. The proposed technique is also compared with an implicit multi moment-matching method based on Galerkin projection.


Pending publications

Rational-based model order reduction of Helmholtz frequency response problems with adaptive finite element snapshots

Francesca Bonizzoni, Davide Pradovera, and Michele Ruggeri

Submitted, 2021



We introduce several spatially adaptive model order reduction approaches tailored to non-coercive elliptic boundary value problems, specifically, parametric-in-frequency Helmholtz problems. The offline information is computed by means of adaptive finite elements, so that each snapshot lives on a different discrete space that resolves the local singularities of the solution and is adjusted to the considered frequency value. A rational surrogate is then assembled adopting either a least-squares or an interpolatory approach, yielding the standard rational interpolation method (SRI), a vector- or function-valued version of it ($\mathcal{V}$-SRI), and the minimal rational interpolation method (MRI). In the context of building an approximation for linear or quadratic functionals of the Helmholtz solution, we perform several numerical experiments to compare the proposed methodologies. Our simulations show that, for interior resonant problems (whose singularities are encoded by poles on the real axis), the spatially adaptive $\mathcal{V}$-SRI and MRI work comparably well. Instead, when dealing with exterior scattering problems, whose frequency response is mostly smooth, the $\mathcal{V}$-SRI method seems to be the best-performing one.


Theses and project reports

Model order reduction based on functional rational approximants for parametric PDEs with meromorphic structure

PhD thesis, supervised by F. Nobile, Scientific Computing and Uncertainty Quantification, EPFL, 2021


Many engineering fields rely on frequency-domain dynamical systems for the mathematical modeling of physical (electrical/mechanical/etc.) structures. With the growing need for more accurate and reliable results, the computational burden incurred by frequency sweeps has increased too: in many practical cases, a direct frequency-response analysis over a wide range of frequencies is prohibitively expensive. In this respect, model order reduction (MOR) methods are very appealing, as they allow to replace the costly solves of the original problem with a cheap-to-evaluate surrogate model.
In this work, we describe a MOR approach, dubbed "minimal rational interpolation" (MRI), that builds a rational interpolant of the frequency response of the dynamical system. In MRI, we build a surrogate model in a data-driven fashion, starting from only few (very expensive) solves of the original problem at well-chosen frequencies. Notably, we do not need any knowledge of (nor access to) the underlying structure of the original problem, so that MRI can be described as a "non-intrusive" method. We perform a theoretical analysis of MRI, showing that it converges to the exact frequency response in a quasi-optimal way, in an "approximation theory" sense. We also describe how this approach can be complemented by adaptive sampling strategies, which, relying on \emph{a posteriori} error estimators, automatically select the "best" sampling frequencies.
Oftentimes, the underlying problem does not depend on frequency alone, but also on additional parameters, which might represent uncertain features of the physical system or design parameters that have to be optimized. This is the so-called "parametric" case, which is much more complex than the non-parametric one, especially if a modest number of parameters is involved. As a way to tackle the parametric setting, we propose a MOR approach based on marginalization: we use MRI to build local frequency surrogates at different parameter configurations, and then we combine these local surrogates to obtain a global reduced model. Several issues arise when carrying out this "combination" step. In this thesis, we propose a practical algorithm for this, relying on matching the partial fraction expansions of the local surrogates term-by-term.
Several numerical experiments are carried out as a way to showcase the effectiveness of our proposed approaches, both in the non-parametric and parametric settings. Our "case studies" are selected as simplified versions of problems of practical interest. Notably, we include examples of resonant behavior of mechanical structures with uncertain material properties, and of impedance modeling of distributed electrical circuits with a modest number of design parameters.

Randomized low-rank approximation of matrices and tensors

Master thesis, supervised by D. Kressner, Numerical Analysis and High-Performance Computing Group, EPFL, 2017


Low-rank matrix and tensor approximation is a central topic in computational linear algebra, with countless applications in scientific computing and data analysis. In this thesis, we explore the possibility to compute low-rank approximations of matrices and tensors by using randomized techniques. In particular, we provide an overview of the existing literature on the topic, and we derive several extensions to tackle two variations of the compression problem: the computation of low-rank approximations with fixed rank and with fixed precision.
This thesis shows that randomized approaches provide a very appealing alternative to deterministic techniques. In particular, in the case of matrix compression, we describe methods which have (approximately) the same complexity as the most efficient deterministic algorithms, but are more appealing because of their robustness.
Regarding tensors, this thesis focuses on the Tensor Train (TT) format, which represents a generalization of the low-rank decomposition for matrices. In particular, we derive a randomized approach which performs better than the state-of-the-art deterministic algorithm for the compression of tensors in the TT format, by exploiting the efficiency with which some operations can be carried out in this format.
Moreover, this thesis extends such procedures to the compression of Hadamard products of matrices and of tensors, which represent fundamental operations in several linear algebra algorithms. In both cases, we devise randomized techniques which outperform their deterministic counterparts. Also, we show that randomness leads to a diminished memory exploitation.
To corroborate all these claims, we provide extensive numerical experiments.

Finite elements-based Padé approximants for Helmholtz frequency response problems

Project report, supervised by F. Nobile, Scientific Computing and Uncertainty Quantification, EPFL, 2017


The focus of this project is the application of rational functions in the approximation of the solution map of a parametric Helmholtz problem with homogeneous Dirichlet boundary conditions. Such an approximation can be applied, for instance, in frequency response problems, where one wants to understand how the solution of the Helmholtz problem changes with respect to the wavenumber.
In this report it is proven that the Helmholtz equation, endowed with Dirichlet, Neumann or mixed Dirichlet-Neumann conditions, is meromorphic in $\mathbb{C}$. Moreover, the regularity of the scattering problem, i.e. the Helmholtz equation coupled with Bohr-Sommerfield radiation conditions, is discussed in some detail. The hypothesis that the solution map of such a problem may still be meromorphic is formulated after the analysis of a simple example.
A rational Padé approximant, which relies on the solution of an optimization problem, is defined for Hilbert space-valued meromorphic functions. Several original algorithms, based on slight variations of the original definition, are described in detail. Numerical tests are used to compare the different algorithms.

Implementation within Akantu of smooth contact mechanics with the mortar method

Project report, supervised by G. Anciaux, Computational Solid Mechanics Laboratory, EPFL, 2016


The scope of this project is the implementation of the mortar method for the enforcement of smooth contact constraints in an open-source finite element framework. Smooth contact constraints are rigorously defined, and two approaches to their enforcement (penalty and Lagrange multipliers) are derived rigorously. A detailed description of a complete algorithm for the application of such methods is provided, with a particular attention to the strategies used to tackle non-linearities. Then a brief presentation of the most important features of the implementation of the algorithm is given. To check the correctness of the implementation, several tests are performed: in particular the numerical results are compared with the hertzian solution for a problem of unilateral smooth contact.