Pith. sign in

REVIEW 3 major objections 5 minor 49 references

A hyperreduced reduced basis element method for reduced-order modeling of component-based nonlinear systems

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper develops a hyperreduced reduced-basis-element method for component-based nonlinear systems that uses an online adaptive, Brezzi-Rappaz-Raviart-informed procedure to select per-component hyperreduction fidelity and guarantee a…

desk verdict A well-executed component-based ROM paper with a genuine new method; the main caveat is that the 'any configuration' guarantee in the abstract is conditional on training representativity and unverified Lipschitz assumptions, so the abstract slightly oversells. read the letter →

arxiv 2501.01621 v1 pith:F6H5OB2R submitted 2025-01-03 math.NA cs.NAphysics.comp-ph

Mehran Ebrahimi , Masayuki Yano This is my paper · ORCID
classification math.NAcs.NAphysics.comp-ph MSC 65N3065N55
keywords reducedbasiselementmethodhyperreductioncomponent-basedmodelreductionempiricalquadratureprocedureBrezzi-Rappaz-RaviarterrorestimateparameterizednonlinearPDEsdomaindecompositionthermalfinproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a reduced-order modeling method for component-based nonlinear systems that are too large or too topology-rich to train as a single model. The idea is to pretrain, once, a library of archetype components, each equipped with a reduced basis and a family of reduced quadrature rules of different fidelities, and then assemble a reduced model online for the particular system configuration by gluing instantiated components together. The central claim is that the hyperreduction error of the assembled system can be controlled quantitatively: an online adaptive procedure informed by the Brezzi-Rappaz-Raviart theorem selects each component's quadrature fidelity so that the error between the reduced and hyperreduced solutions stays within a user-prescribed tolerance in the solution norm, for any topological and parametric configuration that can be assembled from the library. If correct, this removes the need to retrain for each new configuration and makes many-query studies of nonlinear systems with many continuous and topology-varying parameters practical. The method is demonstrated on a nonlinear thermal fin system with up to 225 components and 68 independent parameters, where the adaptive procedure meets a 1% relative error target with about 40x speedup.

What carries the argument

The load-bearing object is the Brezzi-Rappaz-Raviart (BRR) theorem, a nonlinear stability estimate that bounds the distance from a trial point to the nearby zero of a nonlinear map by the map's residual norm divided by the Jacobian's least singular value. The paper specializes it to component-assembled nonlinear systems in Proposition 6, yielding $\|u_{rb}(\mu)-\tilde u_{rb}(\mu)\|_2 \le \bar{\alpha} + \bar{\varepsilon}$ with $\bar{\alpha} = 2 \sum_{c\in C}\sqrt{N_{M(c)}}\,\delta_{R_c} / (\sigma - \sum_{c\in C} N_{M(c)}\,\delta_{J_c})$, and Corollary 7 converts this to the $V$-norm through $\sqrt{\lambda_{\max}}$. Around this bound the method organizes two other pieces: a component-wise LP empirical quadrature procedure that builds, for each archetype component, a family of sparse reduced quadrature rules indexed by hyperreduction tolerances $\delta_{b_c}$, and an online adaptive selection loop (Algorithm 3) that, using $\sigma \approx \sigma_{\min}(\tilde J_{rb}(\tilde u_{rb}))/2$, walks the component fidelities until the BRR condition fits the user tolerance.

What would settle it

Assemble hold-out thermal fin systems drawn from a distribution different from the training one (for instance, port-connection probability $\beta \neq 0.8$ or boundary conditions outside the $[1,250]$ K training range), run Algorithm 3 with a prescribed relative tolerance $\epsilon=0.01$, and compute $\sup_{\mu \in \Xi_{\text{test}}} \|u_{rb}(\mu)-\tilde u_{rb}(\mu)\|_V / \|u_{rb}(\mu)\|_V$. If any such configuration exceeds the tolerance, or if the ratio to the Corollary 7 bound exceeds one, then the representativity assumption or the assumed component-wise bounds fails and the guarantee is not unconditional.

