Pith. sign in

REVIEW 3 major objections 4 minor 44 references

Matching output–control pairs solves constrained calibration.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 19:22 UTC pith:XR46YFSB

load-bearing objection A correct but mostly definitional extension of the disintegration-based calibration program to known control-parameter marginals; the real question is whether the unpaired-data OT surrogate can be trusted, and the paper's evidence on that is thin. the 3 major comments →

arxiv 2607.16975 v1 pith:XR46YFSB submitted 2026-07-18 stat.ME

Non-Parametric Model Calibration with Stochastic Control Parameters

Akshay Prasadan , Samopriya Basu , Faezeh Yazdi , Derek Bingham , Donald Estep This is my paper
classification stat.ME MSC 62F1562G0549Q22
keywords statistical calibrationcomputer model calibrationdisintegration of measureoptimal transportentropic regularizationuncertainty quantificationBayesian inverse problemsnonparametric estimation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tackles the problem of calibrating a computer model when some stochastic inputs have unknown distributions (calibration parameters) while others have known distributions (control parameters). The authors show that the constrained calibration problem—fitting field data while preserving the known marginal of the control parameters—can be reduced to the standard unconstrained calibration problem by augmenting the model output with the control parameter itself. The key theoretical result is that matching the pushforward of the augmented map (Q(λ,u), u) automatically satisfies both constraints: matching the observed data distribution and matching the known control marginal. For the practically important case where control parameter values are not recorded alongside field observations, the paper builds a surrogate joint distribution using entropic optimal transport with a model-aware cost function, then applies the standard disintegration-based estimator. The result is a distributional estimate over the input space that respects both constraints, demonstrated on a quadratic model and a heat-equation example.

Core claim

The paper's central claim is Theorem 1: if a candidate joint distribution on Λ×U has the same pushforward as the trial-generating distribution under the augmented map Q̃(λ,u)=(Q(λ,u),u), then that candidate automatically solves the constrained statistical calibration problem—its pushforward under Q matches the observed field data, and its marginal on U matches the known control distribution. This reduction turns the constrained problem into an ordinary unconstrained SCP on the augmented output space D×U, for which a disintegration-based estimator already exists. When samples are unpaired, the paper supplies the missing joint pushforward by solving an entropically regularized optimal transpor

What carries the argument

The central object is the augmented computer model Q̃(λ,u)=(Q(λ,u),u), which embeds the control parameter into the output space. Any probability measure that matches the trial-generating measure's pushforward under Q̃ automatically satisfies both the data-matching condition and the known-marginal condition. To handle unpaired data, the paper constructs a surrogate for the joint distribution of (Q,u) by solving the entropic Kantorovich problem with the model-aware cost c(q,u)=∫_Λ |q−Q(λ,u)|² ρ_Λ(λ)dλ, regularized by KL divergence to the independent coupling; the Sinkhorn–Knopp algorithm computes the coupling. The disintegration-based estimator is then applied on the augmented space, which req

Load-bearing premise

For unpaired data, the surrogate coupling obtained by optimal transport stands in for the unobserved joint distribution of outputs and control parameters, and the final calibration inherits this assumption without any way to verify it from the data alone.

What would settle it

Construct a synthetic problem where the true coupling has strong negative dependence between Q and u (e.g., u = 1 − Q), run both the paired and unpaired algorithms, and compare the estimated marginal distributions on Λ. If the unpaired estimate substantially differs from the paired estimate despite both matching the desired marginals, then the optimal transport surrogate fails to recover scientifically plausible dependence.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • When paired data exist, the constrained calibration problem is solved by applying the known unconstrained algorithm directly to the augmented model, requiring no new estimation scheme.
  • When only unpaired data exist, the model-aware optimal transport coupling provides a surrogate for the missing joint, and the resulting calibration preserves both the data distribution and the known control marginal by construction.
  • The augmented approach imposes a dimension condition (m ≤ n) but relaxes the Jacobian rank condition to only the λ-derivatives of Q, which may ease verification of regularity conditions.
  • In the examples, the calibration solution is stable across a broad range of the entropy regularization parameter ε, even though the transport coupling itself varies substantially, suggesting an unexpected robustness worth theoretical investigation.
  • The method extends the scope of non-parametric calibration to problems with partially specified stochastic inputs, including physical settings where control parameters follow well-established distributions.

