Pith. sign in

REVIEW 4 major objections 3 minor 161 references

Towards Automatic and Reliable Localized Model Order Reduction

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper claims a complete localized model order reduction pipeline that makes resimulation after local geometry changes nearly independent of the unchanged global mesh.

desk verdict A solid, honest MOR thesis with rigorous localized error bounds and randomized training, but the certified-reliability claim is only demonstrated for coercive problems and the estimator constants are never actually computed. read the letter →

arxiv 1908.02074 v1 pith:QKMLCFAP submitted 2019-08-06 math.NA cs.NA

Andreas Buhr This is my paper · ORCID
classification math.NAcs.NA MSC 65N3065N1565N55
keywords localizedmodelorderreductionreducedbasismethodaposteriorierrorestimationrandomizednumericallinearalgebratransferoperatorwirebasketspacedecompositiononlineenrichmentprintedcircuitboardsimulation
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

The thesis assembles a localized model order reduction pipeline, named ArbiLoMod, for sequences of finite-element simulations in which only small, non-parametric parts of the geometry change between runs. Its central claim is that local training, local a posteriori error estimation, and local online enrichment can be combined so that after a local design edit only the reduced spaces in the changed region need to be recreated, and the global problem is then solved on a small reduced system. The paper further claims a new training algorithm: applying a randomized range finder to a transfer operator on each local patch produces reduced spaces that converge, with high probability, at essentially the singular value decay rate of that operator. If these claims hold, interactive workflows such as signal-integrity simulation of printed circuit boards could resimulate local modifications without re-solving the global high-dimensional problem from scratch.

What carries the argument

The wirebasket space decomposition is the backbone: each degree of freedom is assigned to a coarse mesh entity and extended to neighboring cells by solving the homogeneous equation, giving a direct-sum splitting of the global space into local subspaces. The transfer operator $T$ maps boundary or coupling data on a local training domain to the local solution within it; the randomized range finder approximates the image of its linear part, and the training a priori estimate $\|u-\tilde u\|/\|u\| \le \sqrt{\gamma/\alpha}\, \sqrt{\bar J J^*}\,\max_i \|(1-P_{\tilde V_i})T_i^l\|\,\|P_{S_i}\|$ turns the global error into a bound on local projection errors. The partition-of-unity dual norm localization bounds the residual norm by $\sqrt{\bar J}\,c_{pu,\tilde V}\,(\sum_i \|R_\mu(\tilde u_\mu)\|_{V_i'}^2)^{1/2}$, which makes the error estimator computable from local solves.

What would settle it

For the 2D Maxwell or Olimex A64 example, compute an independently certified lower bound for the inf-sup constant; if the localized error estimator then fails to be an upper bound on the true reduction error, or if the bound cannot be computed without a global solve, the reliability claim fails.

Watch

Extended reading notes

Core claim

ArbiLoMod decomposes the global finite-element space via a wirebasket decomposition: reduced subspaces are attached to coarse vertices, edges, faces, and cells, with basis functions extended into neighboring cells by solving homogeneous local problems. The initial reduced spaces are built by local trainings (random boundary data on a patch around each interface) and by local greedy algorithms; a partition-of-unity based a posteriori error estimator bounds the residual dual norm by the sum (or root-sum-square) of local residual norms, with constants that can be computed or bounded, and an online enrichment loop adds the steepest residual components until the estimated error falls below tolerance. The principal new analytical claims are a probabilistic a priori bound for the randomized training, showing the generated local spaces are nearly as good as the optimal singular-value spaces of the transfer operator, and exponential convergence of the proposed online enrichment algorithms. All ingredients are demonstrated on thermal-channel, two-dimensional Maxwell, and a 65-million-degree-of-freedom printed-circuit-board example; for the inf-sup stable Maxwell problems the author reports that a certified inf-sup lower bound is not provided.

Load-bearing premise

The entire certified error bound rests on having a global lower bound for the coercivity constant (or the inf-sup constant) of the bilinear form, and the thesis computes such a bound only for simple heat-conduction examples, not for the inf-sup stable Maxwell or PCB cases that motivate the work.

Editorial extensions

If this is right

  • After a localized geometry change, only reduced spaces whose training domains intersect the change need to be regenerated; the rest of the offline data can be reused.
  • The localized a posteriori estimator gives a rigorous upper bound on the reduction error with computable constants in coercive problems, and it can steer where to enrich.
  • The randomized training algorithm replaces the user-chosen number of random samples with an adaptive choice, and its convergence rate is tied to the singular-value decay of the transfer operator.
  • The online enrichment loop converges exponentially and can even start from empty local bases, so it provides a fallback when initial training is insufficient.

Reading between the lines

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

  • The missing certified inf-sup lower bound for the Maxwell and PCB examples is the one gap between the proven reliability statement and the stated target application; a localized or randomized inf-sup estimator would close it.
  • Because the training a priori estimate is independent of the number of local subdomains, the method should scale to very large domain decompositions; a three-dimensional PCB example with thousands of subdomains would be a natural stress test.
  • The same randomized range finder could be warm-started with reduced bases from previous geometries, potentially making the reuse after local changes even cheaper than rebuilding from scratch.
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

4 major / 3 minor

Summary. The manuscript is a PhD thesis proposing ArbiLoMod, a localized model order reduction methodology for sequences of PDE simulations with arbitrary local geometry modifications. It develops a wirebasket space decomposition with local training, a localized residual-based a posteriori error estimator via dual norm localization, randomized range finder training algorithms with probabilistic a priori convergence bounds, and online enrichment algorithms. The main claims are that ArbiLoMod delivers a complete localized pipeline, that the localized error estimator is certified with computable constants, and that randomized training converges nearly as fast as the singular value decay of the transfer operator. Numerical experiments cover heat conduction, a 2D Maxwell problem, and a large PCB example, with reproduction scripts.

Significance. The strongest contributions are the abstract localized training configuration with the transfer-operator formulation and the randomized range finder analysis in Chapter 5, including a probabilistic a posteriori norm estimator and an a priori convergence theorem; these are rigorous and are supported by numerical experiments on the Olimex A64 example. The numerically stable offline/online splitting for the reduced basis error estimator in Section 4.5 is a useful practical contribution, and the emphasis on reproducible source code is commendable. If the certification gap described below is resolved, the localized a posteriori estimator would be a significant step toward localized model order reduction for engineering applications. As it stands, however, the certified reliability claim is not established for the targeted Maxwell/PCB application, which is the central weakness of the manuscript.

major comments (4)
  1. [Sections 4.2–4.3, Corollary 4.2.4; Sections 2.6.4 and 3.6.2] The localized error estimator is certified only for coercive problems. Corollary 4.2.4 applies Proposition 4.2.3 to the standard error bound Δ(ũμ) = 1/αμ ‖Rμ(ũμ)‖, which requires a computable lower bound on the coercivity constant αμ. The time-harmonic Maxwell problem in H(curl) (Section 2.6.4) is inf-sup stable but not coercive, and the text states that a lower bound for the inf-sup constant is not computed. Therefore the reliable/certified status claimed in the abstract and in design goal (5) of Section 3.1 is not demonstrated for the main target application; Section 3.6.2 indeed runs the 2D Maxwell experiment without the a posteriori error estimator. The authors should either present an inf-sup lower bound for the Maxwell setting (or its 2D normal-magnetic reduction) or explicitly restrict the certification claim to coercive problems.
  2. [Section 3.6.1, Figure 3.11] The sole numerical demonstration of the localized estimator in the Thermal Channels example is plotted for αμ = cpu,~V = 1. This means the constants whose computability is claimed in the abstract are not actually computed or bounded by their certified estimates in the experiment, and the effectivity of the estimator is not shown. Since the main selling point of Corollary 4.2.4 over the standard global estimator is a localized, computable, certified bound, the demonstration should report the computed or estimated constants, or at least compare the estimator with a certified lower bound using the bounds from Propositions 4.3.2 and 4.3.3.
  3. [Section 5.1, Proposition 5.1.3, Eq. (5.16)] The a priori training estimate rests on the assumption T_a^i ∈ ~V_i (Eq. 5.16), i.e., the affine part of each transfer operator is contained in the reduced local space. In the parameterized ArbiLoMod pipeline, the training algorithms only include snapshots for a finite set Ξ (e.g., Algorithm 3.3, and the randomized range finder in Section 5.2 targets the linear part T_l^i), so this hypothesis is not guaranteed for parameters outside the training set. The text should either prove that the construction enforces Eq. (5.16) for all parameters, or restate the theorem with an additional affine-part/residual term and explain how training controls it.
  4. [Section 3.6.2, Figures 3.16–3.17] For the inf-sup stable Maxwell example, the reduced Galerkin problem is observed to become unstable at certain basis sizes, with the reduced inf-sup constant dropping and the maximum error jumping at a basis size of about 900. The thesis acknowledges this and defers stable Petrov-Galerkin reduction to future work in Section 3.7. This is an honest limitation, but it further means that the method as presented cannot yet be certified for the target inf-sup stable application, and this restriction should be stated prominently in the abstract and conclusions rather than only in the experiment section.
minor comments (3)
  1. [Section 1.4.4 and Bibliography] There are copy-editing problems, for example the sentence fragment in Section 1.4.4 ('CMS Component Mode Synthesis (CMS) component mode synthesis (CMS) introduced in a [Hurty(1965),Bampton and Craig(1968)]') and bibliography entries with 'doi: n/a'; these should be cleaned up before submission.
  2. [Section 3.6.1, Figure 3.11] The text says the factor cpu,~V is 'neglected' while the caption says 'Plotted for αμ = cpu,~V = 1'; the terminology should be unified so that the reader knows whether the plotted quantity is an upper bound, a lower bound, or a heuristic indicator.
  3. [Section 4.3, Proposition 4.3.2] In the proof, the estimate is first obtained with H_i in the denominator and then stated with H = min_i H_i; a one-line clarification that H ≤ H_i justifies the uniform bound would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation chain is self-contained; no circularity found.