Watch

Extended reading notes

Core claim

This work aims to establish that global nonlinearities, online-interchangeable components, and quantitative system-level error control can be combined in one reduced-order method. Previous component-based methods either assumed affine or localizable nonlinearities or lacked a way to control hyperreduction error at the system level. The paper's mechanism is a component-wise extension of LP empirical quadrature: each archetype component is trained offline with several hyperreduction tolerances, and online an adaptive loop (Algorithm 3) chooses, per component, the coarsest reduced quadrature rule such that the Brezzi-Rappaz-Raviart-based bound $\|u_{rb}(\mu) - \tilde u_{rb}(\mu)\|_V \leq (\bar{\alpha} + \bar{\varepsilon})\sqrt{\lambda_{\max}}$ of Proposition 6 and Corollary 7 guarantees the user's tolerance. The component-wise residual and Jacobian tolerances $\delta_{R_c}$ and $\delta_{J_c}$ enter $\bar{\alpha}$, while $\sigma$, the minimum singular value of the assembled RB Jacobian, is approximated online by half the minimum singular value of the hyperreduced Jacobian. On nonlinear thermal fin systems, the paper reports the adaptive procedure meeting a 1% relative error target, with effectivity between about 1.3 and 33.6 depending on system size.

Load-bearing premise

The guarantee rests on the training sample of 100 random subsystems being representative of every system a component will appear in, and on the per-component residual and Jacobian tolerance bounds (15)-(16), together with the Lipschitz condition (18), actually holding for the unseen online configuration; none of these are verified by the online algorithm.

Editorial extensions

If this is right

  • A library is trained once and then reused for many topologies: the online solve time and storage scale with the number of archetypes and instantiated components, not with the truth finite-element mesh size or quadrature count.
  • The user can prescribe an absolute or relative error tolerance on the hyperreduction gap, and the adaptive procedure returns a per-component quadrature fidelity that meets it; in the reported tests it converged in two iterations.
  • Because training is component-wise, no global snapshots of a large assembled system are needed offline, so the approach extends reduced-order modeling to systems too large or too many-parametered for monolithic training.
  • The speedup is limited by the port degrees of freedom, which are not reduced here; the authors identify port reduction as the natural extension to reach the larger speedups reported for linear port-reduced methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A stress test for the method is to assemble hold-out configurations with port-connection statistics that differ from the $\beta=0.8$ training process and check whether the component-wise residual and Jacobian bounds (15)-(16) still hold; the paper's guarantee depends on this representativity rather than on any online verification.
  • The same BRR-guided fidelity selection could be composed with port reduction or with online-adaptive RB selection, which the paper explicitly leaves to future work; the bounds are stated for the assembled system, so they should carry over if the port or RB spaces are enriched adaptively.
  • Because $\bar{\alpha}$ accumulates component tolerances linearly while $\sigma$ is a single global quantity, the adaptive scheme could be made more efficient by allocating tighter tolerances to components with large $N_{M(c)}$ or large residual rather than using a common per-archetype $\delta_c$ as in the numerical study.
  • For convection-dominated problems the BRR bound can become conservative, so a testable prediction is that effectivity of Algorithm 3 worsens as convective transport increases; the authors note this limitation for high-Reynolds-number flows.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces a hyperreduced reduced basis element (HRBE) method for component-based nonlinear parametrized PDEs. Offline, the method builds a library of archetype components, each equipped with a component-wise reduced basis and a family of hyperreduced quadrature rules of varying fidelity, constructed by an extension of the empirical quadrature procedure (EQP) to the component-based setting. Online, an adaptive procedure informed by the Brezzi-Rappaz-Raviart (BRR) theorem selects the hyperreduction fidelity for each component so that the hyperreduction error in the assembled system is claimed to meet a user-prescribed tolerance. The method is demonstrated on two-dimensional nonlinear thermal fin systems with up to 225 components and 68 independent parameters, reporting accuracy and speedups. The central theoretical result is Proposition 6 and Corollary 7, which bound the difference between the RB and HRBE solutions in terms of component-wise residual and Jacobian tolerances and a Lipschitz condition.

