Zurich Colloquium in Applied and Computational Mathematics

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Archive 2017

Date / Time

Speaker

Title

Location

6 March 2017
16:15-17:15

Prof. Dr. Bjorn Engquist
ICES, University of Texas

Event Details

Speaker invited by	Christoph Schwab

Seismic imaging and the Monge-Ampère equation

Y27 H 28

8 March 2017
16:15-17:15

Dr. Carola-Bibiane Schönlieb
University of Cambridge

Event Details

Speaker invited by

Habib Ammari

Abstract

When assigned with the task of reconstructing an image from imperfect data the first challenge one faces is the derivation of a truthful image and data model. In the context of regularised reconstructions, some of this task amounts to selecting an appropriate regularisation term for the image, as well as an appropriate distance function for the data fit. This can be determined by the a-priori knowledge about the image, the data and their relation to each other. The source of this knowledge is either our understanding of the type of images we want to reconstruct and of the physics behind the acquisition of the data or we can thrive to learn parametric models from the data itself. The common question arises: how can we optimise our model choice? In this talk we discuss a bilevel optimization method for learning optimal variational regularisation models. Parametrising the regularisation and data fidelity terms, we will learn optimal total variation type regularisation models for image and video de-noising, and optimal data fidelity functions for pure and mixed noise corruptions. By considering quotients of desirable and undesirable image structures, we will also show how optimization strategies can be used to favour particular structures over others. I will also give an outlook on how such an approach can be used to learn optimal sampling patterns for magnetic resonance tomography. This presentation contains joint work with M. Benning, L. Calatroni, C. Chung, J. C. De Los Reyes, M. Ehrhardt, G. Gilboa, J. Grah, G. Maierhofer, T. Valkonen, and V. Vlacic

Bilevel learning of variational models

Y27 H 25

15 March 2017
16:15-17:15

Prof. Dr. Jean-Michel Coron
Université Pierre et Marie Curie

Event Details

Speaker invited by	Habib Ammari
Abstract	We present various results on the finite-time stabilization of control systems. This includes control systems in finite dimension (with an application to a quadcopter sliding on a plane) as well as control systems modeled by means of partial differential (1-D linear hyperbolic systems and 1-D linear parabolic equations).

Finite-time stabilization

Y27 H25

22 March 2017
16:15-17:15

Dr. Joscha Gedicke
Fakultät für Mathematik, Universität Wien

Event Details

Speaker invited by

Stefan Sauter

Abstract

C^0 interior penalty methods are discontinuous Galerkin methods for fourth order elliptic boundary value problems that are easier to implement than C^1 continuous finite elements. They were recently extended to the obstacle problem of clamped Kirchhoff plates. In this talk we present the residual based a posteriori error analysis for the fourth order obstacle problem following the approach of Braess for second order obstacle problems. We show that the resulting a posteriori error estimator is similar to the one for the fourth order boundary value problem. Moreover, we apply the resulting adaptive finite element algorithm to a specific optimal control problem, that can be reformulated as fourth order obstacle problem. Numerical examples in 2d and 3d illustrate the proven reliability and efficiency of the a posteriori error estimator.

An adaptive C^0 interior penalty method for a biharmonic obstacle problem

Y27 H 25

29 March 2017
16:15-17:15

Prof. Dr. Mathias Fink
ESPCI Paris

Event Details

Speaker invited by

Habib Ammari

Abstract

Because time and space play a similar role in wave propagation, wave control can be achieved or by manipulating spatial boundaries or by manipulating time boundaries. Here we emphasize the role of time boundaries manipulation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire wavefield and can be used to control wavefield and to revisit the holographic principles and the way to create time-reversed waves. Experimental demonstrations of this approach with water waves will be presented and the extension of this concept to acoustic and electromagnetic waves will be discussed. More sophisticated time manipulations can also be studied in order to extend the concept of photonic crystals and wave localization in the time domain.