full rationale

The paper's main derived results are self-contained in the sense that their statements do not coincide by definition with their inputs. The ArbiLoMod training pipeline is analyzed in Proposition 5.1.3, where the reduction error is bounded by the projection error of the transfer operator under the explicit decomposition assumption u = Σ_i T_i P_{S_i} u|_{ω*_i} (Definition 5.1.1, Eq. 5.3); this is an assumption of the framework, not the conclusion of the theorem, and the subsequent randomized range finder bounds that projection error by a standard RandNLA estimate involving the singular values of the operator (Section 5.2). Similarly, the localized a posteriori error estimator in Corollary 4.2.4 is derived from the residual dual norm and the localized stability constants c_{i,~V} and c_{pu,~V}; these constants are defined directly in terms of the reduced space and the space decomposition, and no fitted parameter is renamed as a prediction. The only caveats are explicit limitations: Section 2.6.4 states "we cannot in general estimate the inf-sup constant from below. Obtaining a lower bound for the inf-sup constant is a challenging task which we omit", and Section 3.6.2 applies ArbiLoMod to the 2D Maxwell problem "without ... a posteriori error estimator", so the certified-reliability claim is not demonstrated for the target inf-sup-stable application; Figure 3.11 is plotted for αµ = cpu,~V = 1, so the constants' computability is not numerically demonstrated. These are correctness or validation gaps, not circular reductions. The thesis also cites the author's own prior journal articles (e.g., Buhr and Smetana 2018, Buhr et al. 2017a), but the corresponding analyses are reproduced in the thesis itself and the cited publications are peer-reviewed, so the self-citations are not load-bearing in the derivation chain.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No fitted physical parameters appear in the derivations; the free parameters are algorithmic tolerances that control basis size. The axioms are the structural assumptions of the space decomposition, the affine parameter dependence, availability of stability constants (a known weak point), and conditions used in the training and localization estimates. No new physical entities are introduced.

free parameters (4)
  • M (number of random samples in heuristic training)
    User-chosen in Algorithm 3.3; the improved randomized range finder in Chapter 5 replaces it with an adaptive selection depending on a prescribed accuracy and failure probability.
  • epsilon_train (training tolerance)
    Stopping criterion for the snapshot greedy in local face space training; set to 1e-4 or 1e-5 in experiments, affects basis size but not the mathematical claims.
  • epsilon_greedy (greedy tolerance)
    Stopping criterion for local cell space greedy; set to 1e-3 to 1e-7 in experiments.
  • r (enrichment fraction)
    Doerfler-like fraction in the enrichment loop, set to 0.5 in experiments.
assumptions (6)
  • domain assumption The localizing space decomposition satisfies sum V_i = V and sum P_V_i(phi) = phi for all phi in V.
    Definition 2.4.1; the wirebasket decomposition is designed to satisfy it, but it is a structural requirement for the localized error estimates and training a priori bound.
  • domain assumption The bilinear form has an affine parameter dependence.
    Section 2.3, Eq (2.17); standard in reduced basis methods; used for offline/online splitting.
  • ad hoc to paper A computable lower bound for the coercivity constant alpha_LB (or inf-sup lower bound) is available.
    Used in residual-based estimators (Eq (3.21), Eq (4.1)). The thesis admits such bounds are not available for general problems, in particular for the inf-sup stable Maxwell case (Section 2.6.4).
  • ad hoc to paper The affine part T_a^i of each transfer operator is contained in the reduced space ~V_i (Eq 5.16).
    Needed for the training a priori estimate Prop 5.1.3; if not enforced, the bound has an extra term. The training algorithm must include the right-hand-side response to satisfy this.
  • domain assumption The partition of unity functions rho_i are contained in the reduced space ~V (Prop 4.3.3).
    Used to obtain an H-independent bound on the localization constant; in ArbiLoMod this is fulfilled because vertex spaces (codim 2) are kept full-dimensional.
  • standard math Random coefficient vectors produce inner products with a Gaussian distribution with variance bounded by the extreme eigenvalues of the inner product matrix (Lemma 5.2.1).
    This is a standard property of Gaussian random vectors and is used for the probabilistic norm estimator.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Towards Automatic and Reliable Localized Model Order Reduction." pith.science (2026). https://pith.science/paper/QKMLCFAP

@misc{pith2026190802074,
  author       = {Pith},
  title        = {Pith review of: Towards Automatic and Reliable Localized Model Order Reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKMLCFAP}},
  note         = {Machine review of arXiv:1908.02074}
}
read the original abstract

Finite element based simulation of phenomena governed by partial differential equations is a standard tool in many engineering workflows today. However, the simulation of complex geometries is computationally expensive. Many engineering workflows require multiple simulations with small, non parametric changes in between. The use of localized model order reduction for subsequent simulations of geometries with localized changes is very promising. It produces lots of computational tasks with little dependencies and thus parallelizes well. Furthermore, the possibility to reuse intermediary results in the subsequent simulations can lead to large computational savings. In this thesis, we investigate different aspects of localized model order reduction and propose various improvements. A simulation methodology named ArbiLoMod, comprising a localized training, a localized a posteriori error estimator and an enrichment procedure is proposed. A new localized a posteriori error estimator with computable constants is presented and analyzed. A new training algorithm which is based on a transfer operator is derived. It can be shown to converge nearly as fast as the singular value decay of this operator. The transfer operator's spectrum is observed to decay fast in electromagnetic simulations in printed circuit boards. New online enrichment algorithms are proposed. All results are supported by numerical experiments, for which the source code for reproduction is provided.

Figures

Figures reproduced from arXiv: 1908.02074 by the authors.

