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arxiv: 2604.09325 · v1 · submitted 2026-04-10 · 🧮 math.NA · cs.NA· math.OC

A reduced-order model for parametrized Optimal Transport problems

Elise Bonnet-Weill , Virginie Ehrlacher , Luca Nenna This is my paper

Pith reviewed 2026-05-10 16:26 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC
keywords optimal transportmodel order reductionlinear programminga posteriori error estimationempirical interpolation methodparametrized problemscolor transfer
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The pith

Constraining optimal transport solutions to low-dimensional non-negative subcones produces small linear programs with explicit solvability conditions and a posteriori error bounds.

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

This paper constructs reduced-order models for families of optimal transport problems that vary with parameters. It does so by restricting the primal high-fidelity formulation to a non-negative subcone of small dimension or the dual formulation to a small subspace, which converts the problem into a linear program with far fewer variables and constraints. Explicit conditions are given that guarantee the reduced linear program has at least one solution. Two a posteriori estimators are derived that bound the difference between the reduced optimal value and the high-fidelity optimal value; one of them is made practical through the empirical interpolation method. The approach is tested on a one-dimensional parametrized problem and on color transfer between images, where its performance is compared to the Sinkhorn algorithm.

Core claim

By adding the constraint that the transport plan (or dual potentials) must lie in a non-negative subcone (respectively subspace) of low dimension, the high-fidelity optimal transport linear program is replaced by a much smaller linear program whose optimal value can be bounded relative to the original problem. Under explicit conditions on the subcone, this reduced program is guaranteed to possess at least one feasible solution. Two a posteriori error estimates are proved that quantify the gap to the high-fidelity optimum; the nonlinear estimate is evaluated efficiently by empirical interpolation. Numerical tests on a 1D family and on image color transfer confirm that the reduced models canbe

What carries the argument

The non-negative low-dimensional subcone (or subspace) constraint added to the primal (dual) high-fidelity optimal transport formulation, which produces a small linear program equipped with two a posteriori error estimators.

If this is right

  • The reduced linear program can be solved with a number of variables and constraints that grows only with the chosen dimension of the subcone rather than the mesh size.
  • The a posteriori bounds certify approximation quality without requiring a full high-fidelity solve for every new parameter value.
  • Empirical interpolation makes the nonlinear error estimator inexpensive to evaluate once a small set of basis functions is precomputed.
  • The same reduced space can be reused across many parameter instances once it has been identified.
  • The method supplies an alternative to Sinkhorn iteration whose cost scales with the reduced dimension rather than the full grid size.

Where Pith is reading between the lines

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

  • If snapshot solutions from a few parameter values are used to build the subcone via non-negative matrix factorization or similar techniques, the construction could be made fully data-driven.
  • The same cone-constraint idea might extend to other parametrized linear programs arising in imaging or resource allocation, provided non-negativity is preserved.
  • For problems where optimal solutions vary sharply with the parameter, an adaptive choice of subcone per parameter region could maintain accuracy while keeping dimension low.
  • Combining the reduced model with warm-starting from nearby parameters could further accelerate repeated solves in online settings.

Load-bearing premise

Suitable low-dimensional non-negative subcones or subspaces exist that keep the optimal solutions of the parametrized family well-approximated for the parameter values of interest.

What would settle it

For a concrete parametrized optimal transport family, compute the true high-fidelity optima and the reduced-model optima over a range of parameters; if the gap between them stays larger than the derived error bounds for every choice of low-dimensional subcone, the claim that the reduced model reliably approximates the family is false.

Figures

Figures reproduced from arXiv: 2604.09325 by Elise Bonnet-Weill, Luca Nenna, Virginie Ehrlacher.

