Pith. sign in

REVIEW 3 major objections 5 minor 44 references

An Efficient Augmented Lagrangian Framework for Dynamic Optimal Transport on Surfaces Based on Second-Order Cone Programming Reformulation

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A surface mass-transport problem becomes a cone program whose solver converges with no extra assumptions.

desk verdict A real new approach to surface DOT, but the paper's central SOCP equivalence is invalid as printed: equation (30) is algebraically false and the block operator B has mismatched dimensions, so the proof must be corrected before the convergence claims can stand. read the letter →

arxiv 2506.08988 v2 pith:UGG5S244 submitted 2025-06-10 math.OC

Liang Chen , Youyicun Lin , Yuxuan Zhou This is my paper
classification math.OC MSC 49Q2265K1090C0690-04
keywords dynamicoptimaltransportsecond-orderconeprogrammingaugmentedLagrangianmethodBenamou-BrenierformulationWassersteindistancetriangularmeshsurface
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

Dynamic optimal transport on a curved surface means finding the cheapest way to evolve one density into another over time, with the geometry shaping both the cost and the path. This paper claims that the discretized dual of that problem, set on a triangular mesh in space and a staggered grid in time, can be recast as a linear second-order cone program (SOCP), and that the recast problem can be solved by an inexact proximal augmented Lagrangian method. The authors prove the method converges to a Karush-Kuhn-Tucker point without extra assumptions and that every such point corresponds to a KKT solution of the original discrete surface problem. If correct, this gives a practical route to Wasserstein distances and interpolations on surfaces that runs about 6.9 times faster than the ADMM-based surface solver in the tested cases.

What carries the argument

The load-bearing object is the SOCP reformulation (34), built from a copied-variable operator $\mathcal{F}$, a cone-assembly operator $\mathcal{B}$, and per-vertex second-order cones $\mathcal{K}^{k,v}_{\mathrm{soc}}$, together with two structural facts: $\mathcal{F}^*\mathcal{B}^*\mathcal{B}\mathcal{F}$ is diagonal and $\mathcal{T}$ is a permutation matrix times a diagonal matrix. This structure turns the coupled constraint $A_{k,v}+\frac12 L_t L_s^*(\|\cdot\|^2)\le 0$ into independent cone constraints, so each iteration reduces to linear solves with precomputed factorizations and closed-form cone projections, which is what makes the method fast.

What would settle it

Run the same discrete DOT problem on a mesh where one triangle is deliberately made degenerate, with its three vertices nearly collinear, and monitor the KKT residual; if the residual still reaches the prescribed tolerance, the proof's reliance on $\ker(A_s)=\operatorname{span}\{e_i\otimes\mathbf{1}_{|V|}\}$ is not actually necessary, whereas a stall or divergence would confirm that the non-degeneracy assumption is load-bearing.

Watch

Extended reading notes

Core claim

The central claim is that the discrete Benamou-Brenier dual problem on a triangulated surface is equivalent to a finite-dimensional linear SOCP, obtained by decoupling the coupled inequality constraints through copying operators and area-weighted diagonal maps, and that this SOCP has KKT solutions. The paper proves a correspondence between the KKT systems: from any solution of the SOCP KKT system one can recover a solution of the discrete DOT KKT system. Because of that correspondence, the inexact semi-proximal augmented Lagrangian method proposed for the SOCP converges to a primal-dual solution with no additional hypotheses, and the numerical experiments report an average speedup of about 6.9 times over the ADMM baseline at tolerance $10^{-4}$.

Load-bearing premise

The existence proof assumes every triangle of the mesh is non-degenerate, so the discrete gradient operator has kernel exactly $\operatorname{span}\{e_i\otimes\mathbf{1}_{|V|}\}$; a single degenerate or collinear triangle would invalidate that kernel characterization and with it the strict-feasibility argument.

Editorial extensions

If this is right

  • The quadratic Wasserstein distance on a triangulated surface can be computed by solving one convex SOCP, bringing conic-optimization solvers to surface transport problems.
  • Because convergence to a KKT point holds without extra assumptions, the method applies when the initial and terminal densities vanish in places, a regime where the proximal-gradient surface solver struggles.
  • The decoupled structure makes the dominant per-iteration costs diagonal in $q$ and independent per-vertex projections in $z$, allowing runs on meshes with more than thirty thousand vertices.
  • The same reformulation extends to the congestion-regularized variant (52) with minor changes, giving a tunable parameter that spreads out crowded paths in narrow regions.