Figure 1.1
Figure 1.1. DDR memory channel on a printed circuit board subject to local modification of conductive tracks (own photo, manipulated with GIMP). 1 [PITH_FULL_IMAGE:figures/full_fig_p021_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Visualization of Nédélec ansatz functions (edge elements) for one triangle (reproduc￾tion: Appendix A.1). where the restriction to the boundary is to be understood in the sense of traces. The space H(curl, Ω) is defined as usual as H(curl, Ω) := n ϕ ∈ [PITH_FULL_IMAGE:figures/full_fig_p040_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Geometry of Thermal Block Example: Four blocks of constant, but parameterized, heat conductivity. 2.7.1. Thermal Block Example The first numerical example we denote by Thermal Block Example. It is a simple parametric example which is often used in introductory texts on parameterized model order reduction (for example in [Patera and Rozza(2007)] or [Haasdonk(2017)]). We solve the problem in Definition 2.1.1 with the … view at source ↗
Figures from the paper (48 more)
Figure 2.3
Figure 2.3. Figure 2.3: High-dimensional solution of Equation (2.73) in Thermal Block Example for µ = (0.1, 1.0, 0.4, 0.1). (reproduction: Appendix A.2) [PITH_FULL_IMAGE:figures/full_fig_p044_2_3.png]
Figure 2.4
Figure 2.4. Figure 2.4: First structure in sequence of simulated structures in Thermal Channels Example. Unit square with high and low conductivity regions. White: constant conductivity region (σ = 1), black: parameterized high conductivity region (σµ = 1 + µ). Homogeneous Neumann boundarie…
Figure 2.5
Figure 2.5. Figure 2.5: Sequence of structures simulated in Thermal Channels Example along with solutions for one parameter value. Localized changes in the geometry cause global changes in the solution. (reproduction: Appendix A.3) 25 [PITH_FULL_IMAGE:figures/full_fig_p045_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: Geometries simulated in 2D Maxwell Example. Black area is not part of the domain and treated as Dirichlet zero boundary. Note that the change is topology changing [PITH_FULL_IMAGE:figures/full_fig_p047_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: Example solutions for 2D Maxwell Example for ω = 2π·186 MHz, ω = 2π·561 MHz and ω = 2π · 1 GHz for the first and second geometry. Plotted is |Re(E)|. (reproduction: Appendix A.4) boundary condition. Note that the geometry is slightly asymmetric intentionally, to prod…
Figure 2.8
Figure 2.8. Figure 2.8: Front and perspective view of Olimex A64 Example. 2.7.4. Olimex A64 Example The fourth example is the Olimex A64 Example. Here we solve the problem in Definition 2.1.1 with the bilinear form and the linear form defined as in Equation (2.58) and the space V defined in…
Figure 2.9
Figure 2.9. Figure 2.9: Back and detail view of Olimex A64 Example. In the right picture, the meandering traces between the CPU (top) and the RAM (left) are visible [PITH_FULL_IMAGE:figures/full_fig_p049_2_9.png]
Figure 2.10
Figure 2.10. Figure 2.10: Meshed detail of Olimex A64 Example. Metal parts, modeled as Dirichlet boundaries, are shown in red. A staircase approximation of the geometry is used. air copper prepreg copper FR4 copper prepreg copper FR4 copper prepreg copper air 350 µm 35 µm 127 µm 35 µm 500 µm…
Figure 2.11
Figure 2.11. Figure 2.11: Stackup of Olimex A64 Example: Six layers of metal with insulating layers in between. Data provided by Olimex in private communication. 29 [PITH_FULL_IMAGE:figures/full_fig_p049_2_11.png]
Figure 3.1
Figure 3.1. Figure 3.1: Main components of localized model order reduction • Localized training • Localized a posteriori error estimation • Localized online enrichment Those parts form a global loop, as visualized in [PITH_FULL_IMAGE:figures/full_fig_p052_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Overview of ArbiLoMod. Generation of initial reduced interface spaces by training and of reduced volume spaces by greedy basis generation is subject of Section 3.3 and Chapter 5. Convergence is assessed with the localized a posteriori error estimator presented in Cha…
Figure 3.3
Figure 3.3. Figure 3.3: Visualization of basic spaces V basic i and their extension spaces for Q1 ansatz functions (one DoF per mesh node). Definition 3.2.1 (Basic subspaces). For each element i ∈ {1, . . . , NE } we define a basic subspace V basic i of Vh as: V basic i := span n ψ ∈ B [PI…
Figure 3.4
Figure 3.4. Figure 3.4: Visualization of some example elements of the local subspaces V wb i for inhomogeneous coefficients. The structure in the solution results from variations in the heat conduction coefficient. (a) Value 1 in V basic i (b) Linear in V basic j , j ∈ Υ1 (c) Solving in V b…
Figure 3.5
Figure 3.5. Figure 3.5: Extend operator is executed in two steps for spaces V wb i , i ∈ Υ2. Mesh and spaces as depicted in [PITH_FULL_IMAGE:figures/full_fig_p056_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: Visualization of spaces for a codim 1 coarse mesh entity. Basic space V basic i , training space, and the coupling space of training space for Q1 ansatz functions (one DoF per mesh node). 3.3.2. Local Training for Basis Construction of Reduced Face Spaces To generate…
Figure 3.7
Figure 3.7. Figure 3.7: Data dependencies in ArbiLoMod: Before online enrichment, it is possible to compute all reduced basis function having support on a domain (ω nol 5 here) using only local information about the domain and its surrounding domains. 43 [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 3.8
Figure 3.8. Figure 3.8: Maximum relative H1 -error on training set Ξ in dependence of tolerances in codim 1 training and codim 0 greedy for Thermal Channels Example. Online enrichment disabled. (reproduction: Appendix A.6) work can be distributed on many nodes. Afterwards, only reduced repr…
Figure 3.9
Figure 3.9. Figure 3.9: Relative error over iteration with and without basis reuse for after geometry change for Thermal Channels Example. With codim 1 training. Convergence criterion: kRµ(ueµ)kVh 0 < 10−4 , greedy tolerance: εgreedy = 10−5 . See also [PITH_FULL_IMAGE:figures/full_fig_p065…
Figure 3.10
Figure 3.10. Figure 3.10: Relative H1 error over iterations with and without basis reuse for after geometry change for Thermal Channels Example. Without codim 1 training. Convergence criterion: kRµ(ueµ)kV 0 h < 10−4 , greedy tolerance: εgreedy = 10−5 . See also [PITH_FULL_IMAGE:figures/full…
Figure 3.11
Figure 3.11. Figure 3.11: Error estimators ∆rel(ueµ) and ∆ g loc,rel(ueµ) over iterations, compared to the relative error for Thermal Channels Example. Plotted for αµ = c pu,Ve = 1. Simulation performed with a convergence criterion of kRµ(ueµ)kV 0 h < 10−6 , a training tolerance of εtrain = …
Figure 3.12
Figure 3.12. Figure 3.12: Distribution of local basis sizes after initial training for Thermal Channels Exam￾ple. Relative reduction error at this configuration: 1.3 · 10−3 . (reproduction: Ap￾pendix A.11) computing environment. Especially when the solution has to be calculated at multiple p…
Figure 3.13
Figure 3.13. Figure 3.13: inf-sup and continuity constant of bilinear form for 2D Maxwell Example. Linear and logarithmic. (reproduction: Appendix A.12) 0 20 40 60 80 100 10−15 10−7 101 basis size max rel. projection error geometry 1 geometry 2 [PITH_FULL_IMAGE:figures/full_fig_p069_3_13.png]
Figure 3.14
Figure 3.14. Figure 3.14: Error when approximating the solution set for all µ ∈ Ξ with an n-dimensional basis obtained by greedy approximation of this set. This is an upper bound for the Kolmogorov n-width. (2D Maxwell Example) (reproduction: Appendix A.13) excitation [PITH_FULL_IMAGE:figur…
Figure 3.15
Figure 3.15. Figure 3.15: Domain decomposition used for 2D Maxwell Example. 49 [PITH_FULL_IMAGE:figures/full_fig_p069_3_15.png]
Figure 3.16
Figure 3.16. Figure 3.16: Maximum error when solving 2D Maxwell Example with a localized basis, generated by global solves. (reproduction: Appendix A.14) 0 1,000 2,000 10−9 10−6 10−3 100 basis size maximum relative error 0 1,000 2,000 10−4 10−3 10−2 10−1 100 basis size eβ 170 MHz 340 MHz 700…
Figure 3.17
Figure 3.17. Figure 3.17: Comparison of maximum error over all frequencies with inf-sup constant of reduced system at selected frequencies for geometry 1. Basis generated by global solves. Increased error and reduced inf-sup constant at basis size of ca. 900 (2D Maxwell Example). (reproducti…
Figure 3.18
Figure 3.18. Figure 3.18: Maximum error over all frequencies for both geometries of 2D Maxwell Example. Basis generated by global solves vs. basis generated by local training. (reproduction: Appendix A.16) Dahmen et al.(2014)] to the localized setting is not straightforward. Properties of Tr…
Figure 3.19
Figure 3.19. Figure 3.19: Impact of geometry change of 2D Maxwell Example: 5 domains contain changes, 14 domain spaces and 20 interface spaces have to be regenerated. geo 1: geo 2 [PITH_FULL_IMAGE:figures/full_fig_p072_3_19.png]
Figure 3.20
Figure 3.20. Figure 3.20: Basis size distribution in 2D Maxwell Example. (reproduction: Appendix A.11) have to be regenerated. All other bases can be reused, which results in large computational savings compared to a simulation from scratch. However, the Galerkin projection leads to an unsta…
Figure 4.1
Figure 4.1. Figure 4.1: Evaluation of proposed offline/online splitting: maximum relative reduction errors and estimated reduction errors (H1 -norm) for numerical example Thermal Block Example. (reproduction: Appendix A.18) space Ve is constructed from the linear span of solutions to Defini…
Figure 5.1
Figure 5.1. Figure 5.1: Geometry of Rangefinder Example 1 and Rangefinder Example 2 Analytic interface problem To analyze the behavior of the proposed algorithm, we first apply it to an analytic problem where the singular values of the transfer operator are known. We refer to this numerical…
Figure 5.2
Figure 5.2. Figure 5.2: Comparison of optimal basis functions with the basis functions generated by Algo￾rithm 5.1 for Rangefinder Example 1 . Basis functions are normalized to an L 2 (Γin) norm of one. (reproduction: Appendix A.19) 0 5 10 100 10−5 10−10 10−15 basis size n [PITH_FULL_IMAGE…
Figure 5.3
Figure 5.3. Figure 5.3: Projection error supξ∈S\{0} infζ∈Ren kTξ−ζkR kξkS = [PITH_FULL_IMAGE:figures/full_fig_p105_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: Projection error supξ∈S\{0} infζ∈Ren kTξ−ζkR kξkS = [PITH_FULL_IMAGE:figures/full_fig_p107_5_4.png]
Figure 5.5
Figure 5.5. Figure 5.5: h dependency in Rangefinder Example 1 ; Statistics over 10,000 samples. (reproduc￾tion: Appendix A.23) 0 5 10 15 20 10−15 10−10 10−5 100 i σi+1 k = 0 k = 10 k = 20 k = 30 k = 40 0 5 10 15 20 0 20 10− 40 15 10−10 10−5 100 i κ σi+1 [PITH_FULL_IMAGE:figures/full_fig_p1…
Figure 5.6
Figure 5.6. Figure 5.6: Singular value decay for Rangefinder Example 2 . The red line at i = k/π in the right plot marks the observed length of the plateau. (reproduction: Appendix A.24) Γout and the boundary conditions are the same as in Section 5.3, only the operator A differs and is defi…
Figure 5.7
Figure 5.7. Figure 5.7: Projection error supξ∈S\{0} infζ∈Ren kTξ−ζkR kξkS = [PITH_FULL_IMAGE:figures/full_fig_p109_5_7.png]
Figure 5.8
Figure 5.8. Figure 5.8: Projection error supξ∈S\{0} infζ∈Ren kTξ−ζkR kξkS = [PITH_FULL_IMAGE:figures/full_fig_p109_5_8.png]
Figure 5.9
Figure 5.9. Figure 5.9: Domain decomposition used for Olimex A64 Example with 40 × 28 domains. For each domain ωi (e.g. ω816, yellow) we introduce an oversampling domain ω ∗ i (e.g. ω ∗ 816, green) [PITH_FULL_IMAGE:figures/full_fig_p110_5_9.png]
Figure 5.10
Figure 5.10. Figure 5.10: Example randomized solution in oversampling domain of ω816 of Olimex A64 Example (approx. 350.000 unknowns). For simplicity, we use the euclidean norm of the coefficient vectors in V basic 816 and compare with the singular values of the matrix of T l 816. In [PITH_…
Figure 5.11
Figure 5.11. Figure 5.11: Singular value decay of T816 and decay of randomized norm estimator for Olimex A64 Example. (reproduction: Appendix A.27) 815 820 825 830 835 840 845 850 855 0 200 400 600 domain time/s assembly factorization testvector generation basis generation [PITH_FULL_IMAGE:…
Figure 5.12
Figure 5.12. Figure 5.12: Runtimes for randomized range finder on Olimex A64 Example. (reproduction: Appendix A.27) 92 [PITH_FULL_IMAGE:figures/full_fig_p112_5_12.png]
Figure 6.1
Figure 6.1. Figure 6.1: Geometry of Thermal Channels Variant. Left: Coefficient field σ. White is 1, black is 105 . Middle: right hand side q. Black is −105 , gray is 0, white is 105 . Right: reference solution u. (reproduction: Appendix A.28) . . . . . . . . [PITH_FULL_IMAGE:figures/full…
Figure 6.2
Figure 6.2. Figure 6.2: Overlapping domain decomposition ωi (green and blue) constructed from 2×2 patches of non-overlapping domain decomposition (red) for Thermal Channels Variant. a variant of the Thermal Channels Example but driven by a right hand side instead of inhomoge￾neous boundary …
Figure 6.3
Figure 6.3. Figure 6.3: Decay of relative energy error during iteration for Thermal Channels Variant (zoom). (reproduction: Appendix A.29) 0 100 200 300 400 500 600 10−9 10−6 10−3 100 [PITH_FULL_IMAGE:figures/full_fig_p122_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: Decay of relative energy error during iteration. (reproduction: Appendix A.29) 102 [PITH_FULL_IMAGE:figures/full_fig_p122_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: Convergence of online enrichment in Thermal Channels Variant. 1 would be conver￾gence within one iteration, 0 would be stagnation. (reproduction: Appendix A.29) 0 100 200 300 400 500 600 100 103 106 numerical noise Iteration n equation (6.6) equation (6.7) equation (…
Figure 6.6
Figure 6.6. Figure 6.6: Sharpness of inequalities in Algorithm 6.1: For equation (6.6),  kuen − uk 2 a − kR(uen)k 2 O0 k  /  kuen+1 − uk 2 a  is plotted. For equation (6.7),  kR(uen)k 2 V 0 k  /  1 NVi PNVi i=1 kR(uen)k 2 V 0 k  is plotted. For equation (6.8),  c pu,Ve 2 PNVi i=1 k…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

