Generalization of the Neville-Aitken Interpolation Algorithm on Grassmann Manifolds : Applications to Reduced Order Model

Abdallah El Hamidi; Antoine Falaize; Aziz Hamdouni; Rolando Mosquera

arxiv: 1907.02831 · v1 · pith:TPDZQC3Enew · submitted 2019-07-05 · 🧮 math.NA · cs.NA· math-ph· math.MP

Generalization of the Neville-Aitken Interpolation Algorithm on Grassmann Manifolds : Applications to Reduced Order Model

Rolando Mosquera , Abdallah El Hamidi , Aziz Hamdouni , Antoine Falaize This is my paper

Pith reviewed 2026-05-25 01:58 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath-phmath.MP

keywords Grassmann manifoldNeville-Aitken algorithmPOD subspace interpolationreduced order modelgeodesic barycenterparametric PDECFD applications

0 comments

The pith

The Neville-Aitken algorithm extends to Grassmann manifolds by replacing linear averages with recursive geodesic barycenters of POD subspaces.

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

The paper generalizes the classical Neville-Aitken interpolation procedure from vector spaces to the Grassmann manifold of subspaces. Interpolation points are the POD subspaces extracted from solutions of parametric evolution equations at selected parameter values. The extension replaces each linear combination step with the geodesic barycenter between two points, computed recursively in the same tableau structure. The resulting method is applied to three CFD test cases to build reduced-order models at new parameter values. A sympathetic reader would care because the approach promises to obtain accurate flow reconstructions without recomputing the full-order solution at every desired parameter.

Core claim

The Neville-Aitken algorithm is generalized to the Grassmann manifold by performing interpolation recursively via the geodesic barycenter of two points, where the points are the subspaces spanned by the POD bases of the available solutions corresponding to chosen parameter values.

What carries the argument

Recursive geodesic barycenter on the Grassmann manifold, which replaces the linear averaging step of the classical Neville-Aitken tableau.

If this is right

Parametric reduced-order models for the tested flows can be obtained at new parameter values by interpolating only a small set of precomputed POD bases.
The interpolated subspaces produce accurate reconstructions for the Von Karman vortex street, lid-driven cavity with inflow, and rotating-solid flow.
Computation time is reduced relative to solving the full PDE at every new parameter value.
The same recursive structure applies to any set of parameter-dependent POD subspaces whose variation lies on the Grassmann manifold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same geodesic replacement could be tried with other classical polynomial interpolation schemes on the Grassmann manifold.
If the geodesic path matches the actual subspace evolution only locally, accuracy will degrade for large parameter excursions.
The approach suggests testing whether Euclidean interpolation of the basis matrices themselves yields comparable errors on these CFD problems.

Load-bearing premise

The variation of POD subspaces with the chosen parameters can be adequately captured by geodesic interpolation on the Grassmann manifold without significant loss of fidelity in the reconstructed flow fields.

What would settle it

Direct numerical comparison at an intermediate parameter value showing that the flow fields reconstructed from the interpolated POD subspace differ substantially in norm from the full-order solution.

read the original abstract

The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding street, the lid-driven cavity with inflow and the flow induced by a rotating solid. The obtained numerical simulations are pertinent both in terms of the accuracy of results and the time computation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends Neville-Aitken interpolation to Grassmann manifolds by recursive geodesic barycenters on POD subspaces and tests the idea on three parametric CFD problems.

read the letter

The paper's main contribution is extending the Neville-Aitken interpolation algorithm to Grassmann manifolds by performing the recursion with geodesic barycenters of subspaces. This is applied to interpolating POD bases for reduced-order models in parametric flows. They test it on three cases: von Karman vortex shedding, lid-driven cavity with inflow, and flow from a rotating solid. The results are presented as accurate with acceptable compute time. This gives a concrete tool for someone who already has a few full-order solutions at different parameters and wants to generate bases at new parameter values without extra full simulations. The recursive structure mirrors the original algorithm, which is a nice touch. The soft spot is that we don't see any derivation of why the geodesic barycenter is the right choice or any error estimates. The central assumption that the subspaces vary in a way that geodesic interpolation works well could fail if the parameter dependence has topology changes or sharp turns. The stress-test concern holds if the paper doesn't show comparisons where the interpolated model is checked against a held-out full simulation at an intermediate parameter. If the full text has the implementation details and the accuracy metrics are solid, that would strengthen it. As is, the evidence is the three examples. This paper is for researchers in numerical methods for parametric PDEs who use manifold techniques for ROMs. It could be useful for someone looking for a simple interpolation scheme on the Grassmannian. I think it deserves peer review because it offers a new algorithmic variant with practical demonstrations, even if more analysis would help.