Where Pith is reading between the lines

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

  • The surrogate coupling for unpaired data is fundamentally unidentifiable from the two marginals alone; any coupling with the given marginals yields a valid constrained SCP solution, but only the true coupling captures the actual physical dependence, so the calibrated distribution on Λ can be misleading even when both constraints are satisfied.
  • Different choices of the transport cost (e.g., copula-based or domain-informed costs) would produce different—still constraint-satisfying—calibrations, meaning the optimal transport step is not just computational but encodes a modeling assumption about the joint structure.
  • The known continuity results for the unconstrained SCP suggest a limiting argument: as the control distribution concentrates to a point, the constrained solution should degenerate into a conditional calibration at that fixed control value, bridging to classical fixed-control calibration.
  • A testable extension is to run the unpaired algorithm on synthetic data where the true coupling is deliberately anti-correlated with the model-aware cost, then compare the resulting Λ-marginal against the paired-data benchmark to reveal potential bias from the surrogate coupling.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper extends non-parametric, measure-disintegration-based computer model calibration to the setting where the input space contains both calibration parameters with unknown distribution and control parameters with a known marginal distribution. The constrained SCP (Definition 3.1) requires the solution to match both the data pushforward and the known control marginal. The paper's formal device is to augment the computer model to \tilde Q=(Q,\pi_U) (Eq. (1)) and prove (Theorem 1) that matching the augmented pushforward of the trial-generating distribution is sufficient for both constraints. For paired data, the existing SCP algorithm is applied directly with \tilde Q. For unpaired data, the paper constructs a surrogate joint distribution \hat\Pi on D\times U via entropic optimal transport with cost c(q,u)=∫|q-Q(λ,u)|^2ρ_Λ(λ)dλ (Eq. (4)), samples synthetic pairs from \hat\Pi, and then runs the paired algorithm. Two synthetic experiments (a quadratic model and a heat-equation model with a surrogate) are used to argue that the unpaired method performs nearly as well as the paired method.

Significance. If the unpaired-data extension were valid, the paper would contribute a practical method for calibrating computer models with stochastic control parameters whose marginal is known but whose joint dependence with outputs is unobserved. The manuscript has several genuine strengths: it ships reproducible code, Theorem 1 is a clean and correct marginalization argument, Theorem 2 is a standard existence result, and the regularity conditions are discussed carefully in an appendix. However, the central empirical claim — that the unpaired method is nearly as good as paired calibration — is not supported by the evidence presented. The unpaired pipeline relies on an arbitrary optimal-transport surrogate for an unidentifiable joint distribution, and the paper offers no sensitivity analysis, no dependent-TGD experiments, and no quantitative comparison. These issues are load-bearing because the unpaired case is the main methodological novelty; the formal paired case is a straightforward application of existing methods.

major comments (3)
  1. [§3.3, Eq. (4)] In the unpaired case, Algorithm 3 replaces the unavailable augmented pushforward \tilde Q P^t_{\Lambda\times U} with the entropic-OT coupling \hat\Pi computed by Eq. (3) under cost (4). Since only the marginals P_D and P^t_U are identified from unpaired data, every coupling in F(P_D,P^t_U) satisfies the two constraints in Definition 3.1 by construction; the final \hat P_{\Lambda\times U} depends on which coupling is chosen. The paper does not show that \hat P_\Lambda is insensitive to this choice or close to the paired-data solution, and the authors themselves note that the Sinkhorn joint 'certainly differs from the true distribution.' This is load-bearing: the advertised scientific plausibility of the unpaired estimate is not established. I request (i) a sensitivity analysis with respect to the cost function and the regularization \epsilon, and (ii) experiments where the TGD has non-tri
  2. [§4, Figures 2, 5, 6] The central empirical claim that the unpaired method 'yields similar estimates as paired data' rests on two single-run synthetic examples with visual comparison only. No quantitative distance or discrepancy measure between the paired and unpaired \hat P_\Lambda is reported, and there are no repeated simulations with different random seeds or Sinkhorn draws. Since this claim is the main motivation for the unpaired algorithm, the evidence is insufficient as it stands. Please add quantitative comparisons (e.g., L1 or Wasserstein distance between estimated marginal densities of \Lambda), repeated-run variability, and ideally a check of whether the unpaired solution remains close to paired when the surrogate coupling is changed.
  3. [Theorem 1, Definition 3.1] Theorem 1 is correct, but it only reduces the constrained problem to exact matching of \tilde Q P^t_{\Lambda\times U}. In the unpaired pipeline this target is replaced by the surrogate \hat\Pi, so Theorem 1 does not validate Algorithm 3. The U-marginal agreement of the final solution is imposed by construction in the OT step and then reproduced by Algorithm 2; it is not a data-informed property. The paper should state this limitation explicitly and separate the formal contribution (Theorem 1 and the paired Algorithm 2) from the heuristic unpaired extension, which currently requires an unvalidated modeling assumption.
minor comments (4)
  1. [§4.2] The text says 'We return to the heat transfer problem of Example 4.2' but the running example is Example 1.1.
  2. [§3.3] The statement that without the KL term '\hat\Pi would overfit to the graph of Q and not admit a density in Lebesgue measure' is imprecise for the discrete implementation described in Algorithm 3. Clarify whether this refers to the continuous population problem or to the discrete Sinkhorn approximation.
  3. [Algorithms 1, 2] The histogram estimator has undefined 0/0 ratios in bins that contain no prior samples. The paper should state how zero-count bins are handled in the implementation.
  4. [§4.2] The heat-equation example uses an XGBoost surrogate for the PDE solver, and Section 5 acknowledges that this introduces additional statistical error and may violate regularity conditions. A diagnostic of surrogate accuracy (e.g., held-out error) would help the reader assess the example.