161 extracted references · 66 canonical work pages

  1. [1]

    Aarnes, T

    J. Aarnes, T. Y. Hou . Multiscale domain decomposition methods for elliptic problems with high aspect ratios. Acta Mathematicae Applicatae Sinica, English Series, 18 (2002) (1), pp. 63--76. doi:10.1007/s102550200004

    work page doi:10.1007/s102550200004 2002

  2. [2]

    A. Abdulle . On a priori error analysis of fully discrete heterogeneous multiscale FEM . Multiscale Modeling and Simulation, 4 (2005) (2), pp. 447--459. doi:10.1137/040607137

    work page doi:10.1137/040607137 2005

  3. [3]

    Albrecht, M

    F. Albrecht, M. Ohlberger . The localized reduced basis multi-scale method with online enrichment. Oberwolfach Reports, 7 (2013), pp. 406--409. doi:10.4171/OWR/2013/07

    work page doi:10.4171/owr/2013/07 2013

  4. [4]

    Albrecht, B

    F. Albrecht, B. Haasdonk, M. Ohlberger, S. Kaulmann . The Localized Reduced Basis Multiscale Method. Proceedings of Algoritmy 2012, Conference on Scientific Computing, Vysoke Tatry, Podbanske, September 9-14, 2012, (2012), pp. 393--403. doi:n/a

    2012

  5. [5]

    M. S. Aln s, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, G. N. Wells . The FEniCS Project Version 1.5. Archive of Numerical Software, 3 (2015) (100). doi:10.11588/ans.2015.100.20553

    doi:10.11588/ans.2015.100.20553 2015

  6. [6]

    Alzetta, D

    G. Alzetta, D. Arndt, W. Bangerth, V. Boddu, B. Brands, D. Davydov, R. Gassmoeller, T. Heister, L. Heltai, K. Kormann, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, D. Wells . The deal.II Library, Version 9.0 . Journal of Numerical Mathematics, 26 (2018) (4), pp. 173--183. doi:10.1515/jnma-2018-0054

    doi:10.1515/jnma-2018-0054 2018

  7. [7]

    P. F. Antonietti, P. Pacciarini, A. Quarteroni . A discontinuous G alerkin Reduced Basis Element method for elliptic problems . ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016) (2), pp. 337--360. doi:10.1051/m2an/2015045

    arXiv 2016

  8. [8]

    Babuska, R

    I. Babuska, R. Lipton . Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems. Multiscale Modeling & Simulation, 9 (2011) (1), pp. 373--406. doi:10.1137/100791051

    doi:10.1137/100791051 2011

Show all 161 references
  1. [9]

    Balay, S

    S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, H. Zhang . PETS c W eb page . http://...

    2018

  2. [10]

    M. C. C. Bampton , R. R. Craig , Jr. Coupling of substructures for dynamic analyses. AIAA Journal, 6 (1968) (7), pp. 1313--1319. doi:10.2514/3.4741

    1968 doi

  3. [11]

    Bastian, M

    P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, O. Sander . A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework. Computing, 82 (2008 a ) (2-3), pp. 103--119. doi:10.1007/s00607-008-0003-x

    2008 doi

  4. [12]

    Bastian, M

    P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Kl \"o fkorn, R. Kornhuber, M. Ohlberger, O. Sander . A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE. Computing, 82 (2008 b ) (2), pp. 121--138. doi:10.1007/s0060...

    2008 doi

  5. [13]

    o ddeke, O. Iliev, O. Ippisch, M. Ohlberger, S. Turek, J. Fahlke, S. Kaulmann, S. M \

    P. Bastian, C. Engwer, D. G \"o ddeke, O. Iliev, O. Ippisch, M. Ohlberger, S. Turek, J. Fahlke, S. Kaulmann, S. M \"u thing, D. Ribbrock . EXA-DUNE: Flexible PDE Solvers, Numerical Methods and Applications. In L. Lopes, J. Z ilinskas, A. Costan, R. G. Cascella, G. Kecskemeti, ...

    2014

  6. [14]

    Bastian, C

    P. Bastian, C. Engwer, J. Fahlke, M. Geveler, D. Göddeke, O. Iliev, O. Ippisch, R. Milk, J. Mohring, S. Müthing, M. Ohlberger, D. Ribbrock, S. Turek . Hardware-Based Efficiency Advances in the EXA - DUNE Project . In Lecture Notes in Computational Science and Engineering, pp. ...

    2016 doi

  7. [15]

    Benner, M

    P. Benner, M. Hess . The Reduced Basis Method for Time-Harmonic Maxwell's Equations. Proceedings in Applied Mathematics and Mechanics, 12 (2012) (1), pp. 661--662. doi:10.1002/pamm.201210319

    2012 doi

  8. [16]

    Benner, M

    P. Benner, M. Ohlberger, A. Cohen, K. E. Wilcox . Model Reduction and Approximation: Theory and Algorithms. SIAM-Society for Industrial & Applied Mathematics, 2017. ISBN 9781611974829

    2017

  9. [17]

    Bourquin

    F. Bourquin . Component mode synthesis and eigenvalues of second order operators: discretization and algorithm. ESAIM : Mathematical Modelling and Numerical Analysis, 26 (1992) (3), pp. 385--423. doi:10.1051/m2an/1992260303851

    1992

  10. [18]

    A. Buhr . Finite Elements in PCB Structures . Master's thesis, TU Darmstadt, Darmstadt, Germany, 2009

    2009

  11. [19]

    A. Buhr . Exponential Convergence of Online Enrichment in Localized Reduced Basis Methods. IFAC-PapersOnLine, 51 (2017) (2), pp. 302--306. doi:n/a. MATHMOD 2018

    2017

  12. [20]

    A. Buhr . Source Code for PhD Thesis by Andreas Buhr, 2019. doi:10.5281/zenodo.2555623

    2019 doi

  13. [21]

    A. Buhr, K. Smetana . Randomized Local Model Order Reduction. SIAM Journal on Scientific Computing, 40 (2018) (4), pp. A2120--A2151. doi:10.1137/17M1138480

    2018 doi

  14. [22]

    A. Buhr, C. Engwer, M. Ohlberger, S. Rave . A Numerically Stable a Posteriori Error Estimator for Reduced Basis Approximations of Elliptic Equations. In X. O. E. Onate, A. Huerta , eds., Proceedings of the 11th World Congress on Computational Mechanics. CIMNE, Barcelona, 2014 ...