Figure 1
Figure 1. Figure 1: Visualization of the parametrized marginals [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left : solutions of the reduced order-model (ROM) for different sizes of reduced bases (RB). [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of the mean error over a test set of 50 random parameters with respect to the [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of precision and time gain between the reduced-order model and the Sinkhorn [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Gap between the fast evaluation and the exact evaluation of the a posteriori error estimation [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the exact error and the a posteriori error estimation. Here we represent a [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Error correction with a multiplicative constant. The values correspond to a mean over 50 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A posteriori error estimation using continuity in function of the size of the training set. [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Error correction with a multiplicative constant. The training set used to approximate the [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Color transfer obtained with a reduced-basis method (line 2) and via. These pictures where [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Detail of the resulting images for α = 0.9. [3] Beatrice Battisti, Tobias Blickhan, Guillaume Enchery, Virginie Ehrlacher, Damiano Lombardi, and Olga Mula. Wasserstein model reduction approach for parametrized flow problems in porous media. ESAIM: Proceedings and Surveys, 73:28–47, 2023. [4] Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the monge￾kantorovich mass transfe… view at source ↗
read the original abstract

In this work, we aim at efficiently solving a parametrized family of optimal transport problems by using model order reduction methods. We propose a reduced-order model by adding to the primal (respectively dual) version of the high-fidelity model the additional constraint to live in a non negative sub cone (resp. in subspaces) of small dimension. The reduced-order model then reads as a linear program with a small number of degrees of freedom and constraints. We identify explicit conditions under which this reduced-order model has at least one solution. We propose two a posteriori error estimations that bounds the error between the optimal values of the high-fidelity problem and the reduced-order model. As one of these estimations requires the computation of non linear terms (with respect to the reduction of dimension), we use an Empirical Interpolation Method (EIM) (see e.g. \cite{maday2007general} or \cite{barrault2004empirical}) to numerically efficiently compute this estimation. We apply the whole methodology on a simple 1D example and on a problem of color transfer between images, and compare its performances to Sinkhorn algorithm.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a reduced-order model (ROM) for a parametrized family of optimal transport (OT) problems. The ROM is obtained by constraining the primal (dual) high-fidelity OT formulation to a low-dimensional non-negative subcone (subspace), yielding a linear program with a small number of degrees of freedom. Explicit conditions are given under which the ROM admits at least one solution. Two a posteriori error estimators are derived that bound the gap between the ROM and high-fidelity optimal values; the estimator involving nonlinear terms is evaluated efficiently via the Empirical Interpolation Method (EIM). The approach is illustrated on a 1D example and a color-transfer problem between images, with performance comparisons to the Sinkhorn algorithm.

Significance. If suitable low-dimensional subcones or subspaces can be identified, the framework supplies a certified, small-scale LP formulation for repeated parametric OT solves together with rigorous a posteriori bounds. The derivation of the error estimators by construction and the incorporation of EIM for computational efficiency are clear strengths. The numerical examples demonstrate feasibility on concrete problems, but the lack of a general offline procedure for constructing the reduced spaces restricts immediate applicability beyond the hand-chosen or snapshot-based cases shown.

major comments (3)
  1. [§3] §3 (Reduced-order model construction): The explicit conditions guaranteeing existence of a solution to the constrained LP are stated, yet these conditions depend on the particular choice of subcone/subspace and no verification procedure is supplied that would hold uniformly over the parameter domain.
  2. [§4] §4 (A posteriori error estimations): The two error bounds are valid by construction because they measure the distance from the high-fidelity optimum to the chosen feasible set; however, the manuscript provides neither numerical verification of the bound values in the examples nor an analysis of how rapidly the bounds deteriorate when the low-dimensional approximation is only moderately accurate.
  3. [§5] §5 (Numerical experiments): In both the 1D test and the color-transfer application the bases are evidently hand-selected or snapshot-derived, but no general offline algorithm (greedy, POD, or reduced-basis-style) is presented that would guarantee a small fixed dimension suffices uniformly for unseen parameter values.
minor comments (2)
  1. [§5] The abstract states that performances are compared to Sinkhorn, yet the numerical section should include explicit tables or plots of wall-clock times, optimality gaps, and reduced-dimension scaling.
  2. [§2] Notation for the non-negative subcone and the subspace should be introduced with explicit definitions and dimension symbols at the beginning of §2 or §3 to improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment point by point below, indicating where the manuscript will be revised.

read point-by-point responses

  1. Referee: [§3] §3 (Reduced-order model construction): The explicit conditions guaranteeing existence of a solution to the constrained LP are stated, yet these conditions depend on the particular choice of subcone/subspace and no verification procedure is supplied that would hold uniformly over the parameter domain.