Referee Report

3 major / 2 minor

Summary. The paper extends the Neville-Aitken interpolation algorithm to Grassmann manifolds for parametric reduced-order modeling. POD subspaces corresponding to sampled parameter values serve as interpolation points on the manifold; the algorithm is generalized by performing recursive interpolation via the geodesic barycenter of pairs of points. The approach is demonstrated on three CFD test cases (Von Karman vortex street, lid-driven cavity with inflow, rotating solid flow), with the abstract asserting that the resulting simulations are accurate and computationally efficient.

Significance. A rigorously derived and verified manifold interpolation method for POD bases could reduce the cost of parametric ROM construction in CFD by avoiding repeated full-order solves and basis recomputations. The numerical examples indicate practical relevance, but the absence of any derivation, error analysis, or quantitative verification prevents a positive assessment of significance at present.

major comments (3)

[Abstract] Abstract: the central claim that the extended algorithm 'produces accurate results' on the three test cases is unsupported because the manuscript supplies neither the explicit recursive formula for the Grassmannian Neville-Aitken step nor any derivation showing that repeated geodesic barycenters reproduce the original algorithm's properties.
[Method] Method section (implied by abstract description): the assumption that POD-subspace variation with parameters can be captured by geodesic interpolation is load-bearing for the claim of fidelity in reconstructed fields, yet no analysis or counter-example test is provided to bound the deviation from true parametric paths on the Grassmannian.
[Numerical results] Numerical results: the three CFD applications are presented as demonstrating accuracy and efficiency, but no error norms, convergence rates with respect to number of interpolation points, or comparison against direct POD recomputation at the target parameters are reported, leaving the weakest assumption (geodesic fidelity) untested.

minor comments (2)

Define the Riemannian metric and geodesic explicitly when first introducing the Grassmann manifold operations.
Clarify whether the reported time savings include the cost of the manifold interpolation itself or only the subsequent ROM solves.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major point below, indicating revisions where the manuscript will be updated to strengthen the presentation and evidence.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the extended algorithm 'produces accurate results' on the three test cases is unsupported because the manuscript supplies neither the explicit recursive formula for the Grassmannian Neville-Aitken step nor any derivation showing that repeated geodesic barycenters reproduce the original algorithm's properties.

Authors: We agree that the abstract claim would benefit from clearer linkage to the method. The revised manuscript will include the explicit recursive formula for the Grassmannian Neville-Aitken step (based on successive geodesic barycenters) in the method section and add a short derivation paragraph explaining the generalization from the Euclidean case, including preservation of the interpolation property at the nodes. The abstract will be revised to reference these elements more precisely. revision: yes
Referee: [Method] Method section (implied by abstract description): the assumption that POD-subspace variation with parameters can be captured by geodesic interpolation is load-bearing for the claim of fidelity in reconstructed fields, yet no analysis or counter-example test is provided to bound the deviation from true parametric paths on the Grassmannian.

Authors: The geodesic interpolation assumption follows from the Riemannian geometry of the Grassmann manifold and is consistent with prior manifold-based ROM literature. We acknowledge the value of bounding the deviation; the revision will add a dedicated paragraph in the method section discussing the approximation properties (citing relevant references on Grassmann geodesics) and note the conditions under which the assumption holds, along with a brief remark on potential limitations. revision: partial
Referee: [Numerical results] Numerical results: the three CFD applications are presented as demonstrating accuracy and efficiency, but no error norms, convergence rates with respect to number of interpolation points, or comparison against direct POD recomputation at the target parameters are reported, leaving the weakest assumption (geodesic fidelity) untested.