    2014

  15. [23]

    A. Buhr, C. Engwer, M. Ohlberger, S. Rave . ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications . SIAM Journal on Scientific Computing, 39 (2017 a ) (4), pp. A1435--A1465. doi:10.1137/15M1054213

    2017 doi

  16. [24]

    A. Buhr, C. Engwer, M. Ohlberger, S. Rave . ArbiLoMod : Local Solution Spaces by Random Training in Electrodynamics , pp. 137--148. Springer International Publishing, Cham. ISBN 978-3-319-58786-8, 2017 b . doi:10.1007/978-3-319-58786-8_9

    2017 doi

  17. [25]

    A. Buhr, L. Iapichino, M. Ohlberger, S. Rave, F. Schindler, K. Smetana . Handbook on Model Order Reduction, chap. Localized model reduction for parameterized problems. Walter De Gruyter GmbH, 2019+. Submitted for publication

    2019

  18. [26]

    Carlberg, C

    K. Carlberg, C. Bou-Mosleh, C. Farhat . Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. International Journal for Numerical Methods in Engineering, 86 (2010) (2), pp. 155--181. doi:10.1002/nme.3050

    2010 doi

  19. [27]

    Casenave

    F. Casenave . Accurate a posteriori error evaluation in the reduced basis method. Comptes Rendus Mathematique, 350 (2012) (9-10), pp. 539--542. doi:10.1016/j.crma.2012.05.012

    2012 doi

  20. [28]

    Casenave, A

    F. Casenave, A. Ern, T. Leli \` e vre . Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. ESAIM : Mathematical Modelling and Numerical Analysis, 48 (2014) (1), pp. 207--229. doi:10.1051/m2an/2013097

    2014

  21. [29]

    J. C \'e a . Approximation variationnelle des probl \`e mes aux limites . Université de Grenoble, 14 (1964) (fasc. 2), pp. 345--444

    1964

  22. [30]

    Y. Chen, J. S. Hesthaven, Y. Maday, J. Rodríguez . A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations. Comptes Rendus Mathematique, 346 (2008) (23), pp. 1295--1300. doi:10.1016/j.crma.2008.10.012

    2008 doi

  23. [31]

    Y. Chen, J. S. Hesthaven, Y. Maday, J. Rodríguez . Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell 's problem . ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009) (6), pp. 1099--1116. doi:10.105...

    2009

  24. [32]

    Y. Chen, J. S. Hesthaven, Y. Maday, J. Rodríguez . Certified Reduced Basis Methods and Output Bounds for the Harmonic Maxwell's Equations. SIAM Journal on Scientific Computing, 32 (2010) (2), pp. 970--996. doi:10.1137/09075250X

    2010 doi

  25. [33]

    Y. Chen, J. S. Hesthaven, Y. Maday . A seamless reduced basis element method for 2 D M axwell's problem: an introduction . In Spectral and high order methods for partial differential equations, vol. 76 of Lect. Notes Comput. Sci. Eng., pp. 141--152. Springer, Heidelberg, 2011....

    2011 doi

  26. [34]

    Y. Chen, J. Jiang, A. Narayan . Robust residual-based and residual-free greedy algorithms for reduced basis methods. arXiv e-prints, (2017). http://arxiv.org/abs/1710.08999v1

    2017 arXiv

  27. [35]

    Y. Chen, J. Jiang, A. Narayan . A robust error estimator and a residual-free error indicator for reduced basis methods. Computers & Mathematics with Applications, (2018). doi:10.1016/j.camwa.2018.11.032

    2018 doi

  28. [36]

    Z. Chen, J. J. Dongarra . Condition numbers of G aussian random matrices . SIAM Journal on Matrix Analysis and Applications, 27 (2005) (3), pp. 603--620. doi:10.1137/040616413

    2005 doi

  29. [37]

    Cheng, Z

    H. Cheng, Z. Gimbutas, P. G. Martinsson, V. Rokhlin . On the compression of low rank matrices. SIAM Journal on Scientific Computing, 26 (2005) (4), pp. 1389--1404. doi:10.1137/030602678

    2005 doi

  30. [38]

    E. T. Chung, Y. Efendiev, W. T. Leung . Generalized multiscale finite element methods for wave propagation in heterogeneous media. Multiscale Modeling and Simulation, 12 (2014) (4), pp. 1691--1721. doi:10.1137/130926675

    2014 doi

  31. [39]

    E. T. Chung, Y. Efendiev, C. S. Lee . Mixed generalized multiscale finite element methods and applications. Multiscale Modeling and Simulation, 13 (2015) (1), pp. 338--366. doi:10.1137/140970574

    2015 doi

  32. [40]

    E. T. Chung, Y. Efendiev, W. T. Leung . An adaptive generalized multiscale discontinuous G alerkin method for high-contrast flow problems . Multiscale Modeling and Simulation, 16 (2018 a ) (3), pp. 1227--1257. doi:10.1137/140986189. 1409.3474

    2018 arXiv

  33. [41]

    E. T. Chung, Y. Efendiev, W. T. Leung . Constraint Energy Minimizing Generalized Multiscale Finite Element Method. Computer Methods in Applied Mechanics and Engineering, 339 (2018 b ), pp. 298--319. doi:10.1016/j.cma.2018.04.010

    2018 doi

  34. [42]

    E. T. Chung, Y. Efendiev, W. T. Leung . Fast Online Generalized Multiscale Finite Element Method using Constraint Energy Minimization. Journal of Computational Physics, 355 (2018 c ), pp. 450--463. doi:10.1016/j.jcp.2017.11.022

    2018 doi

  35. [43]

    Cohen, R

    A. Cohen, R. DeVore, R. H. Nochetto . Convergence rates of AFEM with H^ -1 data . Foundations of Computational Mathematics, 12 (2012) (5), pp. 671--718. doi:10.1007/s10208-012-9120-1

    2012 doi

  36. [44]

    Dahmen, C

    W. Dahmen, C. Plesken, G. Welper . Double greedy algorithms: Reduced basis methods for transport dominated problems. ESAIM : Mathematical Modelling and Numerical Analysis, 48 (2014) (3), pp. 623--663. doi:10.1051/m2an/2013103

    2014

  37. [45]

    J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu . A supernodal approach to sparse partial pivoting. SIAM J. Matrix Analysis and Applications, 20 (1999) (3), pp. 720--755. doi:10.1137/S0895479895291765

    1999 doi

  38. [46]

    Dolean, H

    V. Dolean, H. Fol, S. Lanteri, R. Perrussel . Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods. Journal of Computational and Applied Mathematics, 218 (2008) (2), pp. 435--445. doi:10.1016/j.cam.2007.05.026. The Proceedings of the Twelfth Int...

    2008 doi

  39. [47]

    Dolean, F

    V. Dolean, F. Nataf, R. Scheichl, N. Spillane . Analysis of a Two-level Schwarz Method with Coarse Spaces Based on Local Dirichlet-to-Neumann Maps. Computational Methods in Applied Mathematics, 12 (2012) (4). doi:10.2478/cmam-2012-0027

    2012 doi

  40. [48]

    D \"o rfler

    W. D \"o rfler . A convergent adaptive algorithm for Poisson's equation. SIAM Journal on Numerical Analysis, 33 (1996) (3), pp. 1106--1124. doi:10.1137/0733054

    1996 doi

  41. [49]

    Drineas, R

    P. Drineas, R. Kannan, M. Mahoney . Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication. SIAM Journal on Computing, 36 (2006 a ) (1), pp. 132--157. doi:10.1137/S0097539704442684

    2006 doi

  42. [50]

    Drineas, R

    P. Drineas, R. Kannan, M. Mahoney . Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix. SIAM Journal on Computing, 36 (2006 b ) (1), pp. 158--183. doi:10.1137/S0097539704442696

    2006 doi

  43. [51]

    Drineas, R

    P. Drineas, R. Kannan, M. W. Mahoney . Fast M onte C arlo Algorithms for Matrices. III . C omputing a compressed approximate matrix decomposition . SIAM Journal on Computing, 36 (2006 c ) (1), pp. 184--206. doi:10.1137/S0097539704442702

    2006 doi

  44. [52]

    W. E, B. Engquist . The Heterognous Multiscale Methods. Communications in Mathematical Sciences, 1 (2003) (1), pp. 87--132. doi:10.4310/cms.2003.v1.n1.a8

    2003 doi

  45. [53]

    W. E, B. Engquist . The Heterogeneous Multi-Scale Method for Homogenization Problems. In Lecture Notes in Computational Science and Engineering, vol. 44 of Lect. Notes Comput. Sci. Eng., pp. 89--110. Springer-Verlag, Berlin, 2005. doi:10.1007/3-540-26444-2_4

    doi

  46. [54]

    Efendiev, T

    Y. Efendiev, T. Y. Hou . Multiscale Finite Element Methods: Theory and Applications (Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 4), vol. 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York, 2009. ISBN 0387094954. Theory and...