    Authors: The explicit conditions in §3 are formulated in terms of the chosen reduced subcone or subspace (e.g., non-negativity and inclusion of specific vectors that guarantee feasibility of the reduced LP). Because the reduced space is selected by the user, the conditions necessarily depend on that choice; we state them explicitly so that they can be verified once the space is fixed. We do not supply an automatic, parameter-uniform verification algorithm because the paper focuses on the ROM formulation and error analysis rather than on an automated space-construction procedure. In the numerical examples the conditions are satisfied by construction for the selected spaces. We will add a short clarifying paragraph in §3 explaining how the conditions reduce to simple checks once the reduced space is given. revision: partial

  2. Referee: [§4] §4 (A posteriori error estimations): The two error bounds are valid by construction because they measure the distance from the high-fidelity optimum to the chosen feasible set; however, the manuscript provides neither numerical verification of the bound values in the examples nor an analysis of how rapidly the bounds deteriorate when the low-dimensional approximation is only moderately accurate.

    Authors: We agree that the bounds are valid by construction. The manuscript does not currently display numerical values of the estimators alongside the true errors, nor does it examine their sharpness for moderately accurate reduced spaces. In the revised version we will add tables and/or plots in §5 that compare the computed a-posteriori bounds with the actual optimality gaps for both the 1D and color-transfer examples. We will also include a brief discussion of how the bounds behave as the dimension of the reduced space is decreased. revision: yes

  3. Referee: [§5] §5 (Numerical experiments): In both the 1D test and the color-transfer application the bases are evidently hand-selected or snapshot-derived, but no general offline algorithm (greedy, POD, or reduced-basis-style) is presented that would guarantee a small fixed dimension suffices uniformly for unseen parameter values.

    Authors: The numerical sections illustrate the ROM and the EIM-based error estimator on two concrete problems; the reduced spaces are therefore chosen accordingly (hand-selected for the simple 1D case, snapshot-based for the image color-transfer problem). The manuscript does not claim or develop a general offline procedure for constructing reduced spaces that works uniformly for arbitrary parameter values. Such a procedure would constitute a separate, substantial contribution. We will add a sentence in the introduction and conclusions clarifying the scope of the present work and identifying the automated construction of reduced spaces as an interesting direction for future research. revision: partial

Circularity Check

0 steps flagged

No significant circularity: derivation proceeds by direct constraint imposition and independent a posteriori bounds.

full rationale

The reduced-order model is formed by adding explicit non-negativity subcone (or subspace) constraints to the standard primal/dual OT linear programs, yielding a smaller LP whose solvability conditions are stated explicitly. The two a posteriori error estimators bound the optimality gap via standard duality arguments and are independent of any fitted parameters or self-referential definitions; one estimator's nonlinear terms are evaluated via the external EIM technique with citations to Maday et al. and Barrault et al. No load-bearing step reduces a claimed result to its own inputs by construction, and no self-citation chain is invoked to justify uniqueness or ansatz choices. The approach is therefore self-contained against the high-fidelity OT formulations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the existence of low-dimensional approximating cones or subspaces whose choice is problem-dependent; standard linear-programming theory is invoked for the reduced problem.

free parameters (1)
  • reduced dimension
    The dimension of the subcone or subspace is a user-chosen parameter that controls the trade-off between speed and accuracy.
axioms (1)
  • domain assumption Existence of at least one solution for the reduced linear program under the stated explicit conditions
    The paper identifies conditions but treats them as holding for the target applications.

pith-pipeline@v0.9.0 · 5503 in / 1225 out tokens · 78316 ms · 2026-05-10T16:26:50.752178+00:00 · methodology

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    Relation between the paper passage and the cited Recognition theorem.

