arXiv subjects

Ludovic Chamoin

Publications and source records attributed to Ludovic Chamoin.

At least 19 recordsLinked to original sources

arXiv2026

Certification of PGD reduced-order models with separated spatial variables

Model order reduction techniques have become an attractive approach for obtaining fast approximations of multidimensional problems. Besides computational efficiency, ensuring the reliability of the resulting approximations is of primary importance. This work focuses on the certification of PGD-based reduced-order models based on the separation of spatial variables, which are particularly well suited to plate and shell geometries. Considering diffusion problems defined in plate-like domains, we introduce a guaranteed global error estimate associated with the PGD approximation. To this end, the error bounds are derived from the Constitutive Relation Error (CRE) method. The main difficulty of this approach lies in the construction of equilibrated fluxes, for which a dedicated procedure is proposed. Based on the resulting estimator, an adaptive strategy is developed to control both the discretization error and the number of PGD modes. This certification procedure is further extended to the error control in quantities of interest. We provide several numerical examples illustrating the reliability and efficiency of our procedure.

math.NA↗

arXiv2026

Towards a new PGD strategy for the simulation of slender structures

Effective models for slender structures derived from well-known plate (or shell) theories are justified within the limit of a small thickness, and may therefore prove limited for intermediate slenderness. On the other hand, direct 3D simulation of such structures is sub-optimal because it does not take advantage of the presence of small dimensions in some directions and is sometimes too costly and ill-conditioned. In this context, the Proper Generalized Decomposition (PGD) method, a model order reduction method based on a modal representation of the solution with separation of variables, makes it possible to obtain a 3D solution with 2D resolution complexity. In this work, an analysis of the links between the PGD reduced order model and the solution provided by plate theory is carried out using asymptotic expansion. It is shown that, in the limit of large slenderness, the first mode of the PGD exhibits Kirchhoff-Love type kinematics, but only corresponds to the asymptotic solution in very special cases of loading and boundary conditions. To capture the asymptotic solution, a new PGD strategy is introduced consisting of computing the first two modes simultaneously. We also demonstrate that the PGD is subject to shear locking, and we show how to deal with it. Numerical experiments are provided, demonstrating the interest of this approach and confirming the theoretical analysis.

math.NA↗

arXiv2025

Hybrid twinning using PBDW and DeepONet for the effective state estimation and prediction on partially known systems

The accurate estimation of the state of complex uncertain physical systems requires reconciling theoretical models, with inherent imperfections, with noisy experimental data. In this work, we propose an effective hybrid approach that combines physics-based modeling with data-driven learning to enhance state estimation and further prediction. Our method builds upon the Parameterized Background Data-Weak (PBDW) framework, which naturally integrates a reduced-order representation of the best-available model with measurement data to account for both anticipated and unanticipated uncertainties. To address model discrepancies not captured by the reduced-order space, and learn the structure of model deviation, we incorporate a Deep Operator Network (DeepONet) constrained to be an orthogonal complement of the best-knowledge manifold. This ensures that the learned correction targets only the unknown components of model bias, preserving the interpretability and fidelity of the physical model. An optimal sensor placement strategy is also investigated to maximize information gained from measurements. We validate the proposed approach on a representative problem involving the Helmholtz equation under various sources of modeling error, including those arising from boundary conditions and source terms.

cs.LG↗

arXiv2024

Non-intrusive model combination for learning dynamical systems

In data-driven modelling of complex dynamic processes, it is often desirable to combine different classes of models to enhance performance. Examples include coupled models of different fidelities, or hybrid models based on physical knowledge and data-driven strategies. A key limitation of the broad adoption of model combination in applications is intrusiveness: training combined models typically requires significant modifications to the learning algorithm implementations, which may often be already well-developed and optimized for individual model spaces. In this work, we propose an iterative, non-intrusive methodology to combine two model spaces to learn dynamics from data. We show that this can be understood, at least in the linear setting, as finding the optimal solution in the direct sum of the two hypothesis spaces, while leveraging only the projection operators in each individual space. Hence, the proposed algorithm can be viewed as iterative projections, for which we can obtain estimates on its convergence properties. To highlight the extensive applicability of our framework, we conduct numerical experiments in various problem settings, with particular emphasis on various hybrid models based on the Koopman operator approach.

math.DS↗

arXiv2024

One-Step Early Stopping Strategy using Neural Tangent Kernel Theory and Rademacher Complexity

