Error estimate of the non-intrusive reduced basis method with finite volume schemes

Grosjean, Elise; Maday, Yvon

doi:10.1051/m2an/2021044

Cited by 8 publications

(16 citation statements)

References 24 publications

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“…This coarse approximation is, of course, not of sufficient precision, but it is calculated much faster than the HF one. The NIRB post-processing then improves precision significantly by projecting u H (µ) on the RB in a very short runtime [8,6,13,9]. The classical NIRB approximation is given by…”

Section: Reminders On the Nirb Two-grid Methods For Stationnary Problemsmentioning

confidence: 99%

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Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

Grosjean¹,

Maday²

2022

Preprint

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Reduced Basis Methods (RBMs) are frequently proposed to approximate parametric problem solutions. They can be used to calculate solutions for a large number of parameter values (e.g. for parameter fitting) as well as to approximate a solution for a new parameter value (e.g. real time approximation with a very high accuracy). They intend to reduce the computational costs of High Fidelity (HF) codes. They necessitate well-chosen solutions, called snapshots, that have been previously computed (e.g. offline) with a HF classical method, involving, for instance a fine mesh (finite element or finite volume) and generally require a profound modification of the HF code, in order for the online computation to be performed in short (or even real) time. We will focus on the Non-Intrusive Reduced Basis (NIRB) two-grid method. Its main advantage is that it uses the HF code exclusively as a "black-box," as opposed to other so-called intrusive methods that require code modification. This is very convenient when the HF code is a commercial one that has been purchased, as is frequently the case in the industry. The effectiveness of this method relies on its decomposition into two stages, one offline (classical in most RBMs as presented above) and one online. The offline part is time-consuming but it is only performed once. On the contrary, the specificity of this NIRB approach is that, during the online part, it solves the parametric problem on a coarse mesh only and then improves its precision. As a result, it is significantly less expensive than a HF evaluation. This method has been originally developed for elliptic equations with finite elements and has since been extended to finite volume. In this paper, we extend the NIRB two-grid method to parabolic equations. We recover optimal estimates in L ∞ (0, T; H 1 (Ω)) using as a model problem, the heat equation. Then, we present numerical results on the heat equation and on the Brusselator problem.

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Section: Reminders On the Nirb Two-grid Methods For Stationnary Problemsmentioning

confidence: 99%

“…Furthermore, it has been numerically shown that for a large range of H, the rectification post-treatment allows for the recovery of the fine solution's accuracy. This two-grid method has also been generalized and analyzed in the context of finite volume schemes such as [13], in which a surrogate to Aubin-Nitsche's is used.…”

Section: Motivation and Earlier Workmentioning

confidence: 99%

Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

Grosjean¹,

Maday²

2022

Preprint

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“…In this paper, we aim at computing the sensitivities with respect to some parameters of interest µ ∈ G, with the direct and adjoint methods combined with NIRB techniques. We focus on the NIRB two-grid method [7,18,8,49] (see also different NIRB methods [6,2,16] from the two-grid method). Like most RBMs, the NIRB two-grid method relies on the assumption that the manifold of all solutions S = {u(µ), µ ∈ G} has a small Kolmogorov width [36] (in what follows, u h (µ) will refer to the HF solution for the parameter µ).…”

Section: Introductionmentioning

confidence: 99%

“…The two-grid algorithm can be employed for a variety of PDEs and is simple to implement. It has been studied with FEM in the context of elliptic equations [7] and parabolic equations [20] (see also [18] for finite volume schemes). Furthermore, because it is non-intrusive, it is suitable for a wide range of problems.…”

Section: Introductionmentioning

confidence: 99%

The non-intrusive reduced basis two-grid method applied to sensitivity analysis

Grosjean¹,

Simeon²

2023

Preprint

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This paper deals with the derivation of Non-Intrusive Reduced Basis (NIRB) techniques for sensitivity analysis, more specifically the direct and adjoint state methods. For highly complex parametric problems, these two approaches may become too costly. To reduce computational times, Proper Orthogonal Decomposition (POD) and Reduced Basis Methods (RBMs) have already been investigated. The majority of these algorithms are however intrusive in the sense that the High-Fidelity (HF) code must be modified. To address this issue, non-intrusive strategies are employed. The NIRB two-grid method uses the HF code solely as a "black-box", requiring no code modification. Like other RBMs, it is based on an offline-online decomposition. The offline stage is time-consuming, but it is only executed once, whereas the online stage is significantly less expensive than an HF evaluation. In this paper, we propose new NIRB two-grid algorithms for both the direct and adjoint state methods. On the direct method, we prove on a classical model problem, the heat equation, that HF evaluations of sensitivities reach an optimal convergence rate in L ∞ (0, T; H 1 (Ω)), and then establish that these rates are recovered by the proposed NIRB approximation. These results are supported by numerical simulations. We then numerically demonstrate that a Gaussian process regression can be used to approximate the projection coefficients of the NIRB two-grid method. This further reduces the computational costs of the online step while only computing a coarse solution of the initial problem. All numerical results are run with the model problem as well as a more complex problem, namely the Brusselator system.

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“…According to the results obtained in various techniques, the mesh-free finite volume (MFV) approach stays a pledging practice to conveniently solve diverse equations concerning the composite plates. The efficiency of the MFV has been approved in several studies (Asadi and Ataie-Ashtiani, 2021; Dong and Kang, 2022; Ejabati and Fallah, 2021, 2022; Fallah and Delzendeh, 2018; Grosjean and Maday, 2021; Jamshidi and Fallah, 2021; Lashkariani and Firoozjaee, 2021; Molaee and Shahrezaee, 2022; Pan et al, 2022). Regarding vibration control, Jamshidi and Jafari (2021) examined the vibration of piezoelectric layers.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic reliability analysis for active vibration control of piezoelectric laminated composite plates using mesh-free finite volume method

Badeleh

Fallah

2022

Journal of Intelligent Material Systems and Structures

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The implementation of deterministic design approaches in the presence of uncertainty is an adequate design task. Probabilistic methods are then used in structural design procedures, offering structural safety predictions. In this study, reliability-based analysis of a piezoelectric laminated composite plate subjected to the mechanical loads was investigated, accounting for the epistemic source of uncertainty. Material properties of the plate were considered temperature-dependent. The sophisticated mesh-free finite volume approach was employed to actively control the vibration of the temperature-dependent piezoelectric laminated composite plate through the first-order shear deformation principles. The induced vibration was treated as a non-stationary random process, and the generalized failure index was estimated using the first-passage probability concept by adopting the novel asymptotic sampling technique. The veracity of the suggested model was corroborated by alternate approaches in the literature, comprising the finite element approach. Furthermore, the temperature-dependent piezoelectric laminated composite plate with different ply orientations was considered to assess the safety index variation against changes in the ambient temperature and the laminate stacking orientation. The obtained results revealed that with increasing temperature, the reliability of the composite plate reduced. It was also realized that the reliability index is directly correlated with the piezoelectric voltage.

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Error estimate of the non-intrusive reduced basis method with finite volume schemes

Cited by 8 publications

References 24 publications

Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

The non-intrusive reduced basis two-grid method applied to sensitivity analysis

Probabilistic reliability analysis for active vibration control of piezoelectric laminated composite plates using mesh-free finite volume method

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