Reading between the lines

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

  • Beyond the paper, the same reformulation could be carried over to Wasserstein gradient flows and mean-field games on surfaces whenever the cost remains quadratic, because those share the Benamou-Brenier dual structure the method exploits.
  • A natural stress test the paper does not run is to compare the inexact ALM against a carefully implemented interior-point method with more memory than the off-the-shelf solvers it cites, since the reported speed advantage is against ADMM and the cited commercial solvers.
  • The convergence guarantee is for the discrete KKT system, not for the gap to the continuous optimal transport solution; mesh refinement is still needed to turn the computed discrete accuracy into an approximation of the continuous problem.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes an SOCP-based augmented Lagrangian method for dynamic optimal transport on triangulated surfaces. It discretizes the dual Benamou-Brenier problem (14), reformulates it as a linear SOCP (34), proves KKT existence (Proposition 1) and a correspondence between KKT points of the two formulations (Proposition 4), and solves the SOCP with an inexact semi-proximal ALM (Algorithm 1) whose convergence is stated in Theorem 1. Numerical experiments on several surfaces report an average 6.9x speedup over the ADMM baseline at Tol = 1e-4, and an open-source package is provided.

Significance. If the SOCP reformulation is made correct, the paper would be a valuable computational contribution: the variable-decoupling idea is natural, the convergence result is intended to be unconditional relative to the discretized problem, the implementation details (diagonal q-update, closed-form cone projections, precomputed LU decompositions) are efficient, and the extensive experiments are supported by a public software package. The reliance on prior work [11,12] is appropriate and not circular. However, the core Section 4 contains algebraic and dimensional errors that currently invalidate the central equivalence; these are fixable but must be addressed before the theoretical claims can be accepted.

major comments (3)
  1. [Section 4, Eq. (30)] The displayed identity A_{k,v} + S/12 = (1+A_{k,v})^2 + S/3 - (1-A_{k,v})^2, with S := sum_i(||T_i^{k,v}D_i^+||^2 + ||T_i^{k,v}D_i^-||^2), is algebraically false: since (1+A)^2 - (1-A)^2 = 4A, the right-hand side equals 4A + S/3. For example, A=1 and S=0 gives 1 = 4. This identity is the bridge to the cone equivalence (32) and therefore to the SOCP reformulation (34). The correct equivalence for A + S/12 <= 0 is (1-A)^2 >= (1+A)^2 + S/3, which can be encoded by a second-order cone with z1 = 1-A, z2 = (1/sqrt(3))(D+;D-), and z3 = 1+A. This equation and all subsequent steps depending on it must be corrected.
  2. [Section 4, block operator B before Eq. (31)] The displayed block operator B has only two block rows, while the definitions of z and T^{k,v} in (31) require a three-block output z = (z1;z2;z3) with z2 in R^{6|Gst_time||T|3}. With the displayed B, BFq+d = (1-A + (sqrt(3)/3)D; 1+A), so the middle block z2 is not generated and the top block incorrectly mixes A and D. Consequently T^{k,v}(BFq+d) does not implement the required cone condition, and the claim in Proposition 2 that F^*B^*BF is diagonal is false for the displayed B because the off-diagonal block -(sqrt(3)/3) contributes. Please supply the corrected three-block B, e.g., block rows [-I_A, 0], [0, (1/sqrt(3))I_D], [I_A, 0], and re-verify the diagonal q-update in Section 5.1.
  3. [Section 4.1, Proposition 4 and Theorem 1] The proof of Proposition 4 relies on (32) and on the block structure z = BFq+d, in particular the identities (z1)_{k,v} = 1-A_{k,v}, (z3)_{k,v} = 1+A_{k,v}, and the placement of D in z2. With the incorrect identity (30) and the mismatched B, those identities and the subsequent derivation of (41)-(44) do not follow. As a result, the claimed correspondence between the KKT systems (38) and (17), and the convergence guarantee of Theorem 1 for Algorithm 1, are not established as printed. The reformulation and the proof of Proposition 4 must be re-derived with a corrected B and identity before the main theoretical claims can be accepted.
minor comments (5)
  1. [Abstract] The abstract states that Gurobi and MOSEK 'either fail to solve the same SOCP reformulation due to out-of-memory or require substantially prolonged computation times', but Section 6 contains no such experiments. Please add the comparison or remove the claim.
  2. [Conclusions] The Conclusions claim a speedup of 'at least 7 times', but Table 2 shows several examples with speedups between roughly 4.8 and 6.8 times (e.g., Examples 4-2, 5-2, 6-2, 7-2, and 8). The reported average of 6.9x is accurate, so the text should say 'several times' or report the average explicitly.
  3. [Table 4] The rows for Example 10 list theta = 0 twice; the second row should presumably be theta = 0.01.
  4. [Section 3.2, Proposition 1] The proof of Proposition 1 uses the full-rank Jacobian assumption and the characterization ker(A_s) = span{e_i ⊗ 1_|V|} from (21). This is acceptable under the maintained regular-triangular-mesh hypothesis, but the abstract's phrase 'without imposing any additional assumptions' should be qualified by this mesh-regularity condition.
  5. [Section 6.2] The 'slight modification' to the ADMM stopping criterion from [27] should be specified precisely, and the rows in Table 3 where ADMM reaches the 3.6e4-second cap should be described as not having reached the tolerance, so that the Tol = 1e-5 comparison is not misleading.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SOCP reformulation is derived algebraically, the ALM convergence theorem is external general-purpose support, and the speedups are benchmarked against external solvers.