Significance. If the claimed guarantee holds, the paper makes a valuable contribution: it extends component-based reduced basis methods from linear or locally nonlinear settings to globally nonlinear problems, while providing a quantitative, system-level mechanism for controlling hyperreduction error. The derivation of the BRR-based error estimate in Proposition 6 is careful and the numerical validation is thorough, including effectivity tables, parameter sweeps, and scaling studies across system sizes. The component-wise EQP training is a natural and useful extension of prior work, and the reported speedups are meaningful even without port reduction. The main gap is between the conditional theory and the unverified online algorithm; the numerical evidence is strong but does not by itself establish the 'for any topological and parametric configuration' guarantee stated in the abstract.

major comments (3)
  1. [Section 5.2, Algorithm 3 and Proposition 6] Algorithm 3's stopping criterion does not verify the hypotheses of Proposition 6 at the online configuration. Conditions (15) and (16) are enforced during offline training only on the training snapshots through constraints (27)–(28), and the online component-wise residual and Jacobian deviations at eurb(µ) are never computed. Condition (18) on the Lipschitz constant L(α) is not checked at all. Moreover, the algorithm applies Proposition 6 with ¯urb = urb (as stated in Section 5.2), but urb is not available online; only eurb is. Consequently, the algorithm can terminate while the assumptions of Proposition 6 are false, so the bound (22) is not certified. The numerical results in Table 5 demonstrate that the target is met on the tested configurations, but they do not support the unconditional claim in the abstract and in contribution 4 that the error tolerance is met 'for any topological and parametric configuration.'
  2. [Section 5.2, Eq. (29) and Lemma 10] The replacement of σmin(Jrb(urb(µ); µ)) by σmin(eJrb(eurb(µ); µ))/2 is heuristic. The factor 1/2 relies on inequality (29), which follows from the BRR theorem only if the BRR hypotheses hold for G = Rrb and v = eurb(µ); these hypotheses are not verified by Algorithm 3. Lemma 10 bounds the difference between σmin(Jrb(eurb)) and σmin(eJrb(eurb)), but the algorithm requires control at urb(µ), not merely at eurb(µ). The numerical evidence in Tables 4 and 6 supports the approximation empirically for the cases tested, but it does not establish the 'for any configuration' guarantee claimed in the paper.
  3. [Section 4.1 and abstract] The guarantee for arbitrary topological and parametric configurations rests on the 'fundamental assumption' stated in Section 4.1 that the randomly generated training subsystems in Algorithm 1 (with Nsample = 100 and β = 0.8) sufficiently represent all potential online configurations. The paper acknowledges this assumption explicitly in Section 4.1, but the abstract and contribution 4 present the guarantee without this caveat. The authors should either weaken the claims to conditional guarantees or provide an additional mechanism for verifying or estimating the representativity and the BRR conditions online; as written, the stated certainty is not supported.
minor comments (5)
  1. [Section 3.2, Corollary 7 and Eq. (21)] In the definitions of λmin and λmax in (21), the infimum and supremum are taken over V, but the coordinate norm ∥v∥2 is only defined for functions whose generalized coordinates are available, i.e., functions in Vrb (or possibly Vh). This should be clarified, for instance by stating that the constants are computed over the truth space Vh and hence provide conservative bounds for the RB subspace.
  2. [Section 4.4, proof of Proposition 6] In inequality (20), the term δJ_{M(c)} should be δJc for consistency with the hypotheses of Proposition 6, since δJc is defined for each instantiated component c ∈ C; the same notational slip appears in the definition of ε just below the display following (20).
  3. [Section 5.2, Algorithm 3] The initial hyperreduction tolerances δc in Line 3 of Algorithm 3 are 'selected' without guidance. A sentence or small remark on how to choose these values (e.g., starting from the coarsest available tolerance and refining) would improve reproducibility.
  4. [Section 6.5, Table 5] Table 5 reports the maximum relative errors and the text gives effectivities from 1.315 to 33.602, but the table itself has no effectivity column. Adding such a column would help the reader assess the sharpness of the error bound across system sizes.
  5. [Throughout] The notation for the reduced quadrature rule, rendered as eQr and eQf, is unusual and slightly awkward; a tilde-based notation (e.g., \(\widetilde{Q}^r\)) would be more conventional and easier to read.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the system-level error bound is derived from the BRR theorem and component-wise residual/Jacobian inequalities, not fitted; numerical validation against independently solved truth systems breaks any definitional loop.

