Schedule for: 17w5078 - Generated Jacobian Equations: from Geometric Optics to Economics

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The general framework for Generated Jacobian Equations (GJE) was first studied in a systematic fashion by Neil Trudinger in 2014, motivated by near field geometric optics problems. These equations involve the prescription of the Jacobian of a transformation derived from a scalar potential, which itself arises as the envelope of a some family of "generating" functions. As such, they also appear in areas such as economics, geometric optics, and optimal transport --which, to use the economic literature terminology, corresponds to the special case of "quasilinear" generating functions. I will discuss these examples as well as the elements of the theory of GJE, and provide a partial overview of the growing literature on the field.

In the Euclidean space, the barycentre of a measure is uniquely defined due to the strict convexity of the functional it minimizes. This is no longer true in more general metric spaces, including Riemannian manifolds. In this short talk, we explain an attempt (joint work with Brendan Pass) to define a canonical notion of unique barycentre of a measure on a metric measure space.

Full regularity results for solutions to the SBVP for GJEs require strong geometric conditions on the domains of the problem as well as higher order structure conditions on the generating function. For example, in the optimal transport problem, these are $c$-convexity and $c^*$-convexity restrictions on the source and target domains respectively and the MTW conditions. We extend the partial regularity result of De Philippis and Figalli from the optimal transport setting to the general GJE setting and show that without any geometric conditions on the domains or additional structure conditions, akin to the MTW conditions, on the generating function, solutions are smooth outside a closed singular set of measure zero. This result is especially relevant to the general GJE framework when applied to problems in geometric optics: in the reflector shape design problem, Karakhanyan and Wang show that smooth data may yield many solutions each with different regularity properties.

Multi-marginal optimal transport with Coulomb cost arises as a dilute limit of density functional theory, which is a widely used electronic structure model. The number $N$ of marginals corresponds to the number of particles. I will discuss the question whether ''Kantorovich minimizers'' must be ''Monge minimizers'' (yes for $N=2$, open for $N\gt 2$, no for $N=\infty$), and derive the surprising phenomenon that the extreme correlations of the minimizers turn into independence in the large $N$ limit. I will also discuss work in progress on the next order term.
The talk is based on joint works with Gero Friesecke (TUM), Claudia Klueppelberg (TUM), Brendan Pass (Alberta) which appeared in CPAM (2013) and Calc.Var.PDE (2014), and on joint work in progress with Mircea Petrache on the next order term.

It has been recently shown that a damped Newton algorithm allows to solve the optimal transport problem between an absolutely continuous measure and a discrete one when the cost satisfies a discrete version of the Ma-Trudinger-Wang condition. I will consider here the case where the source measure is not supported on a set of maximal dimension. More precisely, under genericity and connectedness conditions, I will show the convergence of the damped Newton algorithm to solve the optimal transport problem for the quadratic cost in $\mathbb{R}^d$ when the source measure is supported on a simplex soup, each simplex being of arbitrary dimension greater than $2$.

I will describe a simple iterative method for approximating weak solutions to generated Jacobian equations and discuss conditions under which this method converges in a finite number of steps. Some applications to optimal mass transport and geometric optics will also be provided.

We introduce a framework for constructing monotone approximations of Monge-Ampère type equations on general meshes or point clouds. These schemes easily handle complex geometries and non-uniform distributions of discretization points. The schemes fit within the Barles-Souganidis convergence framework for approximation of viscosity solutions. However, the PDE itself does not always satisfy the strong form of comparison principle required by this convergence proof. For the Dirichlet problem, which admits non-continuous viscosity solutions, we describe a modified comparison principle that guarantees convergence in the interior of the domain.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Motivated by problems from economics, I will present a framework for "Equilibrium Transportation", which embeds the Monge-Kantorovich "Optimal Transportation" problem, but is more general, and more natural in some applications. In the discrete case, this framework allows for a unified description of Gale and Shapley's stable marriage problem, as well as Koopmans and Beckmann's optimal assignment problem. I will sketch the link with "Galois connections" and recent results by Trudinger on the local theory of prescribed Jacobian equations. I will then turn to computational issues, and will present an extension of the Iterated Projections algorithm that allows for efficient approximate computation of these problems.

I will present a model in which one looks for equilibrium prices on a quality good market where consumers maximize utility subject to a budget constraints and producers maximize their profit which is quasi-linear in the price. This is a non transferable case for which optimal transport techniques cannot be used directly. I will give an existence result, discuss some special cases and a possible computation algorithm.

We use the theory of abstract convexity to study adverse-selection principal-agent problems and two-sided matching problems, departing from much of the literature by not requiring quasilinear utility. We formulate and characterize a basic underlying implementation duality. We show how this duality can be used to obtain a sharpening of the taxation principle, to obtain a general existence result for solutions to the principal-agent problem, to show that (just as in the quasilinear case) all increasing decision functions are implementable under a single crossing condition, and to obtain an existence result for stable outcomes featuring positive assortative matching in a matching model.

A monopolist wishes to maximize her profits by finding an optimal price policy. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his own utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this talk, we describe a necessary and sufficient condition for the convexity or concavity of the principal's problem, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e. quasilinear) function of the price paid. Concavity when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences. This talk represents joint work with my supervisor Robert McCann.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Causal transport plans between two Polish filtered probability spaces \((X,(F^X_t)_t,\mu)\) and $(Y,(F^Y_t)_t,\nu)$, are transport plans (in the classical Kantorovic sense) such that, in addition, the regular conditional kernel with respect to the first coordinate satisfies a certain measurability condition. Roughly speaking: the amount of mass transported to a subset of the target space $Y$ belonging to $F^Y_t$, is solely determined by the information contained in $F^X_t$. 1) to obtain a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration; 2) to give an estimate of the value of having additional information, for some classical stochastic optimization problems. This talk is based on a joint work with Julio Backhoff and Anastasiia Zalashko.