Wave Control and Holography with Time Transformations

Y27 H 25

5 April 2017
16:15-17:15

Prof. Dr. Eitan Tadmor
University of Maryland, ETH-ITS

Event Details

Speaker invited by

Remi Abgrall

Abstract

Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such variational models leads to the question of representing general images as the divergence of uniformly bounded vector fields. We construct uniformly bounded solutions of div(U)= F for general F’s in the critical regularity space L^d(T^d). The study of this equation and related problems was motivated by recent results of Bourgain & Brezis. The intriguing aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations U = \sum_j u_j which we introduced earlier in the context of image processing. The u_j’s are constructed recursively as proper minimizers, yielding a multi-scale decomposition of “images” U.

Hierarchical construction of images and the problem of Bourgain-Brezis

Y27 H 25

12 April 2017
16:15-17:15

Dr. Máté Gerencsér
Institute of Science and Technology, Austria

Event Details

Speaker invited by	Arnulf Jentzen
Abstract	We discuss Feynman-Kac formulae for linear stochastic PDEs. Due to the adapted randomness of the equations to be represented, the associated backward flow does not make (Itô) sense, and hence the temporal inversion has to be replaced by a spatial inversion. Some applications of such formulae to numerics and the theory of SPDEs will be outlined. Based on joint work with I. Gyöngy.

Characteristics of SPDEs

Y27 H 25

26 April 2017
16:15-17:15

Prof. Dr. Xue-Mei Li
University of Warwick

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

Gaussian upper and lower bounds for heat kernels are the basic tools for large deviation estimates. There are two well known characterisations on the derivatives of heat semi-grouops: the lower bound of the Ricci curvature by gradient bounds on the heat semi-group; and the validity of the Logarithmic Sobolev inequality for the distributions of the Brownian motion by bounds on the Ricci curvature. What can we say about their second order derivatives? What can we say about the kernels of the self-adjoint operator, which is the sum of the Laplace-Beltrami operator plus a gradient vector field and a potential function? This talk will not be technical. We will discuss why the stochastic damped parallel translation and the doubly damped stochastic parallel translation equation are the natural companions for the heat equations, we will also discuss the associated estimates, the second order Feynman-Kac formulas, and the role of the Brownian bridges and the semi-classical Brownian bridges.

Weighted heat kernels and 'Brownian bridges'

Y27 H 25

3 May 2017
16:15-17:15

Prof. Dr. Dmitri Kuzmin
Technische Universität Dortmund

Event Details

Speaker invited by

Remi Abgrall

Abstract

This talk presents the first extensions of the flux-corrected transport (FCT) methodology to discontinuous and continuous high-order finite element discretizations of scalar conservation laws. Using Bernstein polynomials as local basis functions, we constrain the variation of the numerical solution by imposing local discrete maximum principles on the coefficients of the Bezier net. The design of accuracy-preserving FCT schemes for high-order Bernstein-Bezier finite elements requires a major upgrade of algorithms tailored for linear and multilinear Lagrange elements. The proposed ingredients include (i) a new discrete upwinding strategy leading to low order approximations with compact stencils, (ii) a variational stabilization operator based on the difference between two gradient approximations, and (iii) new localized limiters for antidiffusive element contributions. The optional use of a smoothness indicator based on a second derivative test makes it possible to avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1D and 2D. This is a joint work with R. Anderson, V. Dobrev, Tz. Kolev, C. Lohmann, M. Quezada de Luna, S. Mabuza, R. Rieben, J.N. Shadid, and V. Tomov.

Flux-Corrected Transport Schemes for High-Order Bernstein Finite Elements

Y27 H 25

10 May 2017
16:15-17:15

Prof. Dr. Ivan Oseledets
INM RAS and SkolTech, Moscow

Event Details

Speaker invited by

Christoph Schwab

Abstract

In this talk I overview recent results in the algorithms and theory for the approximation of multivariate functions using low-rank tensor decompositions and deep neural networks (DNN), outline connections between two areas and also discuss open problems that need to be addressed. Tensor decompositions can be applied in DNN in several ways: first, they can be used to compress layers of DNN; second, DNN can be viewed as a generalized tensor network. A separate part will be denoted to the generalization ability of DNN, which is not fully described by the standard methods, and I will show our recent experimental study of the existence of "bad" local minima for neural networks.

Deep learning and tensors for the approximation of multivariate functions: recent results and open problems.

