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A variational multiscale approach to PDE-constrained optimization problems arising in data-driven computational mechanics.

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Title: A variational multiscale approach to PDE-constrained optimization problems arising in data-driven computational mechanics.
Authors: Codina, Ramon1,2 (AUTHOR), Ausas, Roberto F.3 (AUTHOR), Bazon, Pedro B.3 (AUTHOR), Gebhardt, Cristian G.1,4 (AUTHOR) cristian.gebhardt@uib.no
Source: Computer Methods in Applied Mechanics & Engineering. Jul2026, Vol. 456, pN.PAG-N.PAG. 1p.
Subjects: Finite element method, Reaction-diffusion equations, Multiscale modeling, Numerical analysis, Computational mechanics, Mathematical optimization
Abstract: We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous setting, we establish well-posedness of such concomitant formulations. Then, we propose stable and consistent finite element approximations for these underlying primal and dual problems relying on the Variational MultiScale framework. For quasi-uniform finite element partitions, we investigate approximations' general properties and establish well-posedness for two canonical choices of the sub-grid scales, i.e., the Algebraic Sub-Grid Scale and Orthogonal Sub-Grid Scale. Moreover, for continuous finite element functions, we are able to move back and forth between the discrete primal and dual formulations only by changing the design of the stabilization parameters. To conclude, we stress-test the proposed approximations through a series of progressively sophisticated cases, providing both a comparative and qualitative assessment of their numerical performance. [ABSTRACT FROM AUTHOR]
Copyright of Computer Methods in Applied Mechanics & Engineering is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 193055980
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
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  Data: A variational multiscale approach to PDE-constrained optimization problems arising in data-driven computational mechanics.
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  Data: <searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Reaction-diffusion+equations%22">Reaction-diffusion equations</searchLink><br /><searchLink fieldCode="DE" term="%22Multiscale+modeling%22">Multiscale modeling</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Computational+mechanics%22">Computational mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink>
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  Data: We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous setting, we establish well-posedness of such concomitant formulations. Then, we propose stable and consistent finite element approximations for these underlying primal and dual problems relying on the Variational MultiScale framework. For quasi-uniform finite element partitions, we investigate approximations' general properties and establish well-posedness for two canonical choices of the sub-grid scales, i.e., the Algebraic Sub-Grid Scale and Orthogonal Sub-Grid Scale. Moreover, for continuous finite element functions, we are able to move back and forth between the discrete primal and dual formulations only by changing the design of the stabilization parameters. To conclude, we stress-test the proposed approximations through a series of progressively sophisticated cases, providing both a comparative and qualitative assessment of their numerical performance. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computer Methods in Applied Mechanics & Engineering is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.cma.2026.118944
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      – Code: eng
        Text: English
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      – SubjectFull: Finite element method
        Type: general
      – SubjectFull: Reaction-diffusion equations
        Type: general
      – SubjectFull: Multiscale modeling
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      – SubjectFull: Numerical analysis
        Type: general
      – SubjectFull: Computational mechanics
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      – SubjectFull: Mathematical optimization
        Type: general
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      – TitleFull: A variational multiscale approach to PDE-constrained optimization problems arising in data-driven computational mechanics.
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            NameFull: Codina, Ramon
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            NameFull: Ausas, Roberto F.
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            NameFull: Bazon, Pedro B.
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            – D: 01
              M: 07
              Text: Jul2026
              Type: published
              Y: 2026
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