I will discuss joint work with Robert McCann on the optimal transport problem between densities supported on manifolds with different dimensions. We show that the problem is equivalent to a non-local analog of the Monge-Ampere equation. We also show that, under certain topological conditions, the solution is smooth if and only if a local variant of the equation admits a smooth, uniformly elliptic solution.

In this talk I will discuss a classical topic in geometry and point out some connections with optimal transport and reflector/refractor design. Blaschke's rolling ball theorem will be the starting point. Roughly speaking it says that local inclusion of convex sets implies global inclusion. Among other things, I will prove a generalisation of Blaschke's theorem for $c$-convex sets with $c$ satisfying the MTW condition.

Recently the detailed study of "unbalanced" transport problems between measures of different total mass has attracted considerable attention. A prominent example is the Wasserstein-Fisher-Rao distance which can be interpreted as an interpolation between the Wasserstein-2 and the Hellinger distance. In this talk we study analogous extensions of unbalanced Wasserstein-1 transport by generalizing the Kantorovich-Rubinstein formula. These are numerically attractive since they preserve the compact formulation as min-cost-flow problems. In addition to these "static" models we analyze "dynamic" models in the spirit of Benamou and Brenier. It turns out that the particular structure of the $W_1$-transport cost leads to a decoupling of transport and mass change and allows a reduction of dynamic models to static equivalents. Joint work with Benedikt Wirth.

For a range of physical and biological processes—from dynamics of granular media to biological swarming—the evolution of a large number of interacting agents is modeled according to the competing effects of pairwise attraction and (possibly degenerate) diffusion. In the slow diffusion limit, the degenerate diffusion formally becomes a hard height constraint on the density of the population, as arises in models of pedestrian crown motion.
Motivated by these applications, we bring together new results on the Wasserstein gradient flow of nonconvex energies with the theory of free boundaries to study a model of Coulomb interaction with a hard height constraint. Our analysis demonstrates the utility of Wasserstein gradient flow as a tool to construct and approximate solutions, alongside the strength of viscosity solution theory in examining their precise dynamics. By combining these two perspectives, we are able to prove quantitative estimates on convergence to equilibrium, which relates to recent work on asymptotic behavior of the Keller-Segel equation. This is joint work with Inwon Kim and Yao Yao.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

We show existence of a lens, when its lower surface is given, such that it refracts radiation emanating from a planar source, with a given field of directions, into the far field that preserves given distribution of energies. Conditions are shown under which the lens obtained is physically realizable. It is shown that the upper surface of the lens satisfies a pde of Monge-Ampère type. This is joint work with Ahmad Sabra. References: 1. Aspherical lens design and imaging, (with A. Sabra), SIAM Journal on Imaging Sciences, Vol. 9, No.1, pp. 386-411, 2016., 2. Freeform Lens Design for Scattering Data With General Radiant Fields, (with A. Sabra), preprint.

Optical surfaces suggested using the Minkowski optimization might face physical constraints and be impractical in industrial design. We discuss the limitation of ray obstruction. Suggested reflective models in the mathematical literature might obstruct the rays before reaching the target. This phenomena is more common in the reflective designs since the incident and reflective rays lie in the same medium. In this talk, we present different ways to overcome this limitation, and also construct a near field non concave reflector where all the rays reach a given target with prescribed energy conditions. Results presented in this talk are joint work with C.E. Gutiérrez.

It is well known that by using the Brenier's map, one can give a simple proof of the classical isoperimetric inequality with optimal transport method. It is however an open question if the general Alexandrov-Fenchel inequality can be proved in a similar manner. This relies on the solvability of $\sigma_k$ Hessian equation with a suitable boundary condition. In this talk, I will discuss the solvability of $\sigma_k$ Hessian equation with various boundary conditions. If time permits, I will also talk about the conformal invariant properties for the $k$-Yamabe problem with boundary, which shed light on how this PDE problem of the $\sigma_k$ Hessian operator is interplaying with the geometry of the (convex) body. This is joint work with Jeffrey Case.

In this talk, we show that the classical Sobolev and Sobolev trace inequalities are embedded into the same one-parameter family of sharp constrained Sobolev inequalities on half-spaces. Using a new variation of a mass transportation argument introduced by Cordero-Erausquin, Nazaret, and Villani, we prove each inequality in this one-parameter family and characterize the equality cases. The case $p=2$ corresponds to a family of variational problems on conformally flat metrics, whose absolute minimizers interpolate between conformally flat spherical and hyperbolic geometries, passing through the Euclidean geometry defined by the fundamental solution of the Laplacian. This is joint work with Francesco Maggi.

Given an optimal transportation problem between two manifolds, Kim and McCann offered a pseudo-Riemannian metric on the product manifold, which captures some of the geometry of the problem. By modifying this metric depending on the mass, the graph of the solution is a minimal surface. It is natural to ask then, how mean curvature flow behaves on these manifolds. Work of Li-Salavessa shows that even in high-codimension, mean curvature flow in pseudo-Riemannian spaces can behave remarkably well. We explore this question in both the background-flat case, and in the curved case, where, not surprisingly, we find the Kim-McCann expression of the Ma-Trudinger-Wang condition.

In recent joint work with Károly Böröczky, Yong Huang, Erwin Lutwak, Deane Yang, Gaoyong Zhang, and Yiming Zhao, dual curvature measures of convex bodies in $\mathbb{R}^n$, which are conceptually dual to Federer's curvature measures, were constructed. This leads naturally to a Minkowski-type problem, which we call the dual Minkowski problem and which is equivalent to a Monge-Ampere PDE.
In particular, the dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this when $1\lt q\lt n$ will be presented.
The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.