Optimal Transport Research Articles

(Digital Presentation) The Effect of Coating Quality on Water Transport in the Gas Diffusion Layer of PEM Fuel Cells

Present study examines the impact of the quality of the GDL non-wetting coating on water transport. It is found that there are two distinct regions defined by the coating quality which present fundamentally distinct behaviours, one region characterized by few coating defects with the second marked by significant coating degradation. Simulations consider the influence of heterogeneous wetting and pinning conditions on protruding water droplets in contact with the GDL fibrous matrix. The impact is assessed by examining the breakthrough pressure as a function of protruding droplet volumes. When a non-wetting coating such as PTFE is applied, inherent coating defects determine a mixed-non-wetting (MNW) behaviour of water passing through the pores of the GDL. The MNW behaviour for a quasi-uniform non-wetting coating having less than 25% defects is characterized by a high breakthrough pressure, yielding improved transport properties in the stable displacement regime. The MNW meniscus is critical to establishing a high breakthrough pressure and a small characteristic droplet volume, parameters considered to be important in the design of a GDL that promotes optimal water transport. The coating degrades under normal operating conditions and as a consequence it changes the water transport characteristics of the porous medium. In this range where coating defects are predominant it is found that the water breakthrough pressure changes significantly, rendering poor performance below that of a non-coated wetting GDL. Various degrees of coating degradation are considered and the corresponding Laplace pressure is determined as a function of breakthrough droplet volumes.

An optimal transport approach to estimating causal effects via nonlinear difference-in-differences

Abstract We propose a nonlinear difference-in-differences (DiD) method to estimate multivariate counterfactual distributions in classical treatment and control study designs with observational data. Our approach sheds a new light on existing approaches like the changes-in-changes estimator and the classical semiparametric DiD estimator, and it also generalizes them to settings with multivariate heterogeneity in the outcomes. The main benefit of this extension is that it allows for arbitrary dependence between the coordinates of vector potential outcomes and includes higher-dimensional unobservables, something that existing methods cannot provide in general. We demonstrate its utility on both synthetic and real data. In particular, we revisit the classical Card & Krueger dataset, which reports fast food restaurant employment before and after a minimum wage increase. A reanalysis with our methodology suggests that these restaurants substitute full-time labor with part-time labor on aggregate in response to a minimum wage increase. This treatment effect requires estimation of the multivariate counterfactual distribution, an object beyond the scope of classical causal estimators previously applied to this data.

OTMorph: Unsupervised Multi-domain Abdominal Medical Image Registration Using Neural Optimal Transport.

Deformable image registration is one of the essential processes in analyzing medical images. In particular, when diagnosing abdominal diseases such as hepatic cancer and lymphoma, multi-domain images scanned from different modalities or different imaging protocols are often used. However, they are not aligned due to scanning times, patient breathing, movement, etc. Although recent learning-based approaches can provide deformations in real-time with high performance, multi-domain abdominal image registration using deep learning is still challenging since the images in different domains have different characteristics such as image contrast and intensity ranges. To address this, this paper proposes a novel unsupervised multi-domain image registration framework using neural optimal transport, dubbed OTMorph. When moving and fixed volumes are given as input, a transport module of our proposed model learns the optimal transport plan to map data distributions from the moving to the fixed volumes and estimates a domain-transported volume. Subsequently, a registration module taking the transported volume can effectively estimate the deformation field, leading to deformation performance improvement. Experimental results on multi-domain image registration using multi-modality and multi-parametric abdominal medical images demonstrate that the proposed method provides superior deformable registration via the domain-transported image that alleviates the domain gap between the input images. Also, we attain the improvement even on out-of-distribution data, which indicates the superior generalizability of our model for the registration of various medical images. Our source code is available at https://github.com/boahK/OTMorph.