The early stopping strategy consists in stopping the training process of a neural network (NN) on a set $S$ of input data before training error is minimal. The advantage is that the NN then retains good generalization properties, i.e. it gives good predictions on data outside $S$, and a good estimate of the statistical error (``population loss'') is obtained. We give here an analytical estimation of the optimal stopping time involving basically the initial training error vector and the eigenvalues of the ``neural tangent kernel''. This yields an upper bound on the population loss which is well-suited to the underparameterized context (where the number of parameters is moderate compared with the number of data). Our method is illustrated on the example of an NN simulating the MPC control of a Van der Pol oscillator.

cs.LG↗

arXiv2024

Transport Map Coupling Filter for State-Parameter Estimation

Many dynamical systems are subjected to stochastic influences, such as random excitations, noise, and unmodeled behavior. Tracking the system's state and parameters based on a physical model is a common task for which filtering algorithms, such as Kalman filters and their non-linear extensions, are typically used. However, many of these filters use assumptions on the transition probabilities or the covariance model, which can lead to inaccuracies in non-linear systems. We will show the application of a stochastic coupling filter that can approximate arbitrary transition densities under non-Gaussian noise. The filter is based on transport maps, which couple the approximation densities to a user-chosen reference density, allowing for straightforward sampling and evaluation of probabilities.

eess.SP↗

arXiv2023

Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

math.NA↗

arXiv2023

MORe DWR: Space-time goal-oriented error control for incremental POD-based ROM

In this work, the dual-weighted residual (DWR) method is applied to obtain a certified incremental proper orthogonal decomposition (POD) based reduced order model. A novel approach called MORe DWR (Model Order Rduction with Dual-Weighted Residual error estimates) is being introduced. It marries tensor-product space-time reduced-order modeling with time slabbing and an incremental POD basis generation with goal-oriented error control based on dual-weighted residual estimates. The error in the goal functional is being estimated during the simulation and the POD basis is being updated if the estimate exceeds a given threshold. This allows an adaptive enrichment of the POD basis in case of unforeseen changes in the solution behavior which is of high interest in many real-world applications. Consequently, the offline phase can be skipped, the reduced-order model is being solved directly with the POD basis extracted from the solution on the first time slab and -- if necessary -- the POD basis is being enriched on-the-fly during the simulation with high-fidelity finite element solutions. Therefore, the full-order model solves can be reduced to a minimum, which is demonstrated on numerical tests for the heat equation and elastodynamics.

math.NA↗

arXiv2021

Inverse design of reconfigurable piezoelectric topological phononic plates

We present a methodology to perform inverse analysis on reconfigurable topological insulators for flexural waves in plate-like structures. First the unit cell topology of a phononic plate is designed, which offers two-fold degeneracy in the band structure by topology optimization. In the second step, piezoelectric patches bonded over the substrate plate are connected to an external circuit and used appropriately to break space inversion symmetry. The space inversion symmetry breaking opens a topological band gap by mimicking quantum valley Hall effect. Numerical simulations demonstrate that the topologically protected edge state exhibits wave propagation without backscattering and is immune to disorders. Predominantly, the proposed idea enables real-time reconfigurability of the topological interfaces in waveguide applications.

cond-mat.mes-hall↗

arXiv2021

A pedagogical review on a posteriori error estimation in Finite Element computations

This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The review mainly aims at presenting in a unified manner a large set of powerful verification methods, around the concept of equilibrium. Methods based on that concept provide error bounds that are fully computable and mathematically certified. We discuss recovery methods, residual methods, and duality-based methods for the estimation of the whole solution error (i.e. the error in energy norm), as well as goal-oriented error estimation (to assess the error on specific quantities of interest). We briefly survey the possible extensions to non-conforming numerical methods, as well as more complex (e.g. nonlinear or time-dependent) problems. We also provide some illustrating numerical examples on a linear elasticity problem in 3D.

math.NA↗

arXiv2019

Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction

In this work, we present an efficiently computational approach for designing material micro-structures by means of topology optimization. The central idea relies on using the isogeometric analysis integrated with the parameterized level set function for numerical homogenization, sensitivity calculation and optimization of the effective elastic properties. Design variables, which are level set values associated with control points, are updated from the optimizer and represent the geometry of the unit cell. We further improve the computational efficiency in each iteration by employing reduced order modeling when solving linear systems of the equilibrium equations. We construct a reduced basis by reusing computed solutions from previous optimization steps, and a much smaller linear system of equations is solved on the reduced basis. Two- and three-dimensional numerical results show the effectiveness of the topology optimization algorithm coupled with the reduced basis approach in designing metamaterials.