    2009

  47. [55]

    Efendiev, J

    Y. Efendiev, J. Galvis, X.-H. Wu . Multiscale finite element methods for high-contrast problems using local spectral basis functions. Journal of Computational Physics, 230 (2011) (4), pp. 937--955. doi:10.1016/j.jcp.2010.09.026

    2011 doi

  48. [56]

    Efendiev, J

    Y. Efendiev, J. Galvis, T. Y. Hou . Generalized Multiscale Finite Element Methods ( GMsFEM ) . Journal of Computational Physics, 251 (2013), pp. 116--135. doi:10.1016/j.jcp.2013.04.045

    2013 doi

  49. [57]

    J. L. Eftang, A. T. Patera . Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. International Journal for Numerical Methods in Engineering, 96 (2013) (5), pp. 269--302. doi:10.1002/nme.4543

    2013 doi

  50. [58]

    J. L. Eftang, 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, 1 (2014) (3), p. 3. doi:10.1186/...

    2014 doi

  51. [59]

    Eickhorn

    D. Eickhorn . Randomisierte lokalisierte Modellreduktion mit Robin-Transferoperator. Master's thesis, Westfälische Wilhelms Universität Münster, 2019. https://www.uni-muenster.de/AMM/includes/ohlberger/team/eickhorn/MA_Eickhorn.pdf

    2019

  52. [60]

    Engwer, P

    C. Engwer, P. Henning, A. Målqvist, D. Peterseim . Efficient implementation of the localized orthogonal decomposition method. Computer Methods in Applied Mechanics and Engineering, 350 (2019), pp. 123--153. doi:10.1016/j.cma.2019.02.040

    2019 doi

  53. [61]

    Feuers \"a nger

    C. Feuers \"a nger . Manual for Package pgfplots. online, 17 (2011)

    2011

  54. [62]

    J. P. Fink, W. C. Rheinboldt . On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f \"u r Angewandte Mathematik und Mechanik, 63 (1983) (1), pp. 21--28. doi:10.1002/zam...

    1983 doi

  55. [63]

    S. Fu, E. Chung, G. Li . Edge Multiscale Methods for elliptic problems with heterogeneous coefficients. arXiv, (2018). http://arxiv.org/abs/1810.10398v1

    2018 arXiv

  56. [64]

    Galvis, Y

    J. Galvis, Y. Efendiev . Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Modeling & Simulation, 8 (2010) (4), pp. 1461--1483. doi:10.1137/090751190

    2010 doi

  57. [65]

    M. J. Gander, A. Loneland . S HEM : an optimal coarse space for RAS and its multiscale approximation . In Domain decomposition methods in science and engineering XXIII , vol. 116 of Lecture Notes in Computational Science and Engineering, pp. 313--321. Springer, Cham, 2017. doi...

    2017 doi

  58. [66]

    Graham, R

    I. Graham, R. Scheichl . Robust domain decomposition algorithms for multiscale PDEs . Numerical Methods for Partial Differential Equations, 23 (2007) (4), pp. 859--878. doi:10.1002/num.20254

    2007 doi

  59. [67]

    I. G. Graham, P. O. Lechner, R. Scheichl . Domain decomposition for multiscale PDE s . Numerische Mathematik, 106 (2007) (4), pp. 589--626. doi:10.1007/s00211-007-0074-1

    2007 doi

  60. [68]

    J. P. Gram . Ueber die Entwickelung reeller Functionen in Reihen mittelst der Methode der kleinsten Quadrate. Journal für die reine und angewandte Mathematik, 94 (1883), pp. 41--73

  61. [69]

    Guennebaud, B

    G. Guennebaud, B. Jacob, et al. Eigen v3. http://eigen.tuxfamily.org, 2010

    2010

  62. [70]

    Haasdonk

    B. Haasdonk . Chapter 2: Reduced Basis Methods for Parametrized PDEs A Tutorial Introduction for Stationary and Instationary Problems . In Model Reduction and Approximation, pp. 65--136. Society for Industrial and Applied Mathematics, 2017. doi:10.1137/1.9781611974829.ch2

    2017 doi

  63. [71]

    Halko, P.-G

    N. Halko, P.-G. Martinsson, J. A. Tropp . Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53 (2011) (2), pp. 217--288. doi:10.1137/090771806

    2011 doi

  64. [72]

    Heaviside

    O. Heaviside . Electrical Papers. Macmilland And Company, 1892

  65. [73]

    Heinlein, A

    A. Heinlein, A. Klawonn, J. Knepper, O. Rheinbach . Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2D. Electronic Transactions on Numerical Analysis, 48 (2018), pp. 156--182. doi:10.1553/etna_vol48s156

    2018 doi

  66. [74]

    Henning, D

    P. Henning, D. Peterseim . Oversampling for the Multiscale Finite Element Method. Multiscale Modeling & Simulation, 11 (2013) (4), pp. 1149--1175. doi:10.1137/120900332

    2013 doi

  67. [75]

    Henning, A

    P. Henning, A. M lqvist, D. Peterseim . A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM : Mathematical Modelling and Numerical Analysis, 48 (2014 a ) (5), pp. 1331--1349. doi:10.1051/m2an/2013141

    2014

  68. [76]

    Henning, M

    P. Henning, M. Ohlberger, B. Schweizer . An Adaptive Multiscale Finite Element Method. Multiscale Modeling & Simulation, 12 (2014 b ) (3), pp. 1078--1107. doi:10.1137/120886856

    2014 doi

  69. [77]

    M. W. Hess . Reduced Basis Approximations for Electromagnetic Applications. Ph.D. thesis, Otto-von-Guericke Universit \"a t Magdeburg, 2016

    2016

  70. [78]

    J. S. Hesthaven, G. Rozza, B. Stamm . Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics, Springer International Publishing, New York, 2016. ISBN 978-3-319-22469-5; 978-3-319-22470-1. doi:10.1007/978-3-319-22470-1. BC...

    2016 doi

  71. [79]

    Hetmaniuk, A

    U. Hetmaniuk, A. Klawonn . Error estimates for a two-dimensional special finite element method based on component mode synthesis. Electronic Transactions on Numerical Analysis, 41 (2014), pp. 109--132. doi:n/a

    2014

  72. [80]

    U. L. Hetmaniuk, R. B. Lehoucq . A special finite element method based on component mode synthesis. ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010) (3), pp. 401--420. doi:10.1051/m2an/2010007

    2010

  73. [81]

    R. A. Horn, C. R. Johnson . Matrix Analysis. Cambridge University Press, 2012. ISBN 978-0521548236

    2012

  74. [82]

    T. Y. Hou, X.-H. Wu . A Multiscale Finite Element Method For Elliptic Problems In Composite Materials And Porous Media. Journal of Computational Physics, 134 (1997) (1), pp. 169--189. doi:10.1006/jcph.1997.5682

    1997

  75. [83]

    T. J. Hughes . Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Computer Methods in Applied Mechanics and Engineering, 127 (1995) (1--4), pp. 387--401. doi:10.1016/0045-7825(95)00844-9

    1995 doi

  76. [84]

    T. J. Hughes, G. R. Feijóo, L. Mazzei, J.-B. Quincy . The variational multiscale method --- a paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering, 166 (1998) (1), pp. 3--24. doi:10.1016/S0045-7825(98)00079-6

    1998 doi

  77. [85]

    W. C. Hurty . Dynamic analysis of structural systems using component modes. AIAA journal, 3 (1965) (4), pp. 678--685. doi:10.2514/3.2947

    1965 doi

  78. [86]

    Huynh, G

    D. Huynh, G. Rozza, S. Sen, A. Patera . A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants. Comptes Rendus Mathematique, 345 (2007) (8), pp. 473--478. doi:10.1016/j.crma.2007.09.019

    2007 doi

  79. [87]

    Iapichino

    L. Iapichino . Reduced basis methods for the solution of parametrized PDEs in repetitive and complex networks with application to CFD. Ph.D. thesis, École polytechnique fédérale de Lausanne, 2012. doi:10.5075/epfl-thesis-5529

    2012 doi

  80. [88]

    Iapichino, A

    L. Iapichino, A. Quarteroni, G. Rozza . A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Computer Methods in Applied Mechanics and Engineering, 221-222 (2012), pp. 63--82. doi:10.1016/j.cma.2012.02.005

    2012 doi

  81. [89]

    Iapichino, A

    L. Iapichino, A. Quarteroni, G. Rozza . Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries. Computers & Mathematics with Applications, 71 (2016) (1), pp. 408--430. doi:10.1016/j.camwa.2015.12.001

    2016 doi

  82. [90]

    Ihlenburg

    F. Ihlenburg . Finite element analysis of acoustic scattering, vol. 132. Springer Science & Business Media, 2006. doi:10.1007/b98828

    2006 doi

  83. [91]