full rationale

The paper's derivation chain is self-contained rather than circular. The discrete dual DOT problem (14) is transcribed from the Benamou-Brenier dual after a finite-element discretization following [27], and the SOCP reformulation (34) is produced by explicitly defined copying operators F, geometric scalings D_i, the block operator B, and the cone constraints (31)-(32); the equivalence is a derived algebraic reformulation, not an assumed input. Proposition 1 establishes KKT existence through the kernel characterization (21) and Rockafellar's classical theorems, not through the paper's own conclusions. Proposition 4 transfers KKT points from the SOCP system (38) to the discrete DOT system (17) by direct algebraic relations such as (40)-(44). Theorem 1 invokes [11, Theorem 4.2], a peer-reviewed, parameter-free general ALM convergence theorem whose stated assumptions do not include the surface DOT result; although the first author overlaps with the present paper, this is genuine external evidence and does not constitute a self-citation chain that forces the conclusion. The numerical claims are externally falsifiable: they are compared against the ADMM solver of [27] and, in one flat-domain case, against a closed-form solution, and no fitted parameter is renamed as a prediction. The self-citations to [12] and [11] are normal and are not used as unverified premises containing the target result. The reviewer's algebraic objection to Eq. (30) and to the dimensions of B, if sustained, would be a correctness defect in the equivalence proof, not an instance of the derivation reducing to its own inputs by construction; correctness and circularity are distinct concerns. Accordingly, no circular step can be exhibited from the paper's text, and the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central mathematical claim depends on standard convex analysis (Rockafellar), the external convergence theorem [11], the discretization convergence result [26], and a regular connected mesh assumption. No invented entities are introduced. Algorithmic hyperparameters (tau, sigma schedule, theta) are tuning choices, not fitted to force the result.

