arXiv subjects

Yvon Maday

Publications and source records attributed to Yvon Maday.

At least 37 records · Page 2Linked to original sources

arXiv2022

Projection based semi--implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid--Structure Interaction problems

We present a partitioned Model Order Reduction method for multiphysics problems, that is based on a semi-implicit treatment of the coupling conditions, and on a projection scheme. The proposed Reduced Order Method is based on the Proper Orthogonal Decomposition and on a Galerkin projection onto the reduced basis spaces; we aim of addressing both time-dependent and time-dependent, parametrized Fluid-Structure Interaction problems, where the fluid is incompressible and the structure is linear, elastic and two dimensional.

math.NA↗

arXiv2022

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined $hp$ discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined $hp$ space the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.

math.NA↗

arXiv2022

A Kermack-McKendrick model with age of infection starting from a single or multiple cohorts of infected patients

During an epidemic, the infectiousness of infected individuals is known to depend on the time since the individual was infected, that is called the age of infection. Here we study the parameter identifiability of the Kermack-McKendrick model with age of infection which takes into account this dependency. By considering a single cohort of individuals, we show that the daily reproduction number can be obtained by solving a Volterra integral equation that depends on the flow of new infected individuals. We test the consistency our the method by generating data from deterministic and stochastic numerical simulations. Finally we apply our method to a dataset from SARS-CoV-1 with detailed information on a single cluster of patients. We stress the necessity of taking into account the initial data in the analysis to ensure the identifiability of the problem.

math.AP↗

arXiv2022

A flexible smoother adapted to censored data with outliers and its application to SARS-CoV-2 monitoring in wastewater

A sentinel network, Obépine, has been designed to monitor SARS-CoV-2 viral load in wastewaters arriving at wastewater treatment plants (WWTPs) in France as an indirect macro-epidemiological parameter. The sources of uncertainty in such monitoring system are numerous and the concentration measurements it provides are left-censored and contain outliers, which biases the results of usual smoothing methods. Hence the need for an adapted pre-processing in order to evaluate the real daily amount of virus arriving to each WWTP. We propose a method based on an auto-regressive model adapted to censored data with outliers. Inference and prediction are produced via a discretised smoother which makes it a very flexible tool. This method is both validated on simulations and on real data from Obépine. The resulting smoothed signal shows a good correlation with other epidemiological indicators and is currently used by Obépine to provide an estimate of virus circulation over the watersheds corresponding to about 200 WWTPs.

stat.AP↗

arXiv2021

FLEIM: A stable, accurate and robust extrapolation method at infinity for computing the ground state of electronic Hamiltonians

The Kohn-Sham method uses a single model system, and corrects it by a density functional the exact user friendly expression of which is not known and is replaced by an approximated, usable, model. We propose to use instead more than one model system, and use a greedy extrapolation method to correct the results of the model systems. Evidently, there is a higher price to pay for it. However, there are also gains: within the same paradigm, e.g., excited states and physical properties can be obtained.

physics.chem-ph↗

arXiv2021

Error estimate of the Non Intrusive Reduced Basis method with finite volume schemes

The context of this paper is the simulation of parameter-dependent partial differential equations (PDEs). When the aim is to solve such PDEs for a large number of parameter values, Reduced Basis Methods (RBM) are often used to reduce computational costs of a classical high fidelity code based on Finite Element Method (FEM), Finite Volume (FVM) or Spectral methods. The efficient implementation of most of these RBM requires to modify this high fidelity code, which cannot be done, for example in an industrial context if the high fidelity code is only accessible as a "black-box" solver. The Non Intrusive Reduced Basis method (NIRB) has been introduced in the context of finite elements as a good alternative to reduce the implementation costs of these parameter-dependent problems. The method is efficient in other contexts than the FEM one, like with finite volume schemes, which are more often used in an industrial environment. In this case, some adaptations need to be done as the degrees of freedom in FV methods have different meenings. At this time, error estimates have only been studied with FEM solvers. In this paper, we present a generalisation of the NIRB method to Finite Volume schemes and we show that estimates established for FEM solvers also hold in the FVM setting. We first prove our results for the hybrid-Mimetic Finite Difference method (hMFD), which is part the Hybrid Mixed Mimetic methods (HMM) family. Then, we explain how these results apply more generally to other FV schemes. Some of them are specified, such as the Two Point Flux Approximation (TPFA).

math.NA↗

arXiv2021

An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation

The need for multiple interactive, real-time simulations using different parameter values has driven the design of fast numerical algorithms with certifiable accuracies. The reduced basis method (RBM) presents itself as such an option. RBM features a mathematically rigorous error estimator which drives the construction of a low-dimensional subspace. A surrogate solution is then sought in this low-dimensional space approximating the parameter-induced high fidelity solution manifold. However when the system is nonlinear or its parameter dependence nonaffine, this efficiency gain degrades tremendously, an inherent drawback of the application of the empirical interpolation method (EIM). In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear partial differential equations on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation. Two critical ingredients of the scheme are collocation at about twice as many locations as the number of basis elements for the reduced approximation space, and an efficient error indicator for the strategic building of the reduced solution space. The latter, the main contribution of this paper, results from an adaptive hyper reduction of the residuals for the reduced solution. Together, these two ingredients render the proposed R2-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in traditional RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC and its superior stability performance.

math.NA↗

arXiv2020

Epidemiological Forecasting with Model Reduction of Compartmental Models. Application to the COVID-19 pandemic