    Jakobsson, F

    H. Jakobsson, F. Bengzon, M. G. Larson . Adaptive component mode synthesis in linear elasticity. International Journal for Numerical Methods in Engineering, 86 (2011) (7), pp. 829--844

    2011

  84. [92]

    Kaulmann, M

    S. Kaulmann, M. Ohlberger, B. Haasdonk . A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems. Comptes Rendus Mathematique, 349 (2011) (23), pp. 1233--1238. doi:10.1016/j.crma.2011.10.024

    2011 doi

  85. [93]

    Kerfriden, O

    P. Kerfriden, O. Goury, T. Rabczuk, S. Bordas . A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics. Computer Methods in Applied Mechanics and Engineering, 256 (2013), pp. 169--188. doi:10.1016/j.cma.2012.12.004

    2013 doi

  86. [94]

    Klawonn, O

    A. Klawonn, O. B. Widlund . Dual-primal FETI methods for linear elasticity . Communications on Pure and Applied Mathematics, 59 (2006) (11), pp. 1523--1572. doi:10.1002/cpa.20156

    2006 doi

  87. [95]

    Klawonn, P

    A. Klawonn, P. Radtke, O. Rheinbach . A comparison of adaptive coarse spaces for iterative substructuring in two dimensions. Electronic Transactions on Numerical Analysis, 45 (2016), pp. 75--106

    2016

  88. [96]

    Kornhuber, H

    R. Kornhuber, H. Yserentant . Numerical homogenization of elliptic multiscale problems by subspace decomposition. Multiscale Modeling and Simulation, 14 (2016) (3), pp. 1017--1036. doi:10.1137/15M1028510

    2016 doi

  89. [97]

    Kornhuber, J

    R. Kornhuber, J. Podlesny, H. Yserentant . Direct and iterative methods for numerical homogenization. In Domain decomposition methods in science and engineering XXIII , vol. 116 of Lecture Notes in Computational Science and Engineering, pp. 217--225. Springer International Pub...

    2017 doi

  90. [98]

    Kornhuber, D

    R. Kornhuber, D. Peterseim, H. Yserentant . An analysis of a class of variational multiscale methods based on subspace decomposition. Mathematics of Computation, 87 (2018) (314), pp. 2765--2774. doi:10.1090/mcom/3302

    2018 doi

  91. [99]

    M. G. Larson, A. M lqvist . Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimation: Duality Techniques for Elliptic Problems. In Lecture Notes in Computational Science and Engineering, vol. 44 of Lect. Notes Comput. Sci. Eng., pp. 181--193. Springer, Be...

    2005 doi

  92. [100]

    M. G. Larson, A. M lqvist . Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems. Computer Methods in Applied Mechanics and Engineering, 196 (2007) (21-24), pp. 2313--2324. doi:10.1016/j.cma.2006.08.019

    2007 doi

  93. [101]

    M. G. Larson, A. M lqvist . An adaptive variational multiscale method for convection-diffusion problems. Communications in Numerical Methods in Engineering, 25 (2009 a ) (1), pp. 65--79. doi:10.1002/cnm.1106

    2009 doi

  94. [102]

    M. G. Larson, A. M lqvist . A mixed adaptive variational multiscale method with applications in oild reservoid simulation. Mathematical Models and Methods in Applied Sciences, 19 (2009 b ) (07), pp. 1017--1042. doi:10.1142/s021820250900370x

    2009 doi

  95. [103]

    P. Lax, A. Milgram . Contributions to the Theory of Partial Differential Equations, vol. 33, chap. Ix. parabolic equations, pp. 167--190. Princeton University Press, 1954

    1954

  96. [104]

    M. A. Lee . An approach for efficient, conceptual-level aerospace structural design using the static condensation reduced basis element method. Ph.D. thesis, Georgia Institute of Technology, 2018

    2018

  97. [105]

    R. B. Lehoucq, D. C. Sorensen, C. Yang . ARPACK Users' Guide . online, 6 (1998), pp. xvi+142. doi:10.1137/1.9780898719628

    1998 doi

  98. [106]

    X. Li, J. Demmel, J. Gilbert, iL. Grigori, M. Shao, I. Yamazaki . SuperLU Users' Guide . Tech. Rep. LBNL-44289, Lawrence Berkeley National Laboratory, 1999. http://crd.lbl.gov/ xiaoye/SuperLU/. Last update: August 2011

    1999

  99. [107]

    Liberty, F

    E. Liberty, F. Woolfe, P.-G. Martinsson, V. Rokhlin, M. Tygert . Randomized algorithms for the low-rank approximation of matrices. Proceedings of the National Academy of Sciences, 104 (2007) (51), pp. 20167--20172. doi:10.1073/pnas.0709640104

    2007 doi

  100. [108]

    A. E. L vgren, Y. Maday, E. M. R nquist . A reduced basis element method for the steady Stokes problem. ESAIM : Mathematical Modelling and Numerical Analysis, 40 (2006) (3), pp. 529--552. doi:10.1051/m2an:2006021

    2006 doi

  101. [109]

    Maday, E

    Y. Maday, E. M. R nquist . A reduced-basis element method. Journal of scientific computing, 17 (2002) (1-4), pp. 447--459. doi:10.1023/a:1015197908587

    2002 doi

  102. [110]

    Maday, E

    Y. Maday, E. M. R nquist . The Reduced Basis Element Method: Application to a Thermal Fin Problem. SIAM Journal on Scientific Computing, 26 (2004) (1), pp. 240--258 (electronic). doi:10.1137/s1064827502419932

    2004 doi

  103. [111]

    Maier, B

    I. Maier, B. Haasdonk . A D irichlet- N eumann reduced basis method for homogeneous domain decomposition problems . Applied Numerical Mathematics, 78 (2014), pp. 31--48. doi:10.1016/j.apnum.2013.12.001

    2014 doi

  104. [112]

    M lqvist, D

    A. M lqvist, D. Peterseim . Localization of elliptic multiscale problems. Mathematics of Computation, 83 (2014) (290), pp. 2583--2603. doi:10.1090/s0025-5718-2014-02868-8

    2014 doi

  105. [113]

    Mandel, B

    J. Mandel, B. Soused\' i k . Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI - DP iterative substructuring methods . Computer Methods in Applied Mechanics and Engineering, 196 (2007) (8), pp. 1389--1399. doi:10.1016/j.cma.2006.03.010

    2007 doi

  106. [114]

    Martini, G

    I. Martini, G. Rozza, B. Haasdonk . Reduced basis approximation and a-posteriori error estimation for the coupled S tokes- D arcy system . Advances in Computational Mathematics, 41 (2014) (5), pp. 1131--1157. doi:10.1007/s10444-014-9396-6

    2014 doi

  107. [115]

    Martini, G

    I. Martini, G. Rozza, B. Haasdonk . Certified Reduced Basis Approximation for the Coupling of Viscous and Inviscid Parametrized Flow Models. Journal of Scientific Computing, (2017). doi:10.1007/s10915-017-0430-y

    2017 doi

  108. [116]

    Martinsson, V

    P.-G. Martinsson, V. Rokhlin, M. Tygert . A randomized algorithm for the decomposition of matrices. Applied and Computational Harmonic Analysis, 30 (2011) (1), pp. 47--68. doi:10.1016/j.acha.2010.02.003

    2011 doi

  109. [117]

    J. C. Maxwell . On physical lines of force. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21 (1861) (139), pp. 161--175. doi:10.1080/14786431003659180

    doi

  110. [118]

    J. C. Maxwell . A dynamical theory of the electromagnetic field. Proceedings of the Royal Society of London, 13 (1865), pp. 531--536. doi:10.1098/rspl.1863.0098

  111. [119]

    J. M. Melenk, I. Babu s ka . The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139 (1996) (1), pp. 289--314. doi:10.1016/s0045-7825(96)01087-0

    1996 doi

  112. [120]

    J. M. Melenk, K. Gerdes, C. Schwab . Fully discrete hp-finite elements: fast quadrature. Computer Methods in Applied Mechanics and Engineering, 190 (2001) (32--33), pp. 4339--4364. doi:10.1016/S0045-7825(00)00322-4

    2001 doi

  113. [121]

    R. Milk, S. Rave, F. Schindler . py MOR -- G eneric A lgorithms and I nterfaces for M odel O rder R eduction . SIAM Journal on Scientific Computing, 38 (2016) (5), pp. S194--S216. doi:10.1137/15M1026614

    2016 doi

  114. [122]

    R. Milk, F. T. Schindler, T. Leibner . Extending DUNE: The dune-xt modules. Archive of Numerical Software, Vol 5, No 1 (2017), (2017). doi:10.11588/ans.2017.1.27720

    2017 doi

  115. [123]

    P. Monk . Finite element methods for Maxwell's equations. Oxford University Press, 2003. ISBN 9780191545221

    2003

  116. [124]

    Nataf, H

    F. Nataf, H. Xiang, V. Dolean . A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps. Comptes Rendus Mathematique, 348 (2010) (21-22), pp. 1163--1167. doi:10.1016/j.crma.2010.10.007

    2010 doi

  117. [125]

    J. C. Nedelec . Mixed finite elements in R^3 . Numerische Mathematik, 35 (1980) (3), pp. 315--341. doi:10.1007/bf01396415