Circularity Check

0 steps flagged

No significant circularity; the known-marginal constraint is enforced by construction, not presented as an independently predicted quantity.

full rationale

The paper's core reduction, Theorem 1, is a transparent mathematical equivalence rather than a circular derivation: the augmented map is explicitly defined as Qtilde=(Q, pi_U), so matching Qtilde's pushforward is by definition equivalent to matching both the Q-pushforward and the U-marginal. This is a sound reduction, not a hidden reuse of the conclusion. In the paired case, Algorithm 2 directly applies the prior SCP method to samples from Qtilde P^t, so the constrained result follows from the stated assumptions without circularity. In the unpaired case, Algorithm 3 constructs a surrogate joint distribution Pi-hat by entropic optimal transport with fixed marginals P_D and P_U. Consequently, the final solution's U-marginal matches P_U by construction. The paper presents this as a constraint-satisfaction feature, not as a fitted prediction or as evidence that the surrogate coupling is the true joint distribution; indeed it explicitly states that the SK solution 'certainly differs from the true distribution.' The arbitrariness of the OT cost and the unidentifiability of the joint coupling are substantive statistical-modeling limitations, but they are not circularity: the final U-marginal is not claimed to be inferred from the data. The paper also acknowledges surrogate-model regularity concerns in Section 5. Some foundational citations, especially [9], are by overlapping authors and are load-bearing building blocks, but the present contribution is tested on synthetic examples and is not justified solely by a self-citation chain or by an imported uniqueness theorem. Overall, no step in the claimed derivation reduces to its own inputs in a way that would constitute circular reasoning.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claim rests on three pillars: (1) the unconstrained SCP theory of [9], by overlapping authors; (2) regularity conditions that the surrogate-model heat example may violate; and (3) an ad hoc entropic-OT coupling for unpaired data whose choice is not identified by the observed marginals. There are no physically invented entities; the augmented map is a mathematical construction rather than a new postulated object.

free parameters (3)
  • Entropic OT regularization epsilon = epsilon = 1 in both examples
    Chosen by hand; balances the transport cost against the KL penalty. The paper notes tuning is required and too-small values cause numerical issues, but no principled selection or sensitivity analysis is given.
  • Histogram bin counts M, M1, M2 = M=30 (or 20 in heat example), M1=M2=30
    Algorithmic resolutions for estimating P_D and P_U and for the augmented SCP estimator; affect bias and are chosen without a data-driven rule.
  • Monte Carlo sample sizes K3, K4, J = K3=15,000, K4=100,000, J=250,000-300,000
    Used for cost estimation, synthetic-pair sampling, and importance sampling; chosen to be large for numerical stability but not derived from error targets.
axioms (5)
  • domain assumption The unconstrained SCP machinery of [9] - the disintegration formula, Algorithm 1 estimator, and its convergence - is correct and applies to the augmented map \tilde Q.
    The entire method reuses [9] verbatim on \tilde Q; the authors overlap with [9]'s authors (Bingham, Estep), so this is a self-cited load-bearing input rather than independently verified in this paper.
  • domain assumption The regularity conditions in Appendix A hold for Lambda, Q, the prior, the TGD, and the relevant pushforwards (compactness, C^1 a.e., Jacobian rank, densities, boundary conditions).
    Algorithm outputs and theoretical guarantees depend on these conditions. In the heat example, the XGBoost surrogate model is not guaranteed to satisfy them; the paper itself flags this concern in Section 5.
  • ad hoc to paper For unpaired data, the entropic OT coupling \hat\Pi with cost c(q,u) = \int_\Lambda |q-Q(\lambda,u)|^2 \rho_\Lambda(\lambda)d\lambda is an acceptable stand-in for the unobserved joint pushforward \tilde Q P^t_{\Lambda\times U}.
    This is the paper's key heuristic (Eq. 4). The true coupling is unidentifiable from the marginals, and the final distribution over calibration parameters depends on this choice.
  • domain assumption The reference measure for the KL penalty is the independent coupling P_D \times P^t_U, and the prior factorizes as \rho_\Lambda \rho^t_U.
    Used in Eq. (3) and throughout the construction; a reasonable default but not forced by the problem.
  • standard math Theorem 3's measurable selection result depends on the parametrized Kantorovich theorem of Bogachev-Malofeev [43] and lower semicontinuity of KL divergence [44].
    Invoked in Appendix B for the alternative lambda-averaged OT formulation; not used in the main numerical pipeline.

