Schedule for: 24w5198 - Optimal Transport and Dynamics

This talk focuses on recent results concerning various properties of mappings arising in optimal transport problems for non quadratic costs. The cost functions considered have the form $c(x,y)=h(x-y)$, where $h\in C^2(\mathbb{R}^n)$ is convex, positively homogeneous of degree $p\geq 2$, and $D^2h(x)$ has eigenvalues bounded away from zero and infinity for all $x\in \mathbb{S}^{n-1}$. A multivalued mapping $T:\mathbb{R}^n\to \mathcal{P}(\mathbb{R}^n)$ is $c$-monotone if $c(\xi,x)+c(\zeta,y)\leq c(\xi,y)+c(\zeta,x)$ for all $\xi\in Tx,\zeta\in Ty$ for all $x,y\in \mathbb{R}^n$. Optimal maps with respect to the cost $c$ are $c$-monotone. If $h(x)=|x|^2$, then $c$-monotonicity is the standard monotonicity $(\xi-\zeta)\cdot (x-y)\geq 0$ having a large number of applications to optimization and nonlinear evolution PDEs. We prove that $c$-monotone mappings $T$ are single valued a.e. and establish local $L^\infty$-estimates on balls for $u(x)=Tx-Ax-b$ for each matrix $A$ and each vector $b$ in terms of averages of $u$ on a slightly larger ball. As a consequence, we deduce differentiability of $T$ a.e. It is also shown that these maps are related to maps of bounded deformation, and further differentiability and H\"older continuity properties are derived. This is research in collaboration with Annamaria Montanari (U. of Bologna) and it is a continuation of our recent work originated from recent results by M. Goldman and F. Otto concerning partial regularity of optimal transport maps for the quadratic cost.

My talk will mostly draw from the working paper "Repeated Matching Games: An Empirical Framework" with Pauline Corblet and Jeremy Fox, where we introduce a model of dynamic matching with transferable utility, extending the static model of Shapley and Shubik (1971). Forward-looking agents have individual states that evolve with current matches. Each period, a matching market with market-clearing prices takes place. We prove the existence of an equilibrium with time-varying distributions of agent types and show it is the solution to a social planner's problem. We also prove that a stationary equilibrium exists. We introduce econometric shocks to account for unobserved heterogeneity in match formation. We propose two algorithms to compute a stationary equilibrium. We adapt both algorithms for estimation. We estimate a model of accumulation of job-specific human capital using data on Swedish engineers.

I will explain the phenomenon of contact angle pinning/hysteresis at a heuristic level. Then I will discuss some models for rate independent motion of capillary drops under the effects of pinning/hysteresis. We study regularity and other fine properties of solutions including uniqueness. Energy and comparison based formulations provide different advantages. Talk is based on joint works with Inwon Kim and Norbert Pozar, and with Carson Collins.

Generative Adversarial Networks (GANs) was one of the first Machine Learning algorithms to be able to generate remarkably realistic synthetic images. In this presentation, we delve into the mechanics of the GAN algorithm and its profound relationship with optimal transport theory. Through a detailed exploration, we illuminate how GAN approximates a system of PDE, particularly evident in shallow network architectures. Furthermore, we investigate the phenomenon of mode collapse, a well-known pathological behavior in GANs, and elucidate its connection to the underlying PDE framework through an illustrative example.

Capturing data from dynamic processes through cross-sectional measurements is seen in fields from computer graphics to robot path planning and cell trajectory inference. This inherently involves the challenge of understanding and reconstructing the continuous trajectory of these processes from discrete data points, for which interpolation and approximation plays a crucial role. In this talk, we propose a method to compute measure-valued B-splines in the Wasserstein space through consecutive averaging. Our method can carry out approximations with high precision and at a chosen level of refinement, including the ability to accurately infer trajectories in scenarios where particles undergo splitting (division) over time. We rigorously evaluate our method using simulated cell data characterized by bifurcations and merges, comparing its performance against both state-of-the-art trajectory inference techniques and other interpolation methods. The results of our work not only underscore the effectiveness of our method in addressing the complexities of inferring trajectories in dynamic processes but also highlight its proficiency in performing spline interpolation that respects the inherent geometric properties of the data. This is joint work with Amartya Banerjee, Harlin Lee and Nir Sharon.

