Accelerated estimation of quantities of interest via adjoint-based model reduction

Jul 16, 2026

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Introduces an adjoint-based reduced-order model framework for fast, accurate estimation of quantities of interest in many-query linear problems. The approach constructs a reduced model with respect to the adjoint problem, avoiding repeated solution of the primal system and substantially lowering computational cost. It yields a surrogate independent of loading configurations, enabling rapid evaluation across multiple load cases and virtual charts for decision support. Numerical results on the Poisson equation and plane-stress elasticity show rapid convergence, improved accuracy over primal reduction, and practical genericity via kernel-function parameterization.

LanguageEnglish

CategoryResearch & Report

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Pages34

Reading Time1 h 26 min

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Introduction

Reduced-order modeling and related goal-oriented approaches

Focus of this work: many-query loading configurations

arXiv :2607 .04808v 1 [ cs .CE] 6 Jul 2026 Accelerated estimation of quantities of interest via adjoint-based model reduction Cl´ement Vella 1* and Serge Prudhomme2 1 Suqaba, 47 Avenue Victor Hugo, Vanves, 92170, France. 2 Department of Mathematics and Industrial Engineering, Polytechnique Montr´eal, C.P. 6079, succ. Centre-ville, Montr´eal, H3C 3A7, Qu´ebec, Canada. *Corresponding author(s). E-mail(s): [cvella@suqaba.com](cvella@suqaba.com) ; Contributing authors: [serge.prudhomme@polymtl.ca](serge.prudhomme@polymtl.ca) ; Abstract We introduce an adjoint-based reduced-order model framework for fast and accurate estimation of quantities of interest for many-query linear problems. The method builds a reduced-order model with respect to the adjoint problem, thus bypassing the solution of the primal problem and drastically reducing computational cost. It creates a surrogate model that is independent of the loading configurations. It enables fast evaluation across multiple load cases and the generation of virtual charts to support decision-making. Numerical experiments on the Poisson equation and a plane-stress elasticity problem demonstrate that theadjoint reduced-order model converges rapidly, outperforms its primal counterpart, and provides reliable estimates of the quantities of interest. Importantly, it is often more practical to parameterize a kernel function than an entire set of external loads, making the method generic and particularly suited for early-stage prototyping and design optimization. Keywords: Adjoint Problem, Quantities of Interest, Surrogate Modeling, Proper Generalized Decomposition, Virtual Charts. 1 Introduction Problems in design, control, optimization, or uncertainty quantification have substantially increased computational costs [1] . This is driven by the demand for more 1 predictive simulations, which requires accounting for variability in boundary conditions, material properties, and geometry. Reduced-Order Modeling (ROM) provides a framework to address this challenge, offering compact representations of highfidelity models to enable fast evaluations of model outputs. Among the most widely used approaches is the Proper Orthogonal Decomposition (POD), which constructslow-dimensional bases from snapshots of high-fidelity simulations. It is particularly effective when representative training data are available [2–5]. The Reduced Basis (RB) method extends the concept by incorporating error estimation and adaptive sampling, enabling certified reduction in parameterized settings [6–9] . The Proper Generalized Decomposition (PGD) follows a different paradigm: it constructs an a priori separated representation directly from the variational formulation, allowing efficient exploration of high-dimensional parameter spaces without relying on snapshots [10–12] . In many applications, one is often interested in specific quantities of interest (QoIs) of the solution rather than in the full solution field. This has motivated the development of goal-oriented error estimation techniques [13–18], which rigorously control errors with respect to targeted outputs. Coupled with ROM techniques, this perspective enables surrogate models that remain computationally efficient while being tailored to accurately predict the desired QoIs. Several goal-oriented extensions of reduced-order methods have been proposed. A goal-oriented variant of POD was presented in [19], where additional snapshots based on parametric derivatives (socalled sensitivity factors) were incorporated. This approach produces reduced-order solutions that capture parameter variations more accurately than the classical POD. Within the RB framework, goal-oriented strategies have been extensively investigated for a range of problems, including steady linear PDEs [20], unsteady linear parabolic PDEs [21, 22], and nonlinear fluid dynamics [23] . A posteriori error bounds are central to these approaches, and are specifically designed

FAQs

What results validate the method’s performance?

Numerical experiments on the Poisson equation and a plane-stress elasticity problem show that the adjoint reduced-order model converges rapidly, outperforms primal counterparts, and provides reliable quantity-of-interest estimates.

Jul 16, 2026

Upload by Noah