    We propose a reduced-order model by adding to the primal (respectively dual) version of the high-fidelity model the additional constraint to live in a non negative sub cone (resp. in subspaces) of small dimension. ... two a posteriori error estimations that bounds the error between the optimal values

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Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    Automatic choice of global shape functions in structural analysis.Aiaa Journal, 16(5):525–528, 1978

    Bo O Almroth, Perry Stern, and Frank A Brogan. Automatic choice of global shape functions in structural analysis.Aiaa Journal, 16(5):525–528, 1978

    work page 1978

  2. [2]

    An ‘empirical interpolation’method: application to efficient reduced-basis discretization of partial differential equations.Comptes Rendus Mathematique, 339(9):667–672, 2004

    Maxime Barrault, Yvon Maday, Ngoc Cuong Nguyen, and Anthony T Patera. An ‘empirical interpolation’method: application to efficient reduced-basis discretization of partial differential equations.Comptes Rendus Mathematique, 339(9):667–672, 2004. 29 Reduced base Entropic OT Figure 11: Detail of the resulting images forα= 0.9

    work page 2004

  3. [3]

    Wasserstein model reduction approach for parametrized flow problems in porous media.ESAIM: Proceedings and Surveys, 73:28–47, 2023

    Beatrice Battisti, Tobias Blickhan, Guillaume Enchery, Virginie Ehrlacher, Damiano Lombardi, and Olga Mula. Wasserstein model reduction approach for parametrized flow problems in porous media.ESAIM: Proceedings and Surveys, 73:28–47, 2023

    work page 2023

  4. [4]

    A computational fluid mechanics solution to the monge- kantorovich mass transfer problem.Numerische Mathematik, 84(3):375–393, 2000

    Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the monge- kantorovich mass transfer problem.Numerische Mathematik, 84(3):375–393, 2000

    work page 2000

  5. [5]

    Iter- ative bregman projections for regularized transportation problems.SIAM Journal on Scientific Computing, 37(2):A1111–A1138, 2015

    Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, and Gabriel Peyr´ e. Iter- ative bregman projections for regularized transportation problems.SIAM Journal on Scientific Computing, 37(2):A1111–A1138, 2015

    work page 2015

  6. [6]

    Convergence rates for greedy algorithms in reduced basis methods.SIAM journal on mathematical analysis, 43(3):1457–1472, 2011

    Peter Binev, Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova, and Prze- myslaw Wojtaszczyk. Convergence rates for greedy algorithms in reduced basis methods.SIAM journal on mathematical analysis, 43(3):1457–1472, 2011

    work page 2011

  7. [7]

    A registration method for reduced basis problems using linear optimal transport

    Tobias Blickhan. A registration method for reduced basis problems using linear optimal transport. SIAM Journal on Scientific Computing, 46(5):A3177–A3204, 2024

    work page 2024

  8. [8]

    A priori convergence of the greedy algorithmfor the parametrized reduced basis method.ESAIM: Mathematical modelling and numerical analysis, 46(3):595–603, 2012

    Annalisa Buffa, Yvon Maday, Anthony T Patera, Christophe Prud’Homme, and Gabriel Turinici. A priori convergence of the greedy algorithmfor the parametrized reduced basis method.ESAIM: Mathematical modelling and numerical analysis, 46(3):595–603, 2012

    work page 2012

  9. [9]

    Model order reduction for problems with large convection effects

    Nicolas Cagniart, Yvon Maday, and Benjamin Stamm. Model order reduction for problems with large convection effects. InContributions to partial differential equations and applications, pages 131–150. Springer, 2018

    work page 2018

  10. [10]

    Model order reduction by convex displacement interpolation.Journal of Computational Physics, 514:113230, 2024

    Simona Cucchiara, Angelo Iollo, Tommaso Taddei, and Haysam Telib. Model order reduction by convex displacement interpolation.Journal of Computational Physics, 514:113230, 2024

    work page 2024

  11. [11]

    Sinkhorn distances: Lightspeed computation of optimal transport.Advances in neural information processing systems, 26, 2013

    Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport.Advances in neural information processing systems, 26, 2013

    work page 2013

  12. [12]

    Nonlinear reduced basis using mixture wasserstein barycenters: application to an eigenvalue problem inspired from quantum chemistry.Constructive Approximation, pages 1–41, 2026