Regularity of solutions of optimal transportation problems was well studied with its relation to the Monge-Ampere type equations. There are several conditions such as the Twisted condition or MTW condition, in which we need to use the Monge-Ampere type equations to study the regularity of optimal transportation problems. However, we can easily come up with examples which do not satisfy such conditions. In this talk, we consider the optimal transportation problem on a boundary of a convex body with Euclidean distance squared cost function. This problem does not satisfy twisted conditions. We discuss how to get regularity in this case.

The theory of optimal transport has been used successfully to model several freeform lens design problems. A freeform optical surface, refers to an optical surface (lens or mirror) whose shape lacks rotational symmetry. The use of such surfaces allows design of spatially efficient optical devices. In this talk, we exhibit the existence of a far field refracting lens between two anisotropic media by using optimal transport framework.

I will talk about a modification of the Monge-Kantorovich problem taking into account interaction effects via path dependency between particles. We prove the existence of solutions under mild conditions on the data, and after imposing stronger conditions, we characterize the minimizers by relating them to an auxiliary Monge-Kantorovich problem of the more standard kind. With this notion of how particles interact and travel along paths, we produce a dual problem. The main novelty here is to incorporate an interaction effect to the optimal path transport problem. Lastly, our results include an extension of Brenier's theorem on optimal transport maps and a formulation of the celebrated Benamou-Brenier theory with interaction effects.

We will address the problem of assigning optimal routes in a graph that transport two given densities over the nodes. The occupation of each edge at a given time defines a metric over this graph, for which the routes must be geodesics. This model may describe for example the congestion of a city and its solutions are known as Wardrop equilibria. Additionally, a central planner can require that the assignment is efficient, meaning it minimizes the Kantorovich functional arising from this metric. In this presentation, we will characterize this problem in terms of a partial differential equation and illustrate a simple case. This work is a collaboration with Sergio Zapeta Tzul, a former MSc student at CIMAT and current PhD student at the University of Minnesota.

In this talk we will revisit the classical problem on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. We will demonstrate that the theory of optimal transport via gradient flows can bring new perspectives, when it comes to considering confining potentials or nonlocal drift terms within the problem. In particular, we provide quantitative convergence rates in the 2-Wasserstein distance for the singular limit, which are global in time thanks to the contractive property arising from the external potentials. The talk will be based on a recent joint work with Noemi David and Filippo Santambrogio.

It is well-known that the Vlasov-Fokker-Planck equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. In order to better understand this geometric formalism, we propose a time discrete variational scheme whose solution converges to the weak solution of the Vlasov-Fokker-Planck equation. In this talk, we will discuss how the variational scheme can be seen as an implementation of the symplectic Euler scheme in the Wasserstein space. Moreover, we will see that the energy functionals involved in each variational problem are geodesically convex with respect to the metric.

Recent work views adversarial training for binary classification as the minimization of a fidelity term and a non-local perimeter, opening the door to geometric perspectives. As the adversarial budget vanishes, we show that the non-local perimeter Gamma-converges to an anisotropic perimeter that reflects the stability of adversarial training. Interpreting the full adversarial training problem is a bit tricky. We can rely on a source condition or, alternatively, take a dynamic approach. For the latter, we introduce a slight modification of the adversarial training scheme, which can be seen as a minimizing movements scheme for the non-local perimeter functional. From this, we draw rigorous connections to a weighted mean curvature flow, indicating that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. This is joint work with Leon Bungert (Wuerzburg) and Tim Laux (Regensburg).

We study the regularity and comparison principle for a gradient degenerate Neumann problem. The problem is a generalization of the Signorini or thin obstacle problem which appears in the study of certain singular anisotropic free boundary problems arising from homogenization. In scaling terms, the problem is critical since the gradient degeneracy and the Neumann PDE operator are of the same order. We show the (optimal) $C^{1,\frac{1}{2}}$ regularity in dimension $d=2$ and we show the same regularity result in dimension $d>2$ conditional on the assumption that the degenerate values of the solution do not accumulate. We also prove a comparison principle characterizing minimal supersolutions, which we believe will have applications to homogenization and other related scaling limits. This is joint work with Will Feldman.