pith-pipeline@v1.3.0-alltime-deepseek · 17932 in / 14972 out tokens · 141789 ms · 2026-08-01T19:22:29.254847+00:00 · methodology

0 comments
read the original abstract

We present a method for calibrating a computer model using non-parametric techniques where the inputs are stochastic but include calibration parameters whose distributions are unknown and control parameters whose distributions are specified. Our solution gives a distributional estimate over the input space that is consistent with observed field data, while also preserving the distribution of the known marginal of the control parameters. This property is desirable since stochastic inputs often include physical processes affecting the experimental conditions, and a scientifically plausible calibration estimate should preserve well-established distributional properties of these inputs. The method builds on recently developed non-parametric computer model calibration techniques based on the disintegration of measure and Bayesian inference.

Figures

Figures reproduced from arXiv: 2607.16975 by Akshay Prasadan, Derek Bingham, Donald Estep, Faezeh Yazdi, Samopriya Basu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 2 linked inside Pith

  1. [1]

    https://books.google.ca/books?id=VGqbyIUVZw8C

    Fang, K., Li, R., and Sudjianto, A.,Design and Modeling for Computer Experiments, Chapman & Hall/CRC Computer Science & Data Analysis, CRC Press, 2005. https://books.google.ca/books?id=VGqbyIUVZw8C

    2005

  2. [2]

    and O’Hagan, A., Bayesian Calibration of Computer Models,Journal of the Royal Statistical Society Series B: Statistical Methodology, 63(3):425–464, 01 2002

    Kennedy, M.C. and O’Hagan, A., Bayesian Calibration of Computer Models,Journal of the Royal Statistical Society Series B: Statistical Methodology, 63(3):425–464, 01 2002

    2002

  3. [3]

    Breidt, J., Butler, T., and Estep, D., A measure-theoretic computational method for inverse sensitivity problems I: Method and analysis,SIAM Journal on Numerical Analysis, 49(5):1836–1859, 2011

    2011

  4. [4]

    Butler, T., Estep, D., and Sandelin, J., A computational measure theoretic approach to inverse sensitivity problems II: A posteriori error analysis,SIAM Journal on Numerical Analysis, 50(1):22–45, 2012

    2012

  5. [5]

    Butler, T., Estep, D., Tavener, S., Dawson, C., and Westerink, J., A measure-theoretic computational method for inverse sensitivity problems III: Multiple quantities of interest,SIAM/ASA Journal on Uncertainty Quantification, 2(1):174–202, 2014

    2014

  6. [6]

    Butler, T., Jakeman, J., and Wildey, T., Combining push-forward measures and Bayes’ rule to construct consistent solutions to stochastic inverse problems,SIAM Journal on Scientific Computing, 40(2):A984–A1011, 2018

    2018

  7. [7]

    Chi, J., Sliced inverse approach and domain recovery for stochastic inverse problems, PhD thesis, Colorado State University, 2021

    2021

  8. [8]

    Bingham, D., Butler, T., and Estep, D., Inverse problems for physics-based process models,Annual Review of Statistics and Its Application, 11, 2024

    2024

  9. [9]