full rationale

The central claim (contribution 4, Proposition 6/Corollary 7, Algorithm 3) is a conditional bound, not a fitted prediction. The bound ||urb - eurb||_V <= (alpha_bar + eps_bar) sqrt(lambda_max) follows from the external Brezzi-Rappaz-Raviart theorem once component-wise residual and Jacobian bounds (15)-(16) and Lipschitz condition (18) are assumed. The EQP weights are obtained by an LP/ell1 optimization with user-specified tolerances, not by solving for the target error; the online adaptive loop selects RQ fidelity to make the bound's right-hand side equal the user tolerance. This is a theorem-driven selection rule. The only self-citations (Yano/Patera EQP [30,31]) supply the optimization procedure for quadrature weights; the component-wise extension and the system-level BRR bound are derived independently, and the numerical section validates against truth solutions that were not used to set the reported error metrics. The paper's limitations—the 'fundamental assumption' of training representativity in Section 4.1 and the unverified online conditions (15)-(16)/(18) in Algorithm 3—are correctness/certification caveats, not circularity: they weaken the guarantee but do not make any output equal to an input by construction. No step in the derivation reduces to renaming a known result or importing a uniqueness claim from the authors' prior work.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. The central claim rests on training representativity, unverified BRR Lipschitz conditions, and component-wise residual bounds that are assumed to transfer from training to online configurations. The free parameters are training and fidelity hyperparameters, not constants fitted to force the reported error target.

free parameters (5)
  • Nsample = 100
    Number of random sample subsystems per archetype component in Algorithm 1; chosen by hand and controls how well the component library represents online configurations.
  • beta = 0.8
    Probability of connecting each port during training; a training hyperparameter that shapes the snapshot distribution.
  • POD energy threshold = 99.9%
    Sets the bubble RB dimensions (3, 3, 6, 9 for rod, bracket, tee, cross); controls the truth-versus-RB error, which is not controlled in the online phase.
  • delta_bc grid = {1e-4, 1e-3, ..., 1e2}
    Family of hyperreduction tolerances used to build the multi-fidelity reduced quadrature library; the online algorithm selects among these values.
  • Training Dirichlet range = 1 K to 250 K
    Uniform random range for port boundary conditions in Algorithm 1; encodes the expected operating envelope of the components.
assumptions (5)
  • domain assumption The snapshot set generated by random sample subsystems sufficiently represents all potential online solutions and port boundary conditions for each archetype component.
    Section 4.1 calls this the 'fundamental assumption' of the training process; if false, the online residual bounds (15)-(16) are not guaranteed.
  • domain assumption The BRR theorem hypotheses hold for the assembled system: the residual map is C1, the Jacobian is nonsingular at the reference point, and the Lipschitz condition (18) holds.
    Proposition 6 assumes (18), and Algorithm 3 does not compute or verify L(alpha); the certified error bound is therefore conditional on this unverified hypothesis.
  • standard math Truth, RB, and HRBE problems are well-posed for all parameter values in the domain D.
    Assumed in Sections 2.1, 2.2, and 3.1; standard for this literature, but not proved for the specific class of nonlinear PDEs considered.
  • domain assumption Component-wise residual and Jacobian tolerance constraints (15)-(16) hold pointwise on all online parameter and topology configurations.
    EQP training only enforces these constraints on training samples; Algorithm 3 silently sets delta_R = delta_J = delta_c for unseen configurations.
  • domain assumption Geometric mappings are piecewise affine, enabling efficient online computation of lambda_min and lambda_max.
    Remark 9 and Section 6.2 rely on this assumption; it restricts the geometric generality of the method.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A hyperreduced reduced basis element method for reduced-order modeling of component-based nonlinear systems." pith.science (2026). https://pith.science/paper/F6H5OB2R