Y27 H 25

17 May 2017
16:15-17:15

Prof. Dr. Thanasis Fokas
University of Cambridge

Event Details

Speaker invited by

Habib Ammari

Abstract

The unified transform (also referred to as the Fokas method) will be reviewed. In particular, it will be shown that this transform yields unexpected results for such classical problems as the heat equation on the half line which was first investigated by Fourier,as well as for the Laplace equation in the interior of a polygon. Interesting connections of this approach with the Riemann hypothesis has led to the proof of the Lindelof hypothesis for a close variant of the Riemann zeta function.

Revisiting the greats: Fourier, Laplace and Riemann

Y27 H 25

24 May 2017
16:15-17:15

Prof. Dr. Mikhail Shashkov
Los Alamos National Laboratory

Event Details

Speaker invited by

Remi Abgrall

Abstract

Computational experiment is among the most significant developments in the practice of the scientific inquiry in the 21th century. Within last four decades, computational experiment has become an important contributor to all scientific research programs. It is particular important for the solution of the research problems that are insoluble by traditional theoretical and experimental approaches, hazardous to study in the laboratory, or time consuming or expensive to solve by traditional means. Computational experiment includes several important ingredients: creating mathematical model, discretization, solvers, coding, verification and validation, visualization, analysis of the results, etc. In this talk we will describe some aspects of the modern numerical methods for high-speed, compressible, multi-physics, multi-material flows. We will address meshing issues, mimetic discretizations of equations of the Lagrangian gas dynamics and diffusion equation on general polygonal meshes, mesh adaptation strategies, methods for dealing with shocks, interface reconstruction needed for multi-material flows, closure models for multi-material cells, time discretizations, etc.

Modern numerical methods for high-speed, compressible, multi-physics, multi-material flows

KOL G 201

31 May 2017
16:15-17:15

Dr. Maxim Rakhuba
SkTech Institute, Moscow, Russia

Event Details

Speaker invited by

Christoph Schwab

Abstract

This talk focuses on the solution of high-dimensional eigenvalue problems, which arise, e.g. in quantum chemistry and in the modelling of quantum spin systems. The key assumption is that eigenvectors can be approximated in low-rank tensor formats. This a priori knowledge allows us to reduce the number of parameters and to improve the convergence of iterative methods. We present new iterative solvers that explicitly account for the low-rank structure of the eigenvectors and that are capable of computing hundreds of eigenstates in high dimensions. We showcase our method by solving vibrational Scrödinger equation as well as equations arising in density functional theory.

Tensor solvers for high-dimensional eigenvalue problems

Y27 H 25

4 October 2017
16:15-17:15

Prof. Dr. Bertrand Maury
Universite de Paris Sud

Event Details

Speaker invited by

Habib Ammari

Abstract

Modeling collections of living entities like human crowds has become a major challenge among physicists and mathematicians in the last two decades. This domain of investigation shares some features with the modeling of "classical" particle systems, but it also raises new issues from the mathematical standpoint. We shall focus here on very crude models (in terms of individual behavior), assuming everyone tends to selfishly realize a given goal (like exiting a room in fire). The resulting motion is then built as a trade-off between individual tendencies and congestion constraints. This standpoint can be straightforwardly instantiated at both microscopic and macroscopic scales. We shall present how the Wasserstein setting of Optimal Transportation makes it possible to transpose the microscopic framework (differential inclusions in a Hilbert space) at the macroscopic level. Beyond this formal analogy, we shall emphasize the deep differences between the two levels of description. In particular, while the macroscopic model exhibits a good old Laplacian, its microscopic counterpart relies on a bizarre discrete operator which in particular does not verify the maximum principle. We shall explain how the emergence of static jams upstream exits, as well as the so-called "faster is slower effect", can be explained by the very pathological properties of this operator.

Micro-macro issues in crowd motion modeling

HG E 1.2

11 October 2017
16:15-17:15

Prof. Dr. Dirk Bloemker
University of Augsburg, FRG

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

Modulation- or Amplitude-Equations are a universal tool to approximate solutions of complicated systems given by partial or stochastic partial differential equations (SPDEs) near a change of stability, when there is no center manifold theory available. Relying on the natural separation of time-scales at the bifurcation thesolution of the original equation is well approximated by the bifurcatingpattern. For the talk we consider for simplicity the one-dimensional stochastic Swift-Hohenberg equation, which acts as a toy model for the convective instability in Rayleigh-Benard convection. On an unbounded spatial domain the amplitude of the dominating pattern is slowly modulated in time and alsoin space. Furthermore, it solves a stochastic Ginzburg-Landau equation perturbed by an additive space-time white noise. Major problems arise due to the weak regularity of solutions and their unboundedness in space, so that the methods from the theory of deterministic modulation equations all fail.