    Shi, H., Yang, L., Chi, J., Butler, T., Wang, H., Bingham, D., and Estep, D., Nonparametric Bayesian Calibration of Computer Models,arXiv preprint arXiv:2509.22597, 2025, submitted to the SIAM/ASA Journal on Uncertainty Quantification

    Pith/arXiv arXiv 2025

  10. [10]

    Yang, L., Infinite-dimensional stochastic inverse problems, PhD thesis, Colorado State University, 2018

    2018

  11. [11]

    and Pollard, D., Conditioning as disintegration,Statistica Neerlandica, 51(3):287–317, 1997

    Chang, J.T. and Pollard, D., Conditioning as disintegration,Statistica Neerlandica, 51(3):287–317, 1997

    1997

  12. [12]

    Prasadan, A., Estep, D., and Bingham, D., Continuity of the solution of a non-parametric bayesian statistical calibration procedure,arXiv preprint arXiv:2603.20665, 2026, in revision with the SIAM/ASA Journal on Uncertainty Quantification

    arXiv 2026

  13. [13]

    Higdon, D., Kennedy, M., Cavendish, J., Cafeo, J., and Ryne, R., Combining field data and computer simulations for calibra- tion and prediction,SISC, 26:448–466, 2004

    2004

  14. [14]

    Higdon, D., Gattiker, J., Williams, B., and Rightley, M., Computer model calibration using high-dimensional output,JASA, 103:570–583, 2008

    2008

  15. [15]

    Wang, S., Chen, W., and Tsui, K.L., Bayesian validation of computer models,Techno., 51:439–451, 2009

    2009

  16. [16]

    and Wu, J., Efficient calibration for imperfect computer models,Ann

    Tuo, R. and Wu, J., Efficient calibration for imperfect computer models,Ann. Statist., 43:2331–2352, 12 2015

    2015

  17. [17]

    Plumlee, M., Bayesian calibration of inexact computer models,JASA, 112:1274–1285, 2017

    2017

  18. [18]

    Grosskopf, M., Bingham, D., Adams, M., Hawkins, W., and Perez-Nunez, D., Generalized computer model calibration for radiation transport simulation,Techno, 63:27–39, 2021

    2021

  19. [19]

    and Tuo, R., A review of computer model calibration,WIREs Comput

    Sung, C. and Tuo, R., A review of computer model calibration,WIREs Comput. Stat., 16, 02 2024

    2024

  20. [20]

    and Tarantola, A

    Mosegaard, K. and Tarantola, A. Probabilistic approach to inverse problems. In: Lee, W., Kanamori, H., Jennings, P., and Kisslinger, C. (Eds.),International Handbook of Earthquake and Engineering Seismology, Part A, V ol. 81 of International Geophysics, pp. 237–265. Academic Press, 2002

    2002

  21. [21]

    Mosegaard, K.,Monte Carlo Analysis of Inverse Problems, Flemming Nicolaisen, Copenhagen, 2006

    2006

  22. [22]

    and Taylor, A., Multiple models to capture the variability in biological neurons and networks,Nature neuroscience, 14(2):133–138, 2011

    Marder, E. and Taylor, A., Multiple models to capture the variability in biological neurons and networks,Nature neuroscience, 14(2):133–138, 2011

    2011

  23. [23]

    NAS, 110(23):E2098–E2105, 2013

    Britton, O., Bueno-Orovio, A., Van Ammel, K., Lu, H., Towart, R., Gallacher, D., and Rodriguez, B., Experimentally cali- brated population of models predicts and explains intersubject variability in cardiac cellular electrophysiology,Proceed. NAS, 110(23):E2098–E2105, 2013

    2013

  24. [24]

    Schervish, M.,Theory of Statistics, Springer, New York, 1995

    1995

  25. [25]

    (Eds.),Bayesian Nonparametrics, Cambridge Series in Statistical and Prob- abilistic Mathematics, Cambridge University Press, 2010

    Hjort, N., Holmes, C., Muller, P., and Walker, S. (Eds.),Bayesian Nonparametrics, Cambridge Series in Statistical and Prob- abilistic Mathematics, Cambridge University Press, 2010. Calibration with Stochastic Control Parameters19

    2010

  26. [26]

    and van der Vaart, A.,Fundamentals of Nonparametric Bayesian Inference, Cambridge University Press, New York, 2017

    Ghosal, S. and van der Vaart, A.,Fundamentals of Nonparametric Bayesian Inference, Cambridge University Press, New York, 2017

    2017

  27. [27]