We propose a forecasting method for predicting epidemiological health series on a two-week horizon at the regional and interregional resolution. The approach is based on model order reduction of parametric compartmental models, and is designed to accommodate small amount of sanitary data. The efficiency of the method is shown in the case of the prediction of the number of infected and removed people during the two pandemic waves of COVID-19 in France, which have taken place approximately between February and November 2020. Numerical results illustrate the promising potential of the approach.

stat.ME↗

arXiv2020

Weighted analyticity of Hartree-Fock eigenfunctions

We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical solution of such eigenvalue problems.

math.AP↗

arXiv2020

Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters

This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schr{ö}dinger operator with periodic boundary conditions of the form $--$Δ$ + V$ discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

math.NA↗

arXiv2019

Overcoming slowly decaying Kolmogorov n-width by transport maps: application to model order reduction of fluid dynamics and fluid--structure interaction problems

In this work we focus on reduced order modelling for problems for which the resulting reduced basis spaces show a slow decay of the Kolmogorov $n$-width, or, in practical calculations, its computational surrogate given by the magnitude of the eigenvalues returned by a proper orthogonal decomposition on the solution manifold. In particular, we employ an additional preprocessing during the offline phase of the reduced basis method, in order to obtain smaller reduced basis spaces. Such preprocessing is based on the composition of the snapshots with a transport map, that is a family of smooth and invertible mappings that map the physical domain of the problem into itself. Two test cases are considered: a fluid moving in a domain with deforming walls, and a fluid past a rotating cylinder. Comparison between the results of the novel offline stage and the standard one is presented.

math.NA↗

arXiv2019

L1-ROC and R2-ROC: L1- and R2-based Reduced Over-Collocation methods for parametrized nonlinear partial differential equations

The onerous task of repeatedly resolving certain parametrized partial differential equations (pPDEs) in, e.g. the optimization context, makes it imperative to design vastly more efficient numerical solvers without sacrificing any accuracy. The reduced basis method (RBM) presents itself as such an option. With a mathematically rigorous error estimator, RBM seeks a surrogate solution in a carefully-built subspace of the parameter-induced high fidelity solution manifold. It can improve efficiency by several orders of magnitudes leveraging an offline-online decomposition procedure. However, this decomposition, usually through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrading to the online efficiency. In this paper, we augment and extend the EIM approach in the context of solving pPDEs in two different ways, resulting in the Reduced Over-Collocation methods (ROC). These are stable and capable of avoiding the efficiency degradation inherent to a direct application of EIM. There are two ingredients of these methods. First is a strategy to collocate at about twice as many locations as the number of bases for the surrogate space. The second is an efficient approach for the strategic selection of the parameter values to build the reduced solution space for which we study two choices, a recent empirical L1 approach and a new indicator based on the reduced residual. Together, these two ingredients render the schemes, L1-ROC and R2-ROC, online efficient and immune from the efficiency degradation of EIM for nonlinear and nonaffine problems offline and online. Numerical tests on three different families of nonlinear problems demonstrate the high efficiency and accuracy of these new algorithms and their superior stability performance.

math.NA↗

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.

math.NA↗

arXiv2018

A Domain Decomposition Method for the Poisson-Boltzmann Solvation Models

In this paper, a domain decomposition method for the Poisson-Boltzmann (PB) solvation model that is widely used in computational chemistry is proposed. This method, called ddLPB for short, solves the linear Poisson-Boltzmann (LPB) equation defined in $\mathbb R^3$ using the van der Waals cavity as the solute cavity. The Schwarz domain decomposition method is used to formulate local problems by decomposing the cavity into overlapping balls and only solving a set of coupled sub-equations in balls. A series of numerical experiments is presented to test the robustness and the efficiency of this method including the comparisons with some existing methods. We observe exponential convergence of the solvation energy with respect to the number of degrees of freedom which allows this method to reach the required level of accuracy when coupling with quantum mechanical descriptions of the solute.

math.NA↗

arXiv2018

Regularity and $hp$ discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyse their approximation via a discontinuous Galerkin (dG) $hp$ finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non homogeneous weighted Sobolev spaces. Using this result, we prove that the an isotropically graded $hp$ dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the the behaviour of the method for varying discretisation parameters.

math.NA↗

arXiv2018

A coherent derivation of the Ewald summation for arbitrary orders of multipoles: The self-terms

In this work, we provide the mathematical elements we think essential for a proper understanding of the calculus of the electrostatic energy of point-multipoles of arbitrary order under periodic boundary conditions. The emphasis is put on the expressions of the so-called self parts of the \Ewald\, summation where different expressions can be found in literature. Indeed, such expressions are of prime importance in the context of new generation polarizable force field where the self field appears in the polarization equations. We provide a general framework, where the idea of the \Ewald\ splitting is applied to the electric potential and subsequently, all other quantities such as the electric field, the energy and the forces are derived consistently thereof. Mathematical well-posedness is shown for all these contributions for any order of multipolar distribution.

physics.chem-ph↗

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.

math.NA↗

arXiv2017

Reduced-order semi-implicit schemes for fluid-structure interaction problems

POD--Galerkin reduced-order models (ROMs) for fluid-structure interaction problems (incompressible fluid and thin structure) are proposed in this paper. Both the high-fidelity and reduced-order methods are based on a Chorin-Temam operator-splitting approach. Two different reduced-order methods are proposed, which differ on velocity continuity condition, imposed weakly or strongly, respectively. The resulting ROMs are tested and compared on a representative haemodynamics test case characterized by wave propagation, in order to assess the capabilities of the proposed strategies.

math.NA↗