@misc{pith2026250101621,
  author       = {Pith},
  title        = {Pith review of: A hyperreduced reduced basis element method for reduced-order modeling of component-based nonlinear systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F6H5OB2R}},
  note         = {Machine review of arXiv:2501.01621}
}
read the original abstract

We introduce a hyperreduced reduced basis element method for model reduction of parameterized, component-based systems in continuum mechanics governed by nonlinear partial differential equations. In the offline phase, the method constructs, through a component-wise empirical training, a library of archetype components defined by a component-wise reduced basis and hyperreduced quadrature rules with varying hyperreduction fidelities. In the online phase, the method applies an online adaptive scheme informed by the Brezzi-Rappaz-Raviart theorem to select an appropriate hyperreduction fidelity for each component to meet the user-prescribed error tolerance at the system level. The method accommodates the rapid construction of hyperreduced models for large-scale component-based nonlinear systems and enables model reduction of problems with many continuous and topology-varying parameters. The efficacy of the method is demonstrated on a two-dimensional nonlinear thermal fin system that comprises up to 225 components and 68 independent parameters.

Figures

Figures reproduced from arXiv: 2501.01621 by the authors.

Figure 1
Figure 1. (a) Top: an archetype component with two local ports and [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Archetype components in their reference domains. From left to right: rod, bracket, tee and cross. Local ports are [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Decay of POD eigenvalues in the RB construction for the bubble space of different archetype components. [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: RQ points of the archetype components for [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: A 3 × 3 fin system. In (a), red stars mark the components with a volumetric source term. 10-4 10-3 10-2 10-1 100 101 102 /c 10-3 10-2 10-1 s u p 7 2 %test kuh (7)!~urb(7)kV kuh (7)kV (a) truth vs HRBE 10-4 10-3 10-2 10-1 100 101 102 /c 10-7 10-6 10-5 10-4 10-3 10-2 s u…
Figure 6
Figure 6. Figure 6: Maximum relative H1 (Ω)-norm error in the HRBE solution with respect to the truth and RB solutions for different hyperreduction tolerances for the 3 × 3 fin over |Ξtest|= 5 test cases. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Average speedup in wall-clock time relative to solving the truth problem for different hyperreduction tolerances for [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 48 canonical work pages

  1. [1]

    Rozza, D

    G. Rozza, D. B. P. Huynh, and A. T. Patera, “Reduced basis approximation and a posteriori error esti- mation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics,” Archives of Computational Methods in Engineering , vol. 15, no. 3, p. 229, 2008

    work page 2008

  2. [2]

    Quarteroni, A

    A. Quarteroni, A. Manzoni, and F. Negri, Reduced basis methods for partial differential equations: an introduction, vol. 92. Springer, 2015

    work page 2015

  3. [3]

    J. S. Hesthaven, G. Rozza, and B. Stamm, Certified reduced basis methods for parametrized partial differential equations. Springer, 2016

    work page 2016

  4. [4]

    A survey of projection-based model reduction methods for parametric dynamical systems,

    P. Benner, S. Gugercin, and K. Willcox, “A survey of projection-based model reduction methods for parametric dynamical systems,” SIAM Review, vol. 57, no. 4, pp. 483–531, 2015

    2015

  5. [5]

    A. T. Patera and G. Rozza, Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. MIT Cambridge, MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007

    work page 2007

  6. [6]

    Dynamic analysis of structural systems using component modes,

    W. C. Hurty, “Dynamic analysis of structural systems using component modes,” AIAA Journal, vol. 3, no. 4, pp. 678–685, 1965

    work page 1965

  7. [7]

    Component mode synthesis and eigenvalues of second order operators: discretization and algorithm,

    F. Bourquin, “Component mode synthesis and eigenvalues of second order operators: discretization and algorithm,” ESAIM: Mathematical Modelling and Numerical Analysis , vol. 26, no. 3, pp. 385–423, 1992

    work page 1992

  8. [8]