Enhancing supply chain relationships in the circular economy: Strategies for a green centralized supply chain with deteriorating products

This article introduces a green centralized supply chain in a two-stage stochastic programming model using deteriorating products. The model reduces the cost of purchasing, transporting, storing, product recovery and shortages. This cuts down on greenhouse emission related to transportation, product recovery, and recycling programs. On the basis of this, we explore the utilization of the circular economy to the damages that could occur from used products. Furthermore, revenue sharing and quantity discount contracts are examined in the business models between the members of the supply chain and the external manufacturer. Demand is assumed to be uncertain, and scenarios are created to account this. The model specifies the optimal order quantities, transportation modes and contract terms that minimize costs and environmental impacts. Numerical examples analyze the trade-offs between economic and environmental objectives under different supply chain parameters. The results provide insights for circular supply chains that reconcile economic incentives with environmental responsibility for deteriorating product.

Reducing Mode Collapse With Monge-Kantorovich Optimal Transport for Generative Adversarial Networks.

Mode collapse has been a persisting challenge in generative adversarial networks (GANs), and it directly affects the applications of GAN in many domains. Existing works that attempt to solve this problem have some serious limitations: models using optimal transport (OT) strategies (e.g., Wasserstein distance) lead to vanishing or exploding gradients; increasing the number of generators can cause several generators focusing on the same mode; and approaches that modify the loss also do not satisfactorily resolve mode collapse. In this article, we reduce mode collapse by formulating it as a Monge problem of OT map. We show that the Monge problem can be transformed to the distribution transformation problem in GAN, and a rectified affine neural network can be considered as a measurable function. In this way, we propose Monge GAN that uses this measurable function to transform the generated data distribution into the original data distribution. We utilize the Kantorovich formulation to obtain the OT cost, which is regarded as the OT distance between the two distributions. Finally, we conduct extensive experiments on both image and numerical datasets to validate our Monge GAN in reducing model collapse.

Beyond Linear Response: Equivalence between Thermodynamic Geometry and Optimal Transport.

A fundamental result of thermodynamic geometry is that the optimal, minimal-work protocol that drives a nonequilibrium system between two thermodynamic states in the slow-driving limit is given by a geodesic of the friction tensor, a Riemannian metric defined on control space. For overdamped dynamics in arbitrary dimensions, we demonstrate that thermodynamic geometry is equivalent to L^{2} optimal transport geometry defined on the space of equilibrium distributions corresponding to the control parameters. We show that obtaining optimal protocols past the slow-driving or linear response regime is computationally tractable as the sum of a friction tensor geodesic and a counterdiabatic term related to the Fisher information metric. These geodesic-counterdiabatic optimal protocols are exact for parametric harmonic potentials, reproduce the surprising nonmonotonic behavior recently discovered in linearly biased double well optimal protocols, and explain the ubiquitous discontinuous jumps observed at the beginning and end times.

Optimal Transport with Constraints: From Mirror Descent to Classical Mechanics.

Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is limited by the lack of principled and flexible ways to incorporate realistic constraints. We propose a principled physics-based approach to impose constraints flexibly in optimal transport problems. Constraints are included in mirror descent dynamics using the D'Alembert-Lagrange principle from classical mechanics. This results in a sparse, local and linear approximation of the feasible set leading in many cases to closed-form updates.

Velocity model-based adapted meshes using optimal transport

In geophysical numerical models using the finite-element method or its variant, the spectral-element method, to solve seismic wave equations, a mesh is employed to discretize the domain. Generating or adapting a mesh to complex geological properties is a challenging task. To tackle this challenge, we develop an r-adaptivity method to generate or adapt a two-dimensional mesh to a seismic velocity field. Our scheme relies on the optimal transport theory to perform vertices relocation, which generates good-shaped meshes and prevents tangled elements. The mesh adaptation can delineate different regions of interest, like sharp interfaces, salt bodies, and discontinuities. The algorithm has a few user-defined parameters that control the mesh density. With typical seismic velocity examples (e.g., Camembert, SEAM Phase, Marmousi-2), mesh adaptation capability is illustrated within meshes with triangular and quadrilateral elements, commonly employed in seismic codes. Besides its potential use in mesh generation, the method developed can be embedded in seismic inversion workflows like multiscale full waveform inversion to adapt the mesh to the field being inverted without incurring the I/O cost of re-meshing and load rebalancing in parallel computations. The method can be extended to three-dimensional meshes.

Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group

Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group

Free boundary regularity and support propagation in mean field games and optimal transport

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys C1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be Hölder continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

Optimal transport divergences induced by scoring functions

We employ scoring functions, used in statistics for eliciting risk functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport problem. This gives rise to a rich variety of novel asymmetric MK divergences, subsuming Bregman-Wasserstein divergences. We show that for distributions on the real line, the comonotonic coupling is optimal for the majority of the new divergences. We conclude with two applications to robust optimisation.

Unbalanced Optimal Transport and Maximum Mean Discrepancies: Interconnections and Rapid Evaluation

This contribution presents substantial computational advancements to compare measures even with varying masses. Specifically, we utilize the nonequispaced fast Fourier transform to accelerate the radial kernel convolution in unbalanced optimal transport approximation, built upon the Sinkhorn algorithm. We also present accelerated schemes for maximum mean discrepancies involving kernels. Our approaches reduce the arithmetic operations needed to compute distances from On2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {O}}}\\left( n^{2}\\right) $$\\end{document} to Onlogn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathcal {O}}}}\\left( n \\log n \\right) $$\\end{document}, opening the door to handle large and high-dimensional datasets efficiently. Furthermore, we establish robust connections between transportation problems, encompassing Wasserstein distance and unbalanced optimal transport, and maximum mean discrepancies. This empowers practitioners with compelling rationale to opt for adaptable distances.

Opportunity Analysis of Phosphorus Recovery from Municipal Wastewater for Cropland Based on the Simulated Vehicle Transport Distance in the Yangtze River Delta, China.

With the rapid depletion of phosphate rocks and increasing agricultural demand, establishing a phosphorus (P) flow "loop" rather than a one-way trajectory between cropland and urban areas was imperative. Recovering P from municipal wastewater stood as a viable strategy to mitigate reliance on traditional P-containing chemical fertilizer. This study analyzed the intricate relationships between the potentials of P recovery from municipal wastewater and the P demand of croplands in the populated Yangtze River Delta (YRD), China. An indicator of the P vehicle transport distance was constructed and calculated to estimate the potential to recover and reuse P in agriculture, applying the simulated annealing (SA) algorithm and road networks obtained from OpenStreetMap (OSM). The results indicated that, on a regional scale, recovered P from municipal wastewater could fulfill 14.0% of the cropland P demands in the YRD, with a median P vehicle transport distance of 3.1 km/Mg of P. Notably, the P vehicle transport distance varied largely depending upon the cropland distributions, road density, and P recovery potential from municipal wastewater. The novel methodology developed here determined the optimal transportation routes for P recovery from wastewater treatment plants (WWTPs) to cropland, which played a crucial role in refining the wastewater management strategies aligned with the United Nations Sustainable Development Goals.

Subdivision scheme for discrete probability measure-valued data

This paper deals with the approximation of discrete probability measure-valued data by a new subdivision scheme. Its construction relies on a coupling between linear subdivision and optimal transport. A mathematical analysis is performed to study its convergence. Two test cases are finally described to emphasize its capability: the first one is related to point cloud interpolation while the second one is a first attempt in the framework of image approximation.

A Paradox of Telecommuting and Staggered Work Hours in the Bottleneck Model

We study the long- and short-term effects of telecommuting (TLC), staggered work hours (SWH), and their combined scheme on peak-period congestion and location patterns. In order to enable a unified comparison of the schemes’ long- and short-term effects, we develop a novel equilibrium analysis approach that consistently synthesizes the long-term equilibrium (location and percentage of telecommuting choice) and short-term equilibrium (preferred arrival time and departure time choice). By exploiting their special mathematical structures similar to optimal transport problems, we derive the closed-form solution to the long- and short-term equilibrium while explicitly considering their interaction. These closed-form solutions elucidate the discrepancies between the effects of each scheme and uncover a paradoxical finding: the introduction of SWH, in conjunction with TLC, may increase the total commuting costs compared with the scenario with only TLC, without yielding any improvement in worker utility. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: This work was supported by Council for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), the 3rd period of SIP “Smart Infrastructure Management System” [Grant JPJ012187] (Funding agency: PublicWorks Research Institute, Japan). This work was also supported by Japan Society for the Promotion of Science (JSPS) KAKENHI [Grants JP20J21744, JP21H01448, JP24K00999, JP20K14843, and JP23K13418] and the Support Program for Urban Studies of the Obayashi Foundation. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0520 .