Authors: We will expand the numerical results section to report quantitative error norms (e.g., relative reconstruction errors in velocity/pressure fields), convergence behavior as the number of interpolation points increases, and direct comparisons against POD bases computed at the target parameters for at least one test case. This will provide explicit verification of the geodesic interpolation fidelity. revision: yes

Circularity Check

0 steps flagged

No circularity; algorithmic extension is self-contained

full rationale

The paper describes a direct generalization of the Neville-Aitken recursion to Grassmann manifolds via geodesic barycenters of POD subspaces, with numerical illustrations on three CFD cases. No equations or procedures reduce a claimed prediction or result to a fitted input or self-referential definition by construction. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear in the derivation. The central claim is an independent algorithmic construction whose validity is assessed externally through accuracy on the test applications rather than by tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract supplies insufficient detail to enumerate free parameters or invented entities; the method rests on the standard domain assumption that POD bases form points on the Grassmann manifold.

axioms (1)

domain assumption POD bases of parametric solutions lie on the Grassmann manifold and their parameter dependence can be interpolated geodesically
Invoked implicitly when the authors state that interpolation points are subspaces spanned by POD bases

pith-pipeline@v0.9.0 · 5664 in / 1120 out tokens · 21707 ms · 2026-05-25T01:58:03.943898+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The well-known Neville-Aitken’s algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

J(y) = 1/2 [(1−α[i,j](λ)) d²(y,yi) + α[i,j](λ) d²(y,yj)]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

P. A. Absil, R. Mahony, R. Sepulchre,Riemannian geometry of Grassmann manifolds with a view on algorithmic computation, Acta Applicandae Mathematicae, 80 (2) (2004) 199-220

work page 2004
[2]

A. C. Aitken,On interpolation by iteration of proportional parts, without the use of diﬀer- ences, Proc. Edinburgh Math. Soc. Ser. 2, 3 (1932) 56-76

work page 1932
[3]

Akkari, A

N. Akkari, A. Hamdouni, E. Liberge, M. Jazar,A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM–POD), for a 2D incompressible ﬂuid ﬂow, Journal of Computational and Applied Mathematics, 270 (2014) 522-530

work page 2014
[4]

Akkari, A

N. Akkari, A. Hamdouni, M. Jazar,Mathematical and numerical results on the sensitivity of the POD approximation relative to the Burgers equation, Applied Mathematics and Computation, 247 (2014) 951-961

work page 2014
[5]

Akkari, A

N. Akkari, A. Hamdouni, E. Liberge, M. Jazar,On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation, Advanced Modeling and Simulation in Engineering Sciences, 2 (2014) 1-16

work page 2014
[6]

Amsallem, C

D. Amsallem, C. Farhat,An interpolation method for adapting reduced order models and application to aeroelasticity, Amer. Inst. Aeronaut. Astronaut., 46 (7) (2008) 1803-1813

work page 2008
[7]

Arnold, F

D. Arnold, F. Brezzi, M. Fortin,A stable ﬁnite element for the stokes equations, CAL- COLO, 21 (4) (1984) 337-344

work page 1984
[8]

Cordier, M

L. Cordier, M. Bergmann,Proper Orthogonal Decomposition: an overview, Lecture series on post-processing of experimental and numerical data, Von Karman Institute for Fluid Dynamics, 4 (2003) 1-46

work page 2003
[9]

Denis de Senneville, A

B. Denis de Senneville, A. El Hamidi, C. Moonen,A direct PCA-based approach for real- time description of physiological organ deformations, IEEE Transactions on Medical Imag- ing, 34 (4) (2015) 974-982

work page 2015
[10]

Falaize, E

A. Falaize, E. Liberge, A. Hamdouni,POD-based reduced order model for ﬂows induced by rigid solids in forced rotation, Journal of Fluids and Structures, (In press). Preprint submitted to the Journal of Fluids and Structures (https://hal.archives- ouvertes.fr/hal-01874892) On the interpolation on Grassmann Manifolds : Neville-Aitken’s Algorithm 19

work page
[11]