    Maxime Dalery, Genevi` eve Dusson, Virginie Ehrlacher, and Alexei Lozinski. Nonlinear reduced basis using mixture wasserstein barycenters: application to an eigenvalue problem inspired from quantum chemistry.Constructive Approximation, pages 1–41, 2026

    work page 2026

  13. [13]

    Stori, 2024.https://colab.research.google.com/github/storimaging/Notebooks/blob/ main/ContrastAndColor/TP_color_transfer_with_Sinkhorn.ipynb

    Julie Delon, Bruno Galerne, Agnes Desolneux, Valentin De Bortoli, and Luc´ ıa Bouza. Stori, 2024.https://colab.research.google.com/github/storimaging/Notebooks/blob/ main/ContrastAndColor/TP_color_transfer_with_Sinkhorn.ipynb

    work page 2024

  14. [14]

    Sparse wasserstein barycenters and application to reduced order modeling.Journal of Scientific Computing, 102(3):64, 2025

    Minh-Hieu Do, Jean Feydy, and Olga Mula. Sparse wasserstein barycenters and application to reduced order modeling.Journal of Scientific Computing, 102(3):64, 2025

    work page 2025

  15. [15]

    Nonlinear model reduction on metric spaces

    Virginie Ehrlacher, Damiano Lombardi, Olga Mula, and Fran¸ cois-Xavier Vialard. Nonlinear model reduction on metric spaces. application to one-dimensional conservative pdes in wasserstein spaces. ESAIM: Mathematical Modelling and Numerical Analysis, 54(6):2159–2197, 2020. 30

    work page 2020

  16. [16]

    Regularized discrete optimal transport.SIAM Journal on Imaging Sciences, 7(3):1853–1882, 2014

    Sira Ferradans, Nicolas Papadakis, Gabriel Peyr´ e, and Jean-Fran¸ cois Aujol. Regularized discrete optimal transport.SIAM Journal on Imaging Sciences, 7(3):1853–1882, 2014

    work page 2014

  17. [17]

    Interpolating between optimal transport and mmd using sinkhorn divergences

    Jean Feydy, Thibault S´ ejourn´ e, Fran¸ cois-Xavier Vialard, Shun-ichi Amari, Alain Trouv´ e, and Gabriel Peyr´ e. Interpolating between optimal transport and mmd using sinkhorn divergences. InThe 22nd international conference on artificial intelligence and statistics, pages 2681–2690. PMLR, 2019

    work page 2019

  18. [18]

    Cupid’s invisible hand: Social surplus and identification in matching models.The Review of Economic Studies, 89(5):2600–2629, 2022

    Alfred Galichon and Bernard Salani´ e. Cupid’s invisible hand: Social surplus and identification in matching models.The Review of Economic Studies, 89(5):2600–2629, 2022

    work page 2022

  19. [19]

    On h¨ older continuity-in-time of the optimal transport map towards measures along a curve.Proceedings of the Edinburgh Mathematical Society, 54(2):401–409, 2011

    Nicola Gigli. On h¨ older continuity-in-time of the optimal transport map towards measures along a curve.Proceedings of the Edinburgh Mathematical Society, 54(2):401–409, 2011

    work page 2011

  20. [20]

    Efficient reduced- basis treatment of nonaffine and nonlinear partial differential equations.ESAIM: Mod´ elisation math´ ematique et analyse num´ erique, 41(3):575–605, 2007

    Martin A Grepl, Yvon Maday, Ngoc C Nguyen, and Anthony T Patera. Efficient reduced- basis treatment of nonaffine and nonlinear partial differential equations.ESAIM: Mod´ elisation math´ ematique et analyse num´ erique, 41(3):575–605, 2007

    work page 2007

  21. [21]

    Martin A Grepl and Anthony T Patera. A posteriori error bounds for reduced-basis approxima- tions of parametrized parabolic partial differential equations.ESAIM: Mod´ elisation math´ ematique et analyse num´ erique, 39(1):157–181, 2005

    work page 2005

  22. [22]

    Convergence rates of the pod–greedy method.ESAIM: Mathematical mod- elling and numerical Analysis, 47(3):859–873, 2013