    Li, Q., Wang, L., and Yang, Y ., Differential equation–constrained optimization with stochasticity,SIAM/ASA JUQ, 12(2):549– 578, 2024

    2024

  28. [28]

    Vadeboncoeura, A., Girolamia, M., and Stuart, A., Efficient deconvolution in populational inverse problems,arXiv preprint arXiv:2505.19841v1, 2025

    Pith/arXiv arXiv 2025

  29. [29]

    Prob., 30:110301, 2014

    Calvetti, D., Kaipio, J., and Somersalo, E., Inverse problems in the Bayesian framework,Inv. Prob., 30:110301, 2014

    2014

  30. [30]

    Stuart, A., Inverse problems: A Bayesian perspective,Acta Num., 19:451–559, 2010

    2010

  31. [31]

    Tarantola, A.,Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, 2005

    2005

  32. [32]

    https://jmlr.org/papers/v23/19-843.html

    Lin, T., Ho, N., Cuturi, M., and Jordan, M.I., On the complexity of approximating multimarginal optimal transport,Journal of Machine Learning Research, 23(65):1–43, 2022. https://jmlr.org/papers/v23/19-843.html

    2022

  33. [33]

    Nelsen, R.B.,An introduction to copulas, Springer, 2006

    2006

  34. [34]

    87, Springer, 2015

    Santambrogio, F.,Optimal transport for applied mathematicians, V ol. 87, Springer, 2015

    2015

  35. [35]

    338, Springer, 2008

    Villani, C.,Optimal transport: old and new, V ol. 338, Springer, 2008

    2008

  36. [36]

    Nutz, M., Introduction to entropic optimal transport,Lecture notes, Columbia University, 2021

    2021

  37. [37]

    and Sinkhorn, R., Concerning nonnegative matrices and doubly stochastic matrices.,Pacific Journal of Mathematics, 21(2):343 – 348, 1967

    Knopp, P. and Sinkhorn, R., Concerning nonnegative matrices and doubly stochastic matrices.,Pacific Journal of Mathematics, 21(2):343 – 348, 1967

    1967

  38. [38]

    https://jmlr.org/papers/v22/20-451.html

    Flamary, R., Courty, N., Gramfort, A., Alaya, M.Z., Boisbunon, A., Chambon, S., Chapel, L., Corenflos, A., Fatras, K., Fournier, N., Gautheron, L., Gayraud, N.T., Janati, H., Rakotomamonjy, A., Redko, I., Rolet, A., Schutz, A., Seguy, V ., Sutherland, D.J., Tavenard, R., Tong, A., and Vayer, T., Pot: Python optimal transport,Journal of Machine Learning Re...

    2021

  39. [39]

    POT Python Optimal Transport,

    Flamary, R., Vincent-Cuaz, C., Courty, N., Gramfort, A., Kachaiev, O., Quang Tran, H., David, L., Bonet, C., Cassereau, N., Gnassounou, T., Tanguy, E., Delon, J., Collas, A., Mazelet, S., Chapel, L., Kerdoncuff, T., Yu, X., Feickert, M., Krzakala, P., Liu, T., Fernandes Montesuma, E., Neike, N., Genest, B., Coeurjolly, D., and Corneli, M. POT Python Optim...

  40. [40]

    http://bobby.gramacy.com/surrogates/

    Gramacy, R.B.,Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences, Chapman Hall/CRC, Boca Raton, Florida, 2020. http://bobby.gramacy.com/surrogates/

    2020

  41. [41]

    http://www.ctcms.nist.gov/fipy

    Guyer, J.E., Wheeler, D., and Warren, J.A., FiPy: Partial differential equations with Python,Computing in Science & Engi- neering, 11(3):6–15, 2009. http://www.ctcms.nist.gov/fipy

    2009

  42. [42]

    and Guestrin, C., XGBoost: A scalable tree boosting system, InProceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, pp

    Chen, T. and Guestrin, C., XGBoost: A scalable tree boosting system, InProceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’16, pp. 785–794, New York, NY , USA, 2016. ACM

    2016

  43. [43]

    and Malofeev, I., Nonlinear kantorovich problems with a parameter,The Bulletin of Irkutsk State University

    Bogachev, V .I. and Malofeev, I., Nonlinear kantorovich problems with a parameter,The Bulletin of Irkutsk State University. Series Mathematics, 2022

    2022

  44. [44]

    Posner, E., Random coding strategies for minimum entropy,IEEE Transactions on Information Theory, 21(4):388–391, 1975

    1975