Optimal transport for types and convex analysis for definable predicates in tracial W⁎-algebras

We investigate the connections between continuous model theory, free probability, and optimal transport/convex analysis in the context of tracial von Neumann algebras. In particular, we give an analog of Monge-Kantorovich duality for optimal couplings where the role of probability distributions on Cn is played by model-theoretic types, the role of real-valued continuous functions is played by definable predicates, and the role of continuous function Cn→Cn is played by definable functions. In the process, we also advance the understanding of definable predicates and definable functions by showing that all definable predicates can be approximated by “C1 definable predicates” whose gradients are definable functions. As a consequence, we show that every element in the definable closure of W⁎(x1,…,xn) can be expressed as a definable function of (x1,…,xn). We give several classes of examples showing that the definable closure can be much larger than W⁎(x1,…,xn) in general.

Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

Wasserstein barycenter regression: application to the joint dynamics of regional GDP and life expectancy in Italy

AbstractWe propose to investigate the joint dynamics of regional gross domestic product and life expectancy in Italy through Wasserstein barycenter regression derived from optimal transport theory. Wasserstein barycenter regression has the advantage of being flexible in modeling complex data distributions, given its ability to capture multimodal relationships, while maintaining the possibility of incorporating uncertainty and priors, other than yielding interpretable results. The main findings reveal that regional clusters tend to emerge, highlighting inequalities in Italian regions in economic and life expectancy terms. This suggests that targeted policy actions at a regional level fostering equitable development, especially from an economic viewpoint, might reduce regional inequality. Our results are validated by a robustness check on a human mobility dataset and by an illustrative forecasting exercise, which confirms the model’s ability to estimate and predict joint distributions and produce novel empirical evidence.

Inferring potential landscapes: A Schrödinger bridge approach to maximum caliber

Schrödinger bridges have emerged as an enabling framework for unveiling the stochastic dynamics of systems based on marginal observations at different points in time. The terminology “bridge” refers to a probability law that suitably interpolates such marginals. The theory plays a pivotal role in a variety of contemporary developments in machine learning, stochastic control, thermodynamics, and biology, to name a few, impacting disciplines such as single-cell genomics, meteorology, and robotics. In this work, we extend Schrödinger's paradigm of bridges to account for integral constraints along paths, in a way akin to maximum caliber—a maximum entropy principle applied in a dynamic context. The maximum caliber principle has proven useful to infer the dynamics of complex systems, e.g., model gene circuits and protein folding. We unify these two problems via a maximum likelihood formulation to reconcile stochastic dynamics with ensemble-path data. A variety of data types can be encompassed, ranging from distribution moments to average currents along paths. The framework enables inference of time-varying potential landscapes that drive the process. The resulting forces can be interpreted as the optimal control that drives the system in a way that abides by specified integral constraints. This, in turn, relates to a similarly constrained optimal mass transport problem in the zero-noise limit. Analogous results are presented in a discrete-time, discrete-space setting and specialized to steady-state dynamics. We finish by illustrating the practical applicability of the framework through paradigmatic examples, such as that of bit erasure or protein folding. In doing so, we highlight the strengths of the proposed framework, namely, the generality of the theory, its elegant analytical structure, the ease of computation, and the ability to interpret results in terms of system dynamics. This is in contrast to maximum caliber problems where the focus is typically on updating a probability law on paths. Published by the American Physical Society 2024

Bilevel Optimization of the Kantorovich Problem and Its Quadratic Regularization

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.