Certification of PGD Reduced-Order Models With Separated Spatial Variables

Jul 16, 2026

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Model order reduction techniques deliver fast approximations for multidimensional problems, but their reliability must be certified to support trustworthy engineering decisions. This work certifies Proper Generalized Decomposition (PGD) reduced-order models built on separated spatial variables, tailored to plate and shell geometries. For diffusion in plate-like domains, the study derives a guaranteed global error bound using Constitutive Relation Error (CRE) and proposes a dedicated equilibrated-flux construction. An adaptive strategy controls both discretization error and PGD mode count, and the approach is extended to quantities of interest, validated by numerical examples.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages32

Reading Time1 h 21 min

Introduction and objectives

Certification of PGD reduced-order models (separated spatial variables)

Guaranteed error bounds via CRE

Construction of equilibrated fluxes

Adaptive strategy for discretization and PGD modes

Error control for quantities of interest

Numerical examples

arXiv :2607 .00896v1 [math .NA] 1 Jul 2026 Certification of PGD reduced-order models with separated spatial variables Jean Ruela,∗, Ludovic Chamoinc , Frédéric Legolla,b , Arthur Lebéea a Navier, ENPC, Institut Polytechnique de Paris, Université Gustave Eiffel, CNRS, Marne-la-Vallée, France b MATHERIALS project-team, Inria, Paris, France c Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS, Gif-sur-Yvette, France Abstract 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. Keywords: Model Order Reduction, Proper Generalized Decomposition, Verification, Error estimation, Adaptivity 1. Introduction and objectives Numerical simulation has become an essential component of engineering analysis and design, and various numerical methods are available depending on the problem to be solved. Standard numerical methods can be limited when it comes to simulating multidimensional models in real time, and model order reduction techniques have thus emerged as an effective alternative over the past decades. They exploit the fact that the full-order solution of complex numerical models can often be accurately approximated by the reduced-order solution of surrogate models, so that the dimensionality can be drastically reduced. While enabling complex models to be solved in real time is an important challenge, ensuring the accuracy of the approximations obtained is equally important and can be crucial depending on the application. Among the most widely used approaches are Proper Orthogonal Decomposition (POD) and Reduced Basis (RB) methods, for which extensive work has been devoted to error estimation and certification. ∗ Corresponding author. Email addresses: [jean.ruel@enpc.fr](jean.ruel@enpc.fr) (Jean Ruel), [ludovic.chamoin@ens-paris-saclay.fr](ludovic.chamoin@ens-paris-saclay.fr) (Ludovic Chamoin), frederic.legoll@enpc.fr (Frédéric Legoll), [arthur.lebee@enpc.fr](arthur.lebee@enpc.fr) (Arthur Lebée) In this work, we focus on the Proper Generalized Decomposition (PGD) method [1, 2, 3, 4] . Initially introduced as radial loading approximation [5], it can be seen as a POD extension in which no a priori knowledge on the solution is required. The PGD method is based on a modal representation of the solution with separation of variables (also referred to as a low-rank tensor approximation), and relies on an iterative strategy in which a set of simple problems are solved. Over the past years, many works have been devoted to the offline construction of PGD reduced models with applications to a wide range of parametrized problems, each parameter (related to spatial variables, time, material behavior, geometry, boundary or initial conditions,. . . ) being seen as an extra-coordinate. Indeed, classical numerical methods based on brute force (grid-based) discretization rapidly lead to huge computational costs and storage requirements, as t

FAQs

What does the paper focus on regarding PGD reduced-order models?

It focuses on certifying PGD-based reduced-order models that use separated spatial variables, especially suited to plate and shell geometries.

How are guaranteed error bounds derived?

The paper derives error bounds using the Constitutive Relation Error (CRE) method for diffusion problems in plate-like domains.

What is the main technical difficulty and how is it addressed?

The difficulty is constructing equilibrated fluxes; the paper proposes a dedicated procedure to enable the CRE-based estimator.

How does the proposed certification procedure control accuracy in practice?

An adaptive strategy uses the estimator to control both discretization error and the number of PGD modes, and it is further extended to error control in quantities of interest.

Jul 16, 2026

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