I will start with an overview of some of the key questions in cosmology today. I will then discuss the "reconstruction problem" - inferring the initial conditions of the Universe from our late time observations. Given this, I will then discuss how optimal transport might provide an elegant solution to this. I will conclude with open questions to potentially seed further discussion.

In this talk, I shall present our on-going works in Optimal Transport for cosmology: - Early Universe reconstruction, an inverse problem that aims at reconstructing the trajectories of galaxies, back in time. - Forward simulation of Monge-Ampère gravity, to test some non-linear models of Dark Matter and Dark Energy. This is common work with Roya Mohayaee, Yann Brenier, Farnik Nikakhtar, Sebastian von Hausegger, Ravi Sheth and Nikhil Padmanabhan.

The universe we observe today is dotted with galaxy clusters separated by vast voids, in sharp contrast to its initial state, which was nearly uniform with only minor density fluctuations. The evolution from this early uniformity to today's complex structure of galaxies is a profound transformation, with many intermediate processes still unexplained. This talk focuses on this transformation, aiming to reconstruct both the initial density and the displacement fields of galaxies observed in spectroscopic surveys, and also suggesting a forward modeling approach based on optimal transport theory. This theory deals with moving objects from one place to another while conserving mass and minimizing effort. In a cosmological context, it involves mapping the observed galaxy distribution back to its initial uniform state, minimizing the displacement of galaxies. In this framework, we are able to reconstruct the position and shape of biased tracers in Lagrangian space, in addition to the displacement field, which can be used to reconstruct the initial overdensity fluctuation field. This algorithm also suggests an effective way for field-level inference with forward modeling.

It is known that the log-concavity is preserved by the Dirichlet heat flow in convex domains of Euclidean space. In this talk, I explain that no concavity properties are preserved by the Dirichlet heat flow in a totally convex domain of a Riemannian manifold unless the sectional curvature vanishes everywhere on the domain. This talk is based on joint work with Kazuhiro ISHIGE (U. of Tokyo) and Haruto TOKUNAGA (U. of Tokyo).

Kim and McCann demonstrated the MTW curvature corresponds to the curvature of a $(n,n)$ pseudo-Riemannian Kahler manifold. The metric turns out to have some interesting properties: The graph of an optimal transport map is a Lagrangian space-like submanifold of this manifold, and when given weights depending on the mass distributions, becomes a volume maximizing surface. This leads naturally to the question of flows toward optimal transport plans. When an initial graph is determined by a scalar, the mean curvature flow preserves the Lagrangian property and is controlled by a parabolic Monge-Ampere equation. One can also consider a gradient flow for volume, which is represented by a fourth order quasilinear parabolic equation. We offer a recent survey of this area, and some recent progress.

In this talk, I will present recent joint work with Inwon Kim and Antoine Mellet in which we derive a model for congested transport (a PDE at a macroscopic scale) from particle dynamics (a system of ODEs at the microscopic scale). Such PDEs appear very naturally in the description of crowd motion, tumor growth, and general aggregation phenomena. We begin with a system where the particle trajectories evolve according to a gradient flow constrained to some finite distance of separation from each other. This constraint leads to a Lagrange multiplier which, in the mean field limit (infinite number of particles), generates a pressure variable to enforce the hard-congestion constraint. Our results are confined to one spatial dimension wherein we rely on both the Eulerian and Lagrangian perspectives for the continuum limit.

​​There is a large body of recent work on the approximation of diffusion equations by particle systems. Most of this analysis approaches the problem from a stochastic perspective due to the difficulty of studying deterministic particle trajectories. This is largely due to the fact that in the continuous setting, it may be extremely hard to solve diffusion equations in Lagrangian coordinates. In fact, the existence of Lagrangian solutions to the Porous Media Equation (PME) with general initial data was open until 2022. In this talk, I will discuss how to construct Lagrangian solutions to PME. I will then sketch how this analysis can be used to obtain convergence rates for certain deterministic versions of score matching type algorithms.