    1980 doi

  118. [126]

    J. C. Nedelec . A new family of mixed finite elements in R^3 . Numerische Mathematik, 50 (1986) (1), pp. 57--81. doi:10.1007/bf01389668

    1986 doi

  119. [127]

    Nethercote, J

    N. Nethercote, J. Seward . Valgrind. ACM SIGPLAN Notices, 42 (2007) (6), p. 89. doi:10.1145/1273442.1250746

    2007

  120. [128]

    J. Nečas . Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 16 (1962) (4), pp. 305--326

    1962

  121. [129]

    A. K. Noor, J. M. Peters . Reduced basis technique for nonlinear analysis of structures. AIAA Journal, 18 (1980) (4), pp. 455--462

    1980

  122. [130]

    Ohlberger

    M. Ohlberger . A Posteriori Error Estimates for the Heterogeneous Multiscale Finite Element Method for Elliptic Homogenization Problems. Multiscale Modeling & Simulation, 4 (2005) (1), pp. 88--114. doi:10.1137/040605229

    2005 doi

  123. [131]

    Ohlberger, S

    M. Ohlberger, S. Rave . Localized Reduced Basis Approximation of a Nonlinear Finite Volume Battery Model with Resolved Electrode Geometry. In Model Reduction of Parametrized Systems, vol. 17 of MS&A. Modeling, Simulation and Applications, pp. 201--212. Springer International P...

    2017 doi

  124. [132]

    Ohlberger, F

    M. Ohlberger, F. Schindler . A-Posteriori Error Estimates for the Localized Reduced Basis Multi-Scale Method, vol. 77 of Springer Proc. Math. Stat., pp. 421--429. Springer International Publishing, Cham. ISBN 978-3-319-05684-5, 2014. doi:10.1007/978-3-319-05684-5_41

    2014 doi

  125. [133]

    Ohlberger, F

    M. Ohlberger, F. Schindler . Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment. SIAM Journal on Scientific Computing, 37 (2015) (6), pp. A2865--A2895. doi:10.1137/151003660

    2015 doi

  126. [134]

    Ohlberger, F

    M. Ohlberger, F. Schindler . Non-conforming Localized Model Reduction with Online Enrichment: Towards Optimal Complexity in PDE Constrained Optimization . In Springer Proceedings in Mathematics & Statistics , vol. 200 of Springer Proc. Math. Stat., pp. 357--365. Springer Inter...

    2017 doi

  127. [135]

    Ohlberger, B

    M. Ohlberger, B. Verfürth . A New Heterogeneous Multiscale Method for the Helmholtz Equation with High Contrast. Multiscale Modeling and Simulation, 16 (2018) (1), pp. 385--411. doi:10.1137/16m1108820

    2018 doi

  128. [136]

    Ohlberger, M

    M. Ohlberger, M. Schaefer, F. Schindler . Localized Model Reduction in PDE Constrained Optimization . In V. Schulz, D. Seck , eds., Shape Optimization, Homogenization and Optimal Control, pp. 143--163. Springer International Publishing, Cham. ISBN 978-3-319-90469-6, 2018. doi:...

    2018 doi

  129. [137]

    K. B. lgaard, G. N. Wells . Optimisations for Quadrature Representations of Finite Element Tensors Through Automated Code Generation. ACM Transactions on Mathematical Software, 37 (2010). doi:10.1145/1644001.1644009

    2010

  130. [138]

    Project files for Olimex A64 PCB, 2019

    Olimex . Project files for Olimex A64 PCB, 2019. doi:10.5281/zenodo.2547973

    2019 doi

  131. [139]

    Pacciarini

    P. Pacciarini . Discontinuous Galerkin Reduced Basis Element Methods for Parametrized Partial Differential Equations in Partitioned Domains. Ph.D. thesis, Politecnico Di Milano, Milan, Italy, 2016

    2016

  132. [140]

    Pacciarini, P

    P. Pacciarini, P. Gervasio, A. Quarteroni . Spectral based discontinuous G alerkin reduced basis element method for parametrized S tokes problems . Computers & Mathematics with Applications, 72 (2016) (8), pp. 1977--1987. doi:10.1016/j.camwa.2016.01.030

    2016 doi

  133. [141]

    Patera, G

    A. Patera, G. Rozza . Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Version 1.0, Copyright MIT (2006-2007), to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering., 2007

    2006

  134. [142]

    Pechstein, R

    C. Pechstein, R. Scheichl . Analysis of FETI methods for multiscale PDEs . Numerische Mathematik, 111 (2008) (2), pp. 293--333. doi:10.1007/s00211-008-0186-2

    2008 doi

  135. [143]

    D. B. Phuong Huynh, D. J. Knezevic, A. T. Patera . A Static condensation Reduced Basis Element method: approximation and a posteriori error estimation. ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013) (1), pp. 213--251. doi:10.1051/m2an/2012022

    2013

  136. [144]

    Quarteroni, A

    A. Quarteroni, A. Manzoni, F. Negri . Reduced Basis Methods for Partial Differential Equations, vol. 92 of La Matematica per il 3+2. Springer, 2016. ISBN 978-3-319-15430-5; 978-3-319-15431-2. doi:10.1007/978-3-319-15431-2. An introduction, La Matematica per il 3+2

    2016 doi

  137. [145]

    Sangalli

    G. Sangalli . Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method. Multiscale Modeling & Simulation, 1 (2003) (3), pp. 485--503. doi:10.1137/s1540345902411402

    2003 doi

  138. [146]

    E. Schmidt . Zur Theorie der linearen und nichtlinearen Integralgleichungen. Mathematische Annalen, 63 (1907), pp. 433--476

    1907

  139. [147]

    K. Smetana . A new certification framework for the port reduced static condensation reduced basis element method. Computer Methods in Applied Mechanics and Engineering, 283 (2015), pp. 352--383. doi:10.1016/j.cma.2014.09.020

    2015 doi

  140. [148]

    Smetana, A

    K. Smetana, A. T. Patera . Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures. SIAM Journal on Scientific Computing, 38 (2016) (5), pp. A3318--A3356. doi:10.1137/15M1009603

    2016 doi

  141. [149]

    Smetana, O

    K. Smetana, O. Zahm, A. T. Patera . Randomized Residual-Based Error Estimators for Parametrized Equations. SIAM Journal on Scientific Computing, 41 (2019) (2), pp. A900--A926. doi:10.1137/18m120364x

    2019 doi

  142. [150]

    Sommer, O

    A. Sommer, O. Farle, R. Dyczij-Edlinger . A New Method for Accurate and Efficient Residual Computation in Adaptive Model-Order Reduction. IEEE Transactions on Magnetics, 51 (2015) (3), pp. 1--4. doi:10.1109/tmag.2014.2352812

    2015

  143. [151]

    Spillane, V

    N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl . Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps . Numerische Mathematik, 126 (2014) (4), pp. 741--770. doi:10.1007/s00211-013-0576-y

    2014 doi

  144. [152]

    Strouboulis, K

    T. Strouboulis, K. Copps, I. Babu s ka . The generalized finite element method. Computer Methods in Applied Mechanics and Engineering, 190 (2001) (32), pp. 4081--4193. doi:10.1016/S0045-7825(01)00188-8

    2001 doi

  145. [153]

    Taddei, A

    T. Taddei, A. T. Patera . A Localization Strategy for Data Assimilation; Application to State Estimation and Parameter Estimation. SIAM Journal on Scientific Computing, 40 (2018) (2), pp. B611--B636. doi:10.1137/17m1116830

    2018 doi

  146. [154]

    Toselli, O

    A. Toselli, O. Widlund . Domain decomposition methods---algorithms and theory, vol. 34 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2005. ISBN 3-540-20696-5. doi:10.1007/b137868

    2005 doi

  147. [155]

    Travis E

    O. Travis E . A guide to NumPy. Trelgol Publishing, 2006

    2006

  148. [156]

    Voronin, P.-G

    S. Voronin, P.-G. Martinsson . RSVDPACK: An implementation of randomized algorithms for computing the singular value, interpolative, and CUR decompositions of matrices on multi-core and GPU architectures . Tech. rep., arXiv:1502.05366, 2015. doi:n/a

    2015 arXiv

  149. [157]

    Weinan, B

    E. Weinan, B. Engquist . Multiscale modeling and computation. Notices of the AMS, 50 (2003) (9), pp. 1062--1070. doi:n/a

    2003

  150. [158]

    H. Whitney . Geometric integration theory. Princeton Univ. Press, 1957

    1957

  151. [159]

    J. Xu, L. Zikatanov . Some observations on Babu s ka and Brezzi theories . Numerische Mathematik, 94 (2003) (1), pp. 195--202. doi:10.1007/s002110100308

    2003 doi

  152. [160]

    M. Yano . A Space-Time Petrov--Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations. SIAM Journal on Scientific Computing, 36 (2014) (1), pp. A232--A266. doi:10.1137/120903300

    2014 doi

  153. [161]

    Zaglmayr

    S. Zaglmayr . High Order Finite Elements for Electromagnetic Field Computation. Ph.D. thesis, Universität Linz, 2006

    2006

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.