cs.CE↗

arXiv2019

Goal-oriented error estimation and adaptivity in MsFEM computations

We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.

math.NA↗

arXiv2019

Proper Generalized Decomposition of Parameterized Electrothermal Problems Discretized by the Finite Integration Technique

The proper generalized decomposition is applied to a static electrothermal model subject to uncertainties. A reduced model that circumvents the curse of dimensionality is obtained. The quadratic electrothermal coupling term is non-standard and requires the introduction of a trilinear form. An existing finite integration technique based solver is used to demonstrate the opportunities and difficulties in integrating the proper generalized decomposition in existing codes.

cs.CE↗

arXiv2018

A posteriori error estimation and adaptive strategy for PGD model reduction applied to parametrized linear parabolic problems

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems.

math.NA↗

arXiv2017

A posteriori error estimation and adaptive strategy for the control of MsFEM computations

We introduce quantitative and robust tools to control the numerical accuracy in simulations performed using the Multiscale Finite Element Method (MsFEM). First, we propose a guaranteed and fully computable a posteriori error estimate for the global error measured in the energy norm. It is based on dual analysis and the Constitutive Relation Error (CRE) concept, with recovery of equilibrated fluxes from the approximate MsFEM solution. Second, the estimate is split into several indicators, associated to the various MsFEM error sources, in order to drive an adaptive procedure. The overall strategy thus enables to automatically identify an appropriate trade-off between accuracy and computational cost in the MsFEM numerical simulations. Furthermore, the strategy is compatible with the offline/online paradigm of MsFEM. The performances of our approach are demonstrated on several numerical experiments.

math.NA↗

arXiv2017

On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples

Robust global/goal-oriented error estimation is used nowadays to control the approximate finite element solutions obtained from simulation. In the context of Computational Mechanics, the construction of admissible stress fields (\ie stress tensors which verify the equilibrium equations) is required to set up strict and guaranteed error bounds (using residual based error estimators) and plays an important role in the quality of the error estimates. This work focuses on the different procedures used in the calculation of admissible stress fields, which is a crucial and technically complicated point. The three main techniques that currently exist, called the element equilibration technique (EET), the star-patch equilibration technique (SPET), and the element equilibration + star-patch technique (EESPT), are investigated and compared with respect to three different criteria, namely the quality of associated error estimators, computational cost and easiness of practical implementation into commercial finite element codes. The numerical results which are presented focus on industrial problems; they highlight the main advantages and drawbacks of the different methods and show that the behavior of the three estimators, which have the same convergence rate as the exact global error, is consistent. Two- and three-dimensional experiments have been carried out in order to compare the performance and the computational cost of the three different approaches. The analysis of the results reveals that the SPET is more accurate than EET and EESPT methods, but the corresponding computational cost is higher. Overall, the numerical tests prove the interest of the hybrid method EESPT and show that it is a correct compromise between quality of the error estimate, practical implementation and computational cost.

math.NA↗

arXiv2017

New bounding techniques for goal-oriented error estimation applied to linear problems

The paper deals with the accuracy of guaranteed error bounds on outputs of interest computed from approximate methods such as the finite element method. A considerable improvement is introduced for linear problems thanks to new bounding techniques based on Saint-Venant's principle. The main breakthrough of these optimized bounding techniques is the use of properties of homothetic domains which enables to cleverly derive guaranteed and accurate boundings of contributions to the global error estimate over a local region of the domain. Performances of these techniques are illustrated through several numerical experiments.

math.NA↗

arXiv2017

An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress fields in finite element analyses

In the context of global/goal-oriented error estimation applied to computational mechanics, the need to obtain reliable and guaranteed bounds on the discretization error has motivated the use of residual error estimators. These estimators require the construction of admissible stress fields verifying the equilibrium exactly. This article focuses on a recent method, based on a flux-equilibration procedure and called the element equilibration + star-patch technique (EESPT), that provides for such stress fields. The standard version relies on a strong prolongation condition in order to calculate equilibrated tractions along finite element boundaries. Here, we propose an enhanced version, which is based on a weak prolongation condition resulting in a local minimization of the complementary energy and leads to optimal tractions in selected regions. Geometric and error estimate criteria are introduced to select the relevant zones for optimizing the tractions. We demonstrate how this optimization procedure is important and relevant to produce sharper estimators at affordable computational cost, especially when the error estimate criterion is used. Two- and three-dimensional numerical experiments demonstrate the efficiency of the improved technique.

math.NA↗