J. H. Ferziger, M. Peric,Computational methods for ﬂuid dynamics, Springer Science & Business Media 2012

work page 2012
[12]

Karcher,Riemannian center of mass and molliﬁer smoothing, Comm

H. Karcher,Riemannian center of mass and molliﬁer smoothing, Comm. Pure Appl. Math. 30 (1977) 509-541

work page 1977
[13]

S. E. Kozlov,Geometry of real Grassmannian manifolds, Zap. Nauchn. Semin. POMI, 246 (1997) 108-129

work page 1997
[14]

Kunisch, S

K. Kunisch, S. Volkwein,Control of Burgers equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999) 345-371

work page 1999
[15]

A. Logg, G. N. Wells, J. Hake,DOLFIN: A C++/Python ﬁnite element library, Automated Solution of Diﬀerential Equations by the Finite Element Method, Springer (2012) 173-225

work page 2012
[16]

Longatte, E

E. Longatte, E. Liberge, M. Pomarède, J.F. Sigrist, A. Hamdouni,Parametric study of ﬂow-induced vibrations in cylinder arrays under single-phase ﬂuid cross ﬂows using POD- ROM, Journal of Fluids and Structures 78 (2018) 314-330

work page 2018
[17]

Y. Lu, N. Blal, A. Gravouil,Space time POD based computational vademecums for para- metric studies: application to thermo-mechanical problems, Advanced Modeling and Sim- ulation in Engineering Sciences, 5 (1) (2018) 1-27

work page 2018
[18]

Mosquera, A

R. Mosquera, A. Hamdouni, A. El Hamidi, C. Allery,POD Basis Interpolation via In- verse Distance Weighting on Grassmann Manifolds, Discrete and Continuous Dynamical Systems, Series S. 12 (6) (201 1743–1759

work page
[19]

Haasdonk, M

B. Haasdonk, M. Ohlberger, G. Rozza,A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal., 32 (2008) 145-161

work page 2008
[20]

P. J. Holmes, J. Lumley, J. Berkooz, J. Mattingly, R. Wittenberg,Low dimensional models of coherent structures in turbulence. Phys. Rev. Section Phys. Lett. 4 (1997), 338-384

work page 1997
[21]

Hömberg, S

D. Hömberg, S. Volkwein,Control of laser surface hardening by a reduced-order approach utilizing proper orthogonal decomposition, Math. Comput. Model., 38 (2003) 1003-1028

work page 2003
[22]

Huiling Le,Estimation of Riemannian barycenters, LMS J. Comput. Math., 7 (2004) 193- 200

work page 2004
[23]

Milnor, J

W. Milnor, J. D. Stasheﬀ,Characteristic classes, Ann. Math. Studies, Princeton University Press, 1974

work page 1974
[24]

E. H. Neville,Iterative interpolation, J. Indian Math. Soc. 20 (1934) 87–120

work page 1934
[25]

Patera and G

A.T. Patera and G. Rozza,A Posteriori Error Estimation for Parametrized Partial Dif- ferential Equations, MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007. On the interpolation on Grassmann Manifolds : Neville-Aitken’s Algorithm 20

work page 2007
[26]

Roujol, M

S. Roujol, M. Ries, B. Quesson, C. Moonen, B. Denis de Senneville, Real-time MR- thermometry and dosimetry for interventional guidance on abdominal organs, Magnetic Resonance in Medicine, 63 (2010) 1080-7

work page 2010
[27]

Sirovich,Turbulence and the dynamics of coherent structures, parts I-III, Quart

L. Sirovich,Turbulence and the dynamics of coherent structures, parts I-III, Quart. Appl. Math. 45 (3) (1987) 561-590

work page 1987
[28]

in Nonlin

A.Tallet, C.Allery, C.Leblond, E.Liberge, A minimum residual projection to build coupled velocity-pressure POD-ROM for incompressible Navier-Stokes equations, Comm. in Nonlin. Science and Num. Simulation, 22 (1) (2015) 909-932

work page 2015
[29]

Volkwein,Optimal control of a phase-ﬁeld model using the proper orthogonal decompo- sition, Z