    A reduced-basis element method,

    Y. Maday and E. M. Rønquist, “A reduced-basis element method,” Journal of Scientific Computing , vol. 17, no. 1, pp. 447–459, 2002

    work page 2002

Show all 49 references
  1. [9]

    The reduced basis element method: application to a thermal fin problem,

    Y. Maday and E. M. Ronquist, “The reduced basis element method: application to a thermal fin problem,” SIAM Journal on Scientific Computing , vol. 26, no. 1, pp. 240–258, 2004

    2004

  2. [10]

    A reduced basis element method for the steady Stokes problem,

    A. E. Løvgren, Y. Maday, and E. M. Rønquist, “A reduced basis element method for the steady Stokes problem,” ESAIM: Mathematical Modelling and Numerical Analysis , vol. 40, no. 3, pp. 529–552, 2006

    2006

  3. [11]

    A static condensation reduced basis element method: approximation and a posteriori error estimation,

    D. B. P. Huynh, D. J. Knezevic, and A. T. Patera, “A static condensation reduced basis element method: approximation and a posteriori error estimation,” ESAIM: Mathematical Modelling and Nu- merical Analysis, vol. 47, no. 1, pp. 213–251, 2013

    2013

  4. [12]

    A static condensation reduced basis element method: Complex problems,

    D. B. P. Huynh, D. J. Knezevic, and A. T. Patera, “A static condensation reduced basis element method: Complex problems,” Computer Methods in Applied Mechanics and Engineering , vol. 259, pp. 197–216, 2013

    2013

  5. [13]

    The static condensation algorithm,

    E. L. Wilson, “The static condensation algorithm,” International Journal for Numerical Methods in Engineering, vol. 8, no. 1, pp. 198–203, 1974

    1974

  6. [14]

    Port reduction in parametrized component static condensation: approxi- mation and a posteriori error estimation,

    J. L. Eftang and A. T. Patera, “Port reduction in parametrized component static condensation: approxi- mation and a posteriori error estimation,” International Journal for Numerical Methods in Engineering , vol. 96, no. 5, pp. 269–302, 2013

    2013

  7. [15]

    A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and a posteriori error estimation,

    J. L. Eftang and A. T. Patera, “A port-reduced static condensation reduced basis element method for large component-synthesized structures: approximation and a posteriori error estimation,” Advanced Modeling and Simulation in Engineering Sciences , vol. 1, no. 1, pp. 1–49, 2014

    2014

  8. [16]

    A new certification framework for the port reduced static condensation reduced basis element method,

    K. Smetana, “A new certification framework for the port reduced static condensation reduced basis element method,” Computer Methods in Applied Mechanics and Engineering , vol. 283, pp. 352–383, 2015. 30

    2015

  9. [17]

    A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales,

    N. C. Nguyen, “A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales,” Journal of Computational Physics , vol. 227, no. 23, pp. 9807–9822, 2008

    2008

  10. [18]

    Reduced-basis approach for homogenization beyond the periodic setting,

    S. Boyaval, “Reduced-basis approach for homogenization beyond the periodic setting,” Multiscale Mod- eling & Simulation , vol. 7, no. 1, pp. 466–494, 2008

    2008

  11. [19]

    A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems,

    S. Kaulmann, M. Ohlberger, and B. Haasdonk, “A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems,” Comptes Rendus Mathematique , vol. 349, no. 23-24, pp. 1233–1238, 2011

    2011

  12. [20]

    A multiscale continuous Galerkin method for stochastic simulation and robust design of photonic crystals,

    F. Vidal-Codina, J. Sa` a-Seoane, N.-C. Nguyen, and J. Peraire, “A multiscale continuous Galerkin method for stochastic simulation and robust design of photonic crystals,” Journal of Computational Physics: X , vol. 2, p. 100016, 2019

    2019

  13. [21]

    Component-wise reduced order model lattice-type structure design,

    S. McBane and Y. Choi, “Component-wise reduced order model lattice-type structure design,”Computer Methods in Applied Mechanics and Engineering , vol. 381, p. 113813, 2021