free parameters (3)
  • tau = 1.9
    Step length in Algorithm 1 (Section 6.2); any value in (0,2) preserves convergence, but 1.9 is hand-chosen for fast practical convergence.
  • sigma (penalty) schedule = adaptive, unspecified formula
    Penalty parameter in the augmented Lagrangian, adjusted based on primal-dual residuals (Section 6); no fixed value or formula is stated.
  • theta (congestion) = 0.01, 0.05
    Regularization parameter in variant (52) used for Examples 8-10 (Section 6.3); not part of the central claim, but hand-selected.
assumptions (5)
  • domain assumption The discrete dual formulation (14) is a faithful discretization of the continuous dual DOT problem (3), and its solutions converge to the continuous solution under mesh refinement.
    Invoked in Remark 1 via [26, Section 4]; not proven in this paper.
  • standard math Rockafellar's convex analysis theorems (Theorem 28.2, 30.4, 30.5, Corollary 28.3.1) guarantee Kuhn-Tucker vector existence.
    Used in the proof of Proposition 1.
  • standard math Convergence theorem [11, Theorem 4.2] applies to Algorithm 1 for the restricted problem (45).
    External peer-reviewed result; the paper verifies KKT existence and then reuses this theorem.
  • domain assumption The triangular mesh is regular in the sense of [7, Definition 4.4.13] and connected, and each triangle is non-degenerate.
    Required for the kernel characterization (21) in Proposition 1 and for the finite element gradient construction (5).
  • domain assumption The densities are represented as discrete measures mu_0, mu_N on vertices with total mass 1.
    Standard discrete OT setup, used throughout Section 3.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An Efficient Augmented Lagrangian Framework for Dynamic Optimal Transport on Surfaces Based on Second-Order Cone Programming Reformulation." pith.science (2026). https://pith.science/paper/UGG5S244

@misc{pith2026250608988,
  author       = {Pith},
  title        = {Pith review of: An Efficient Augmented Lagrangian Framework for Dynamic Optimal Transport on Surfaces Based on Second-Order Cone Programming Reformulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UGG5S244}},
  note         = {Machine review of arXiv:2506.08988}
}
read the original abstract

This paper proposes an efficient numerical optimization framework for solving dynamic optimal transport (DOT) problems on surfaces, computing both the quadratic Wasserstein distance and the associated interpolation. Building on the convex DOT model of Benamou-Brenier-Lisini, we first properly reformulate its dual problem, discretized on a triangular mesh in space and a staggered grid in time, into a linear second-order cone programming (SOCP) problem. Then the resulting SOCP is solved via an inexact proximal augmented Lagrangian method with a highly efficient numerical implementation, and the algorithm is guaranteed to converge to a Karush-Kuhn-Tucker point without imposing any additional assumptions. Finally, we implement the proposed framework as an open-source software package. The effectiveness, robustness, and computational efficiency of the software are validated through extensive numerical experiments across diverse datasets, demonstrating that it consistently outperforms state-of-the-art surface DOT solvers by several times in speed, while the commercial solvers Gurobi and MOSEK either fail to solve the same SOCP reformulation due to out-of-memory or require substantially prolonged computation times.

Figures

Figures reproduced from arXiv: 2506.08988 by the authors.

Figure 1
Figure 1. Triangular finite element discretization. The red points represent the vertices. For the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: In each row, from left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: From left to right: the initial distribution [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]
Figure 20
Figure 20. Figure 20: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]
Figure 21
Figure 21. Figure 21: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p026_21.png]
Figure 22
Figure 22. Figure 22: Snapshots of intermediate distribution evolution (from [PITH_FULL_IMAGE:figures/full_fig_p027_22.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

44 extracted references · 42 canonical work pages

  1. [11]

    Chen, X.D

    L. Chen, X.D. Li, D.F. Sun, and K.-C. Toh. On the equivalence of inexact proximal ALM and ADMM for a class of convex composite programming.Math. Program., 185:111–161, 2021. 21

    work page 2021

  2. [1]

    Alizadeh and D

    F. Alizadeh and D. Goldfarb. Second order cone programming.Math. Program., 95:3–51, 2003

    work page 2003

  3. [2]

    Ambrosio, E

    L. Ambrosio, E. Bru´ e, and D. Semola.Lectures on Optimal Transport. Springer, Cham, 2021

    work page 2021

  4. [3]

    Beck and M

    A. Beck and M. Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci., 2(1):183–202, 2009

    work page 2009

  5. [4]

    Benamou and Y

    J.-D. Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem.Numer. Math., 84(3):375–393, 2000

    2000

  6. [5]

    Benamou and G

    J.-D. Benamou and G. Carlier. Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations.J. Optim. Theory Appl., 167:1–26, 2015

    work page 2015

  7. [6]