S. Volkwein,Optimal control of a phase-ﬁeld model using the proper orthogonal decompo- sition, Z. Angew. Math. Mech., 81 (2001) 83-97

work page 2001
[30]

Volkwein,Proper orthogonal decomposition: Theory and reduced-order modeling, Uni- versity of Konstanz, 2013

S. Volkwein,Proper orthogonal decomposition: Theory and reduced-order modeling, Uni- versity of Konstanz, 2013

work page 2013
[31]

Y. C. Wong,Diﬀerential geometry of Grassmann manifolds, Proc Natl Acad Sci U S A. 57 (3) (1967) 589-94

work page 1967

[1] [1]

P. A. Absil, R. Mahony, R. Sepulchre,Riemannian geometry of Grassmann manifolds with a view on algorithmic computation, Acta Applicandae Mathematicae, 80 (2) (2004) 199-220

work page 2004

[2] [2]

A. C. Aitken,On interpolation by iteration of proportional parts, without the use of diﬀer- ences, Proc. Edinburgh Math. Soc. Ser. 2, 3 (1932) 56-76

work page 1932

[3] [3]

Akkari, A

N. Akkari, A. Hamdouni, E. Liberge, M. Jazar,A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM–POD), for a 2D incompressible ﬂuid ﬂow, Journal of Computational and Applied Mathematics, 270 (2014) 522-530

work page 2014

[4] [4]

Akkari, A

N. Akkari, A. Hamdouni, M. Jazar,Mathematical and numerical results on the sensitivity of the POD approximation relative to the Burgers equation, Applied Mathematics and Computation, 247 (2014) 951-961

work page 2014

[5] [5]

Akkari, A

N. Akkari, A. Hamdouni, E. Liberge, M. Jazar,On the sensitivity of the POD technique for a parameterized quasi-nonlinear parabolic equation, Advanced Modeling and Simulation in Engineering Sciences, 2 (2014) 1-16

work page 2014

[6] [6]

Amsallem, C

D. Amsallem, C. Farhat,An interpolation method for adapting reduced order models and application to aeroelasticity, Amer. Inst. Aeronaut. Astronaut., 46 (7) (2008) 1803-1813

work page 2008

[7] [7]

Arnold, F

D. Arnold, F. Brezzi, M. Fortin,A stable ﬁnite element for the stokes equations, CAL- COLO, 21 (4) (1984) 337-344

work page 1984

[8] [8]

Cordier, M

L. Cordier, M. Bergmann,Proper Orthogonal Decomposition: an overview, Lecture series on post-processing of experimental and numerical data, Von Karman Institute for Fluid Dynamics, 4 (2003) 1-46

work page 2003

[9] [9]

Denis de Senneville, A

B. Denis de Senneville, A. El Hamidi, C. Moonen,A direct PCA-based approach for real- time description of physiological organ deformations, IEEE Transactions on Medical Imag- ing, 34 (4) (2015) 974-982

work page 2015

[10] [10]

Falaize, E

A. Falaize, E. Liberge, A. Hamdouni,POD-based reduced order model for ﬂows induced by rigid solids in forced rotation, Journal of Fluids and Structures, (In press). Preprint submitted to the Journal of Fluids and Structures (https://hal.archives- ouvertes.fr/hal-01874892) On the interpolation on Grassmann Manifolds : Neville-Aitken’s Algorithm 19

work page

[11] [11]

J. H. Ferziger, M. Peric,Computational methods for ﬂuid dynamics, Springer Science & Business Media 2012

work page 2012

[12] [12]

Karcher,Riemannian center of mass and molliﬁer smoothing, Comm

H. Karcher,Riemannian center of mass and molliﬁer smoothing, Comm. Pure Appl. Math. 30 (1977) 509-541

work page 1977

[13] [13]

S. E. Kozlov,Geometry of real Grassmannian manifolds, Zap. Nauchn. Semin. POMI, 246 (1997) 108-129

work page 1997

[14] [14]

Kunisch, S

K. Kunisch, S. Volkwein,Control of Burgers equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999) 345-371

work page 1999

[15] [15]