    2021

  14. [22]

    Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction,

    P. Diercks, K. Veroy, A. Robens-Radermacher, and J. F. Unger, “Multiscale modeling of linear elastic heterogeneous structures via localized model order reduction,” International Journal for Numerical Methods in Engineering , vol. 124, no. 20, pp. 4580–4602, 2023

    2023

  15. [23]

    A domain decomposition strategy for reduced order models. application to the incompressible Navier–Stokes equations,

    J. Baiges, R. Codina, and S. Idelsohn, “A domain decomposition strategy for reduced order models. application to the incompressible Navier–Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 267, pp. 23–42, 2013

    2013

  16. [24]

    A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities,

    J. Ballani, D. P. Huynh, D. J. Knezevic, L. Nguyen, and A. T. Patera, “A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities,” Computer Methods in Applied Mechanics and Engineering , vol. 329, pp. 498–531, 2018

    2018

  17. [25]

    Model order reduction for large-scale structures with local nonlinearities,

    Z. Zhang, M. Guo, and J. S. Hesthaven, “Model order reduction for large-scale structures with local nonlinearities,” Computer Methods in Applied Mechanics and Engineering , vol. 353, pp. 491–515, 2019

    2019

  18. [26]

    Domain-decomposition least-squares Petrov–Galerkin (DD- LSPG) nonlinear model reduction,

    C. Hoang, Y. Choi, and K. Carlberg, “Domain-decomposition least-squares Petrov–Galerkin (DD- LSPG) nonlinear model reduction,” Computer Methods in Applied Mechanics and Engineering, vol. 384, p. 113997, 2021

    2021

  19. [27]

    A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity,

    A. Iollo, G. Sambataro, and T. Taddei, “A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity,” Computer Methods in Applied Mechanics and Engineering, vol. 404, p. 115786, 2023

    2023

  20. [28]

    Localized model reduction for nonlinear elliptic partial differential equa- tions: localized training, partition of unity, and adaptive enrichment,

    K. Smetana and T. Taddei, “Localized model reduction for nonlinear elliptic partial differential equa- tions: localized training, partition of unity, and adaptive enrichment,” SIAM Journal on Scientific Computing, vol. 45, no. 3, pp. A1300–A1331, 2023

    2023

  21. [29]

    A fast and accurate domain decomposition nonlinear manifold reduced order model,

    A. N. Diaz, Y. Choi, and M. Heinkenschloss, “A fast and accurate domain decomposition nonlinear manifold reduced order model,” Computer Methods in Applied Mechanics and Engineering , vol. 425, p. 116943, 2024

    2024

  22. [30]

    An LP empirical quadrature procedure for parametrized functions,

    A. T. Patera and M. Yano, “An LP empirical quadrature procedure for parametrized functions,” Comptes Rendus Mathematique , vol. 355, no. 11, pp. 1161–1167, 2017

    2017

  23. [31]

    An LP empirical quadrature procedure for reduced basis treatment of parametrized nonlinear PDEs,

    M. Yano and A. T. Patera, “An LP empirical quadrature procedure for reduced basis treatment of parametrized nonlinear PDEs,” Computer Methods in Applied Mechanics and Engineering , vol. 344, pp. 1104–1123, 2019

    2019

  24. [32]

    Numerical analysis for nonlinear and bifurcation problems,

    G. Caloz and J. Rappaz, “Numerical analysis for nonlinear and bifurcation problems,” Handbook of Numerical Analysis, vol. 5, pp. 487–637, 1997. 31

    1997

  25. [33]

    Karhunen–L` eve procedure for gappy data,

    R. Everson and L. Sirovich, “Karhunen–L` eve procedure for gappy data,”Journal of the Optical Society of America A , vol. 12, no. 8, pp. 1657–1664, 1995

    1995

  26. [34]

    An ‘empirical interpolation’ method: ap- plication to efficient reduced-basis discretization of partial differential equations,