    Bernard Haasdonk. Convergence rates of the pod–greedy method.ESAIM: Mathematical mod- elling and numerical Analysis, 47(3):859–873, 2013

    work page 2013

  23. [23]

    Reduced basis method for finite volume approxima- tionsof parametrizedlinear evolution equations.ESAIM: Mathematical Modelling and Numerical Analysis, 42(2):277–302, 2008

    Bernard Haasdonk and Mario Ohlberger. Reduced basis method for finite volume approxima- tionsof parametrizedlinear evolution equations.ESAIM: Mathematical Modelling and Numerical Analysis, 42(2):277–302, 2008

    work page 2008

  24. [24]

    Springer, 2016

    Jan S Hesthaven, Gianluigi Rozza, Benjamin Stamm, et al.Certified reduced basis methods for parametrized partial differential equations, volume 590. Springer, 2016

    work page 2016

  25. [25]

    Advection modes by optimal mass transfer.Physical Review E, 89(2):022923, 2014

    Angelo Iollo and Damiano Lombardi. Advection modes by optimal mass transfer.Physical Review E, 89(2):022923, 2014

    work page 2014

  26. [26]

    Mapping of coherent structures in parameterized flows by learning optimal transportation with gaussian models.Journal of Computational Physics, 471:111671, 2022

    Angelo Iollo and Tommaso Taddei. Mapping of coherent structures in parameterized flows by learning optimal transportation with gaussian models.Journal of Computational Physics, 471:111671, 2022

    work page 2022

  27. [27]

    On the translocation of masses.Journal of mathematical sciences, 133(4), 2006

    Leonid V Kantorovich. On the translocation of masses.Journal of mathematical sciences, 133(4), 2006

    work page 2006

  28. [28]

    Springer Science & Business Media, 2008

    Howard Karloff.Linear programming. Springer Science & Business Media, 2008

    work page 2008

  29. [29]

    Moaad Khamlich, Federico Pichi, and Gianluigi Rozza. Optimal transport–inspired deep learning framework for slow-decaying kolmogorov-width problems: Exploiting sinkhorn loss and wasser- stein kernel.SIAM Journal on Scientific Computing, 47(2):C235–C264, 2025

    work page 2025

  30. [30]

    Learning to generate wasser- stein barycenters.Journal of Mathematical Imaging and Vision, 65(2):354–370, 2023

    Julien Lacombe, Julie Digne, Nicolas Courty, and Nicolas Bonneel. Learning to generate wasser- stein barycenters.Journal of Mathematical Imaging and Vision, 65(2):354–370, 2023

    work page 2023

  31. [31]

    A numerical algorithm forl {2}semi-discrete optimal transport in 3d.ESAIM: Mathematical Modelling and Numerical Analysis, 49(6):1693–1715, 2015

    Bruno L´ evy. A numerical algorithm forl {2}semi-discrete optimal transport in 3d.ESAIM: Mathematical Modelling and Numerical Analysis, 49(6):1693–1715, 2015

    work page 2015

  32. [32]

    Regularity of potential functions of the optimal transportation problem.Archive for rational mechanics and analysis, 177(2):151–183, 2005

    Xi-Nan Ma, Neil S Trudinger, and Xu-Jia Wang. Regularity of potential functions of the optimal transportation problem.Archive for rational mechanics and analysis, 177(2):151–183, 2005

    work page 2005

  33. [33]

    A general, multipur- pose interpolation procedure: the magic points.Communications on pire and applied analysis, 8(1), 2007

    Yvon Maday, Ngoc Cuong Nguyen, Anthony T Patera, and George SH Pau. A general, multipur- pose interpolation procedure: the magic points.Communications on pire and applied analysis, 8(1), 2007

    work page 2007

  34. [34]

    A multiscale approach to optimal transport

    Quentin M´ erigot. A multiscale approach to optimal transport. 30(5):1583–1592, 2011. 31

    work page 2011

  35. [35]

    Quantitative stability of optimal trans- port maps and linearization of the 2-wasserstein space