    Benamou, G

    J.-D. Benamou, G. Carlier, and M. Laborde. An augmented Lagrangian approach to Wasserstein gradient flows and applications.ESAIM Proc. Surveys, 54:1–17, 2016

    work page 2016

  8. [7]

    Brenner and L.R

    S.C. Brenner and L.R. Scott.The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics. Springer, New York, 3rd edition, 2008

    work page 2008

Show all 44 references
  1. [8]

    Carrillo, K

    J.A. Carrillo, K. Craig, L. Wang, and C.Z. Wei. Primal dual methods for Wasserstein gradient flows. Found. Comput. Math., 22:389–443, 2022

    2022

  2. [9]

    Chambolle and J.P

    A. Chambolle and J.P. Contreras. Accelerated Bregman primal-dual methods applied to optimal transport and Wasserstein barycenter problems.SIAM J. Math. Data Sci., 4(4):1369–1395, 2022

    2022

  3. [10]

    Chambolle and T

    A. Chambolle and T. Pock. A first-order primal-dual algorithm for convex problems with applications to imaging.J. Math. Imaging Vis., 40:120–145, 2011

    2011

  4. [12]

    Chen, Y.Y.C

    L. Chen, Y.Y.C. Lin, and Y.X. Zhou. An efficient second-order cone programming approach for dynamic optimal transport on staggered grid discretization, may 2025

    2025

  5. [13]

    Chen, T.T

    Y.X. Chen, T.T. Georgiou, and M. Pavon. Vector-valued optimal mass transport.SIAM J. Appl. Math., 78(3):1682–1696, 2018

    2018

  6. [14]

    Chen, T.T

    Y.X. Chen, T.T. Georgiou, and M. Pavon. Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schr¨ odinger bridge.SIAM Rev., 63(2):249–313, 2021

    2021

  7. [15]

    Chizat, G

    L. Chizat, G. Peyr´ e, B. Schmitzer, and F.-X. Vialard. Unbalanced optimal transport: Dynamic and Kantorovich formulations.J. Func. Anal., 274(11):3090–3123, 2018

    2018

  8. [16]

    M. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. InAdvances in Neural Information Processing Systems 26, pages 2292–2300, 2013

    2013

  9. [17]

    Fukushima, Z.Q

    M. Fukushima, Z.Q. Luo, and P. Tseng. Smoothing functions for second-order cone complementarity problems.SIAM J. Optim., 12(2):436–460, 2001

    2001

  10. [18]

    Gabay and B

    D. Gabay and B. Mercier. A dual algorithm for the solution of nonlinear variational problems via finite element approximation.Comput. Math. Appl., 2(1):17–40, 1976

    1976

  11. [19]

    Glowinski and A

    R. Glowinski and A. Marroco. Sur l’approximation, par ´ el´ ements finis d’ordre un, et la r´ esolution, par p´ enalisation-dualit´ e d’une classe de probl` emes de Dirichlet non lin´ eaires.Revue fran¸ caise d’automatique, informatique, recherche op´ erationnelle. Analyse num...

    1975

  12. [20]

    Golub and C.F

    G.H. Golub and C.F. Van Loan.Matrix Computations. The Johns Hopkins University Press, Balti- more, 4th edition, 2013

    2013

  13. [21]

    Graber and P

    P.J. Graber and P. Cardaliaguet. Mean field games systems of first order.ESAIM: Control Optim. Calc. Var., 21(3):690–722, 2015

    2015

  14. [22]

    Hestenes

    M. Hestenes. Multiplier and gradient methods.J. Optim. Theory Appl., 4(5):303–320, 1969

    1969

  15. [23]

    Kantorovich

    L.V. Kantorovich. On the transfer of masses (in Russian).Dokl. Akad. Nauk, 37(2):227–229, 1942

    1942

  16. [24]

    Kantorovich

    L.V. Kantorovich. On a problem of Monge.Uspekhi Mat. Nauk, 3:225–226, 1948

    1948

  17. [25]

    Kondratyev, L

    S. Kondratyev, L. Monsaingeon, and D. Voronikov. A new optimal transport distance on the space of finite Radon measures.Adv. Differential Equ., 21(11/12):1117–1164, 2016