Modulation- and Amplitude-Equations for stochastic partial differential equations

HG E 1.2

18 October 2017
16:15-17:15

Prof. Dr. Thomas Kurtz
University of Wisconsin

Event Details

Speaker invited by

Christoph Schwab

Abstract

Stochastic partial differential equations arise naturally as limits of finite systems of weighted interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations for the particle locations and weights. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness, and convergence results. Beginning with the classical McKean-Vlasov limit, the basic results on exchangeable systems will be discussed along with new applications to SPDEs with boundary conditions.

Particle representations for stochastic partial differential equations

HG E 1.2

25 October 2017
16:15-17:15

Prof. Dr. Marlis Hochbruck
Karlsruhe Institute of Technology

Event Details

Speaker invited by

Christoph Schwab

Abstract

In this talk we present an overview on the numerical solution of time-dependent linear Maxwell's equations with an emphasis on the timeintegration. For the space discretization we consider discontinuos Galerkin methods which can handle complex geometries by using unstructured, possibly locally refined meshes. For the time integration we discuss different options, starting with standard explicit and implicit methods. After a general introduction, our main interest is in problems where the spatial mesh contains only a small number of tiny mesh elements (i.e. elements with a very small diameter) while most of the mesh are coarse. Solving such problems with an explicit time integration scheme requires a constraint on the time step size related to the diameter of the smallest mesh element to ensure stability, the well-known CFL condition. This makes the simulation inefficient, in particular if the number of tiny mesh elements is small compared to the total number of elements. A natural way to overcome this restriction is using implicit time integrators but these come with the expense of having to solve a large linear system in each time step. A more suitable approach consists in treating only the tiny mesh elements implicitly while retaining an explicit time integration for the remaining coarse elements. This results in so-called locally implicit methods. We will show how such methods can be constructed and implemented efficiently, present results of a rigorous error analysis, and close with numerical examples.

On the numerical solution of linear Maxwell's equations

HG E 1.2

1 November 2017
16:15-17:15

Prof. Dr. Shi Jin
University of Wisconsin-Madison, U.S.A

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

In this talk we will study the generalized polynomial chaos-stochastic Galerkin (gPC-SG) approach to kinetic equations with uncertain coefficients/inputs, and multiple time or space scales, and show that they can be made asymptotic-preserving, in the sense that the gPC-SG scheme preserves various asymptotic limits in the discrete space. This allows the implementation of the gPC methods for these problems without numerically resolving (spatially, temporally or by gPC modes) the small scales. Rigorous analysis, based on hypocoercivity of the collision operator, will be provided for general kinetic equations to prove uniform convergence toward the local or global equilibrium, and the spectral convergence of the gPC-SG method.

Uncertainty quantification for multiscale kinetic equations with uncertain coefficients

HG E 1.2

8 November 2017
16:15-17:15

Prof. Dr. Richard Craster
Imperial College London

Event Details

Speaker invited by

Habib Ammari

Abstract

It is highly desirable to be able to create continuum equations that embed a known microstructure through effective or averaged quantities such as wavespeeds or shear moduli. The methodology for achieving this at low frequencies and for waves long relative to a microstructure is well-known and such static or quasi-static theories are well developed. However, at high frequencies the multiple scattering by the elements of the microstructure, which is now of a similar scale to the wavelength, has apparently prohibited any homogenization theory. Many interesting features of, say, periodic media: band gaps, localization etc occur at frequencies inaccessible to averaging theories. The materials exhibit effective anisotropy and this leads to topical effects such as cloaking/ invisibility, flat lensing, negative refraction and to inducing directional behaviour of the waves within a structure. Recently we have developed an asymptotic approach that overcomes this limitation, and continuum equations are developed, even though the microstructure and wavelength are now of the same order. The general theory will be described and applications to continuum, discrete and frame lattice structures will be outlined. The results and methodology are confirmed versus various illustrative exact/ numerical calculations showing that theory captures, for instance, all angle negative refraction, ultra refraction and localised defect modes.