A. Logg, G. N. Wells, J. Hake,DOLFIN: A C++/Python ﬁnite element library, Automated Solution of Diﬀerential Equations by the Finite Element Method, Springer (2012) 173-225

work page 2012

[16] [16]

Longatte, E

E. Longatte, E. Liberge, M. Pomarède, J.F. Sigrist, A. Hamdouni,Parametric study of ﬂow-induced vibrations in cylinder arrays under single-phase ﬂuid cross ﬂows using POD- ROM, Journal of Fluids and Structures 78 (2018) 314-330

work page 2018

[17] [17]

Y. Lu, N. Blal, A. Gravouil,Space time POD based computational vademecums for para- metric studies: application to thermo-mechanical problems, Advanced Modeling and Sim- ulation in Engineering Sciences, 5 (1) (2018) 1-27

work page 2018

[18] [18]

Mosquera, A

R. Mosquera, A. Hamdouni, A. El Hamidi, C. Allery,POD Basis Interpolation via In- verse Distance Weighting on Grassmann Manifolds, Discrete and Continuous Dynamical Systems, Series S. 12 (6) (201 1743–1759

work page

[19] [19]

Haasdonk, M

B. Haasdonk, M. Ohlberger, G. Rozza,A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal., 32 (2008) 145-161

work page 2008

[20] [20]

P. J. Holmes, J. Lumley, J. Berkooz, J. Mattingly, R. Wittenberg,Low dimensional models of coherent structures in turbulence. Phys. Rev. Section Phys. Lett. 4 (1997), 338-384

work page 1997

[21] [21]

Hömberg, S

D. Hömberg, S. Volkwein,Control of laser surface hardening by a reduced-order approach utilizing proper orthogonal decomposition, Math. Comput. Model., 38 (2003) 1003-1028

work page 2003

[22] [22]

Huiling Le,Estimation of Riemannian barycenters, LMS J. Comput. Math., 7 (2004) 193- 200

work page 2004

[23] [23]

Milnor, J

W. Milnor, J. D. Stasheﬀ,Characteristic classes, Ann. Math. Studies, Princeton University Press, 1974

work page 1974

[24] [24]

E. H. Neville,Iterative interpolation, J. Indian Math. Soc. 20 (1934) 87–120

work page 1934

[25] [25]

Patera and G

A.T. Patera and G. Rozza,A Posteriori Error Estimation for Parametrized Partial Dif- ferential Equations, MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2007. On the interpolation on Grassmann Manifolds : Neville-Aitken’s Algorithm 20

work page 2007

[26] [26]

Roujol, M

S. Roujol, M. Ries, B. Quesson, C. Moonen, B. Denis de Senneville, Real-time MR- thermometry and dosimetry for interventional guidance on abdominal organs, Magnetic Resonance in Medicine, 63 (2010) 1080-7

work page 2010

[27] [27]

Sirovich,Turbulence and the dynamics of coherent structures, parts I-III, Quart

L. Sirovich,Turbulence and the dynamics of coherent structures, parts I-III, Quart. Appl. Math. 45 (3) (1987) 561-590

work page 1987

[28] [28]

in Nonlin

A.Tallet, C.Allery, C.Leblond, E.Liberge, A minimum residual projection to build coupled velocity-pressure POD-ROM for incompressible Navier-Stokes equations, Comm. in Nonlin. Science and Num. Simulation, 22 (1) (2015) 909-932

work page 2015

[29] [29]

Volkwein,Optimal control of a phase-ﬁeld model using the proper orthogonal decompo- sition, Z

S. Volkwein,Optimal control of a phase-ﬁeld model using the proper orthogonal decompo- sition, Z. Angew. Math. Mech., 81 (2001) 83-97

work page 2001

[30] [30]

Volkwein,Proper orthogonal decomposition: Theory and reduced-order modeling, Uni- versity of Konstanz, 2013

S. Volkwein,Proper orthogonal decomposition: Theory and reduced-order modeling, Uni- versity of Konstanz, 2013

work page 2013

[31] [31]

Y. C. Wong,Diﬀerential geometry of Grassmann manifolds, Proc Natl Acad Sci U S A. 57 (3) (1967) 589-94

work page 1967