    2016

  18. [26]

    Lavenant

    H. Lavenant. Unconditional convergence for discretizations of dynamical optimal transport.Math. Comput., 90:739–786, 2021

    2021

  19. [27]

    Lavenant, S

    H. Lavenant, S. Claici, E. Chine, and J. Solomon. Dynamical optimal transport on discrete surfaces. ACM Trans. Graph., 37(6):1–16, 2018

    2018

  20. [28]

    X.D. Li, D.F. Sun, and K.-C. Toh. A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions.Math. Program., 155:333–373, 2016

    2016

  21. [29]

    X.D. Li, D.F. Sun, and K.-C. Toh. A block symmetric Gauss-Seidel decomposition theorem for convex composite quadratic programming and its application.Math. Program., 175:395–418, 2019

    2019

  22. [30]

    Liero, A

    M. Liero, A. Mielke, and G. Savar´ e. Optimal entropy-transport problems and a new Hellinger- Kantorovich distance between positive measures.Invent. Math., 211(3):969–1117, 2018

    2018

  23. [31]

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

    G. Monge.M´ emoire sur la th´ eorie des d´ eblais et des remblais. Paris, 1781

  24. [32]

    Natale and G

    A. Natale and G. Todeschi. Computation of optimal transport with finite volume.ESAIM: Math. Model. Num., 55:1847–1871, 2021

    2021

  25. [33]

    Natale and G

    A. Natale and G. Todeschi. A mixed finite element discretization of dynamical optimal transport.J. Sci. Comput., 91(38), 2022

    2022

  26. [34]

    Peyr´ e and M

    G. Peyr´ e and M. Cuturi. Computational optimal transport: With application to data science.Found. Trends Mach. Learn., 11:355–607, 2019

    2019

  27. [35]

    M.J.D. Powell. A method for nonlinear constraints in minimization problems. In R. Fletcher, editor, in Optimization, pages 283–298. Academic Press, New York, 1969

    1969

  28. [36]

    Rockafellar.Convex Analysis

    R.T. Rockafellar.Convex Analysis. Princeton University Press, Princeton, 1970

    1970

  29. [37]

    Rockafellar

    R.T. Rockafellar. Augmented Lagrangians and applications of the proximal point algorithm in convex programming.Math. Oper. Res., 1(2):97–116, 1976

    1976

  30. [38]

    Santambrogio.Optimal Transport for Applied Mathematicians

    F. Santambrogio.Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling. Springer, Cham, 2015

    2015

  31. [39]

    Solomon, F

    J. Solomon, F. De Goes, G. Peyr´ e, M. Cuturi, A. Butscher, A. Nguyen, T. Du, and L. Guibas. Convolutional Wasserstein distance: Efficient optimal transportation on geometric domains.ACM Trans. Graph., 34(4):1–11, 2015. 22

    2015

  32. [40]

    Ulrich and P

    P. Ulrich and P. Konrad. Computing discrete minimal surfaces and their conjugates.Experimental Mathematics, 2(1):15–36, 1993

    1993

  33. [41]

    Villani.Optimal Transport: Old and New, volume 338 ofGrundlehren der mathematischen Wis- senschaften

    C. Villani.Optimal Transport: Old and New, volume 338 ofGrundlehren der mathematischen Wis- senschaften. Springer, Berlin, 2009

    2009

  34. [42]

    J.J. Yu, R.J. Lai, W.C. Li, and S. Osher. Computational mean-field games on manifolds.J. Comput. Phys., 484:112070, 2023

    2023

  35. [43]

    J.J. Yu, R.J. Lai, W.C. Li, and S. Osher. A fast proximal gradient method and convergence analysis for dynamic mean field planning.Math. Comput., 93:603–642, 2024

    2024

  36. [44]

    HOT: An efficient Halpern accelerating algorithm for optimal transport problems.IEEE Trans

    Guojun Zhang, Zhexuan Gu, Yancheng Yuan, and Defeng Sun. HOT: An efficient Halpern accelerating algorithm for optimal transport problems.IEEE Trans. Pattern Anal. Mach. Intell., 2025. A Snapshots of density evolutions This part presents the snapshots of the density evolution f...

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.