High frequency homogenization: Connecting the Microstructure to the Macroscale

HG E 1.2

15 November 2017
16:15-17:15

Dr. Felix Voigtlaender
TU Berlin

Event Details

Speaker invited by

Rima Alaifari

Abstract

Recently, machine learning techniques based on deep neural networks have significantly improved the state of the art in tasks like image classification and speech recognition. Nevertheless, a solid theoretical explanation of this success story is still missing. In this talk, we will present recent results concerning the approximation-theoretic propertiesof deep neural networks which help to explain some of the characteristics of such networks; in particular we will see that deeper networks can approximate certain classification functions much more efficiently than shallow networks. We emphasize though that these approximation theoretic properties do not explain why simple algorithms like stochastic gradient descent work so well in practice, or why deep neural networks tend to generalize so well; we purely focus on the expressive power of such networks. Precisely, as a model class for classifier functions we consider the class of (possibly discontinuous) piecewise smooth functions for which the different "smooth regions" are separated by smooth hypersurfaces. Given such a function, and a desired approximation accuracy, we construct a neural network which achieves the desired approximation accuracy, where the error is measured in L². We give precise bounds on the required size (in terms of the number of weights) and depth of the network, depending on the approximation accuracy, on the smoothness parameters of the given function, and on thedimension of its domain of definition.
Finally, we show that this size of the networks is optimal, and that networks of smaller depth wouldneed significantly more weights than the deep networks that we construct, in order to achieve the desired approximation accuracy.

Optimal approximation of piecewise smooth functions using deep ReLU neural networks

HG E 1.2

22 November 2017
16:15-17:15

Dr. Virginie Bonnaillie-Noel
Ecole Normale Superieure

Event Details

Speaker invited by

Habib Ammari

Abstract

In this talk,we would like to analyze the connections between the nodal domains of the eigenfunctions of the Dirichlet-Laplacian and thepartitions of the domain by k open sets D_i which are minimal in the sense that the maximum over the D_i's of the groundstate energy of the Dirichlet realization of the Laplacian is minimal. Instead of considering the maximum among the first eigenvalues, we can also consider the p-norm of the vector composed by the first eigenvaluesof each subdomain.

Minimal k-partition for the p-norm of the eigenvalues

HG E 1.2

29 November 2017
16:15-17:15

Prof. Dr. Marcus Grote
Departement Mathematik, Universität Basel

Event Details

Speaker invited by

Stefan Sauter

Abstract

In the presence of complex geometry, adaptivity and mesh refinement are certainly key for the efficient numerical simulation of wave phenomena. Locally refined meshes, however, impose severe stability constraints on any explicit time-marching scheme, where the maximal time-step allowed by the CFL condition is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small subregion, the use of implicit methods, or a very small time-step in the entire computational domain, are very high a price to pay. Explicit local time-stepping schemes (LTS) overcome the bottleneck due to a few small elements by using smaller time-steps precisely where the smallest elements in the mesh are located. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel.

High-Order Explicit Local Time-Stepping Methods For Wave Propagation

HG E 1.2

6 December 2017
16:15-17:15

Prof. Dr. A. Nouy
Ecole Centrale de Nantes

Event Details

Speaker invited by

Christoph Schwab

Abstract

Tensor methods are among the most prominent tools for the approximation of high-dimensional functions. Such approximation problems naturally arise in statistical learning, stochastic analysis and uncertainty quantification. In many practical situations, the approximation of high-dimensional functions is made computationally tractable by using rank-structured approximations. In this talk, we give an introduction to tree-based (hierarchical) tensor formats and then present adaptive algorithms for the approximation in these formats using statistical methods.

Learning high-dimensional functions with tree-based tensor formats.