    M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, “An ‘empirical interpolation’ method: ap- plication to efficient reduced-basis discretization of partial differential equations,” Comptes Rendus Mathematique, vol. 339, no. 9, pp. 667–672, 2004

    2004

  27. [35]

    Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations,

    M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera, “Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations,” ESAIM: Mathematical Modelling and Numerical Analysis , vol. 41, no. 3, pp. 575–605, 2007

    2007

  28. [36]

    Nonlinear model reduction via discrete empirical interpolation,

    S. Chaturantabut and D. C. Sorensen, “Nonlinear model reduction via discrete empirical interpolation,” SIAM Journal on Scientific Computing , vol. 32, no. 5, pp. 2737–2764, 2010

    2010

  29. [37]

    Efficient and accurate nonlinear model reduction via first-order empirical interpolation,

    N. C. Nguyen and J. Peraire, “Efficient and accurate nonlinear model reduction via first-order empirical interpolation,” Journal of Computational Physics , vol. 494, p. 112512, 2023

    2023

  30. [38]

    Model reduction techniques for parametrized nonlinear partial differential equations,

    N. C. Nguyen, “Model reduction techniques for parametrized nonlinear partial differential equations,” Advances in Applied Mechanics , 2024

    2024

  31. [39]

    Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations,

    K. Carlberg, C. Bou-Mosleh, and C. Farhat, “Efficient non-linear model reduction via a least-squares Petrov–Galerkin projection and compressive tensor approximations,” International Journal for Numer- ical Methods in Engineering , vol. 86, no. 2, pp. 155–181, 2011

    2011

  32. [40]

    Optimizing cubature for efficient integration of subspace deforma- tions,

    S. S. An, T. Kim, and D. L. James, “Optimizing cubature for efficient integration of subspace deforma- tions,” ACM Transactions on Graphics , vol. 27, no. 5, pp. 165:1–165:10, 2008

    2008

  33. [41]

    Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency,

    C. Farhat, P. Avery, T. Chapman, and J. Cortial, “Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency,” International Journal for Numerical Methods in Engineering , vol...

    2014

  34. [42]

    Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models,

    C. Farhat, T. Chapman, and P. Avery, “Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models,” International Journal for Numerical Methods in Engineering...

    2015

  35. [43]

    Dimensional hyper-reduction of nonlinear finite element models via empirical cubature,

    J. A. Hern´ andez, M. A. Caicedo, and A. Ferrer, “Dimensional hyper-reduction of nonlinear finite element models via empirical cubature,” Computer Methods in Applied Mechanics and Engineering , vol. 313, pp. 687–722, 2017

    2017

  36. [44]

    Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds,

    K. Veroy, C. Prud’homme, and A. T. Patera, “Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds,” Comptes Rendus Mathematique , vol. 337, no. 9, pp. 619– 624, 2003

    2003

  37. [45]

    A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation,

    M. Yano, A. T. Patera, and K. Urban, “A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation,” Mathematical Models and Methods in Applied Sciences , vol. 24, no. 09, pp. 1903–1935, 2014

    1903

  38. [46]

    Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced-basis a posteriori error bounds,

    K. Veroy and A. T. Patera, “Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced-basis a posteriori error bounds,” International Journal for Numerical Methods in Fluids , vol. 47, no. 8-9, pp. 773–788, 2005

    2005

  39. [47]

    A space-time Petrov-Galerkin certified reduced basis method: application to the Boussinesq equations,

    M. Yano, “A space-time Petrov-Galerkin certified reduced basis method: application to the Boussinesq equations,” SIAM Journal on Scientific Computing , vol. 36, no. 1, pp. A232–A266, 2014

    2014

  40. [48]

    L. N. Trefethen and D. Bau, Numerical linear algebra. SIAM, 2022. 32

    2022

  41. [49]

    Aluminum 3003-F (UNS A93003)

    National Institute of Standards and Technology, “Aluminum 3003-F (UNS A93003).” https://www. nist.gov/mml/acmd/aluminum-3003-f-unsa93003 . Accessed: June 2024. 33

    2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.