    Quentin M´ erigot, Alex Delalande, and Frederic Chazal. Quantitative stability of optimal trans- port maps and linearization of the 2-wasserstein space. InInternational Conference on Artificial Intelligence and Statistics, pages 3186–3196. PMLR, 2020

    work page 2020

  36. [36]

    Optimal transport: discretization and algorithms

    Quentin Merigot and Boris Thibert. Optimal transport: discretization and algorithms. InHand- book of numerical analysis, volume 22, pages 133–212. Elsevier, 2021

    work page 2021

  37. [37]

    M´ emoire sur la th´ eorie des d´ eblais et des remblais.Mem

    Gaspard Monge. M´ emoire sur la th´ eorie des d´ eblais et des remblais.Mem. Math. Phys. Acad. Royale Sci., pages 666–704, 1781

    work page

  38. [38]

    Reduced basis technique for nonlinear analysis of structures

    Ahmed K Noor and Jeanne M Peters. Reduced basis technique for nonlinear analysis of structures. Aiaa journal, 18(4):455–462, 1980

    work page 1980

  39. [39]

    Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling.SIAM Journal on Scientific Computing, 42(5):A2803–A2836, 2020

    Benjamin Peherstorfer. Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling.SIAM Journal on Scientific Computing, 42(5):A2803–A2836, 2020

    work page 2020

  40. [40]

    Computational optimal transport.Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

    Gabriel Peyr´ e and Marco Cuturi. Computational optimal transport.Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

    work page 2019

  41. [41]

    Automated colour grading using colour distribution transfer.Computer Vision and Image Understanding, 107(1-2):123–137, 2007

    Fran¸ cois Piti´ e, Anil C Kokaram, and Rozenn Dahyot. Automated colour grading using colour distribution transfer.Computer Vision and Image Understanding, 107(1-2):123–137, 2007

    work page 2007

  42. [42]

    Reliable real-time solution of parametrized partial differ- ential equations: Reduced-basis output bound methods.J

    Christophe Prud’Homme, Dimitrios V Rovas, Karen Veroy, Luc Machiels, Yvon Maday, An- thony T Patera, and Gabriel Turinici. Reliable real-time solution of parametrized partial differ- ential equations: Reduced-basis output bound methods.J. Fluids Eng., 124(1):70–80, 2002

    work page 2002

  43. [43]

    Springer, 2015

    Alfio Quarteroni, Andrea Manzoni, and Federico Negri.Reduced basis methods for partial differ- ential equations: an introduction. Springer, 2015

    work page 2015

  44. [44]

    Regularization of transportation maps for color and contrast transfer.2010 IEEE International Conference on Image Processing, pages 1933–1936, 2010

    Julien Rabin, Julie Delon, and Yann Gousseau. Regularization of transportation maps for color and contrast transfer.2010 IEEE International Conference on Image Processing, pages 1933–1936, 2010

    work page 2010

  45. [45]

    Gianluigi Rozza, Dinh Bao Phuong Huynh, and Anthony T Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics.Archives of computational methods in engineering, 15(3):229–275, 2008

    work page 2008

  46. [46]

    Springer, 2015

    Filippo Santambrogio.Optimal transport for applied mathematicians. Springer, 2015

    work page 2015

  47. [47]

    A registration method for model order reduction: data compression and geom- etry reduction.SIAM Journal on Scientific Computing, 42(2):A997–A1027, 2020

    Tommaso Taddei. A registration method for model order reduction: data compression and geom- etry reduction.SIAM Journal on Scientific Computing, 42(2):A997–A1027, 2020

    work page 2020

  48. [48]

    American Mathematical Soc., 2021

    C´ edric Villani.Topics in optimal transportation, volume 58. American Mathematical Soc., 2021

    work page 2021

  49. [49]

    Springer, 2009

    C´ edric Villani et al.Optimal transport: old and new, volume 338. Springer, 2009

    work page 2009

  50. [50]

    Proper orthogonal decomposition: Theory and reduced-order modelling.Lecture Notes, University of Konstanz, 4(4):1–29, 2013

    Stefan Volkwein. Proper orthogonal decomposition: Theory and reduced-order modelling.Lecture Notes, University of Konstanz, 4(4):1–29, 2013. 32

    work page 2013