HG E 1.2

11 December 2017
16:15-17:15

Dr. Sarah Eberle
Mathematisches Institut, Universität Tübingen

Event Details

Speaker invited by

Stefan Sauter

Abstract

In this talk we give an insight in the coupling for interior and exterior problems of wave-type equations as for example the elastic and poroelastic wave equation, where we introduce transparent boundary conditions. In more detail, we start with the background of the Calderon operator, which builds the basis of the stability analysis and the convergence proof for the full discretized problem. After that, we go over to the implementation of the numerical methods and present results of the numerical experiments

FEM-BEM coupling for wave-type equations in 3d

HG E 22

13 December 2017
16:15-17:15

Prof. Dr. Olaf Steinbach
TU Graz, Österreich

Event Details

Speaker invited by

Christoph Schwab

Abstract

For the numerical solution of time-dependent partial differential equations, we consider space-time finite element methods which allow for an adaptive meshing simultaneously in space and time, and for a parallel solution of the global linear system. For the model problem of the heat equation we present a Galerkin--Petrov variational formulation where the test and ansatz spaces are of the same regularity. To prove the related stability condition, we introduce and discuss a Hilbert type transformation operator. This concept is then also used to derive suitable variational formulations for the wave equation. Numerical results are given, and we will discuss future topics, challenges, and applications.

Coercive space-time finite element methods

HG E 1.2

20 December 2017
16:15-17:15

Dr. Gabriel Peyré
Ecole Normale Superieure

Event Details

Speaker invited by

Habib Ammari

Abstract

Optimal transport (OT) has become a fundamental mathematical tool at the interface between calculus of variations, partial differential equations and probability. It took however much more time for this notion to become mainstream in numerical applications. This situation is in large part due to the high computational cost of the underlying optimization problems. There is however a recent wave of activity on the use of OT-related methods in fields as diverse as computer vision, computer graphics, statistical inference, machine learning and image processing. In this talk, I will review an emerging class of numerical approaches for the approximate resolution of OT-based optimization problems. These methods make use of an entropic regularization of the functionals to be minimized, in order to unleash the power of optimization algorithms based on Bregman-divergences geometry (see [2] for a theoretical analysis and a litterature review). This results in fast, simple and highly parallelizable algorithms, in sharp contrast with traditional solvers based on the geometry of linear programming. For instance, they allow for the first time to compute barycenters (according to OT distances) of probability distributions discretized on computational 2-D and 3-D grids with millions of points [1]. This offers a new perspective for the application of OT in machine learning (to perform clustering or classification of bag-of-features data representations) and imaging sciences (to perform color transfer or shape and texture morphing [6]). These algorithms also enable the computation of gradient flows for the OT metric, and can thus for instance be applied to simulate crowd motions with congestion constraints [4]. We will also discus various extensions of classical OT, such as handling unbalanced transportation between arbitrary positive measures [3] (the so-called Hellinger-Kantorovich/Wasserstein-Fisher-Rao problem) and the computation of OT between different metric spaces (the so-called Gromov-Wasserstein problem) [7, 5].
References: [1] J-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, G. Peyré. Iterative Bregman Projections for Regularized Transportation Problems. SIAM Journal on Scientific Computing, 37(2), pp. A1111-A1138, 2015.
[2] G. Carlier, V. Duval, G. Peyré, B. Schmitzer. Convergence of Entropic Schemes for Optimal Transport and Gradient Flows, to appear in SIAM Journal on Mathematical Analysis, 2017.
[3] L. Chizat, G. Peyré, B. Schmitzer, F-X. Vialard. Scaling Algorithms for Unbalanced Transport Problems. Preprint Arxiv:1607.05816, 2016.
[4] G. Peyré. Entropic Approximation of Wasserstein Gradient Flows. SIAM Journal on Imaging Sciences, 8(4), pp. 2323-2351, 2015.
[5] G. Peyré, M. Cuturi, J. Solomon. Gromov-Wasserstein Averaging of Kernel and Distance Matrices. In Proc. ICML’16, pp. 2664-2672, 2016.
[6] J. Solomon, F. de Goes, G. Peyré, M. Cuturi, A. Butscher, A. Nguyen, T. Du, L. Guibas. Convolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains. ACM Transactions on Graphics (Proc. SIGGRAPH 2015), 34(4), pp. 66:1-66:11, 2015.
[7] J. Solomon, G. Peyré, V. Kim, S. Sra. Entropic Metric Alignment for Correspondence Problems. ACM Transactions on Graphics (Proc. SIGGRAPH 2016), 35(4), pp. 72:1-72:13, 2016.

Computational Optimal Transport and its Applications

HG E 1.2

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