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Technical University of Munich

Optimal Transport in Science, Technology, and Society

Initial Principal Investigators

Collaborators

Description

The pictures shows a transport over an evolving network. The picture was published in http://doi.org/10.1142/S0219530515400059.
Transport over an evolving network; copyright by Giacomo Albi, Marco Artina, Massimo Fornasier and Peter A. Markowich; taken from: http://doi.org/10.1142/S0219530515400059

Proposed in 1781 as a mathematical problem by the French mathematician Gaspard Monge, optimal transportation has been at a renewed focus of the scientific community in the last two decades and seen an enormous development. Striking connections to a number of mathematical fields have been established ranging from probability and economics to partial differential equations and Riemannian geometry, where optimal transport is used as a powerful and versatile tool.

This exploration led to the understanding that many phenomena in nature, society, and technology are actually realizing economical principles of optimal transport, e.g., of interacting agents, physical particles or more abstract information.

Stimulated by the collaboration of Prof. Giuseppe Savaré (Univ. Pavia, Italy), vigorous and prominent European scientist in the field, we are developing a three-year research program at TUM-IAS, which embraces classical topics of mathematical optimal transport, its new theoretical challenges, and opens up within TUM to innovative real-life problems in the natural sciences, technology, and society. Within the vast landscape of optimal transport, we will focus on optimal transport on non-smooth structure and more general concepts of optimal transport, e.g., along rate independent evolutions, and mean-field models, mean-field games, and inverse optimal transport. In view of the common ground in the theory of optimal transport, we expect that collaborative research arises naturally at various corners.

In particular, we can foresee potential interactions and breakthroughs where optimal transport, mean-field optimal control, and mean-field games are defined over non-smooth structures (e.g., possibly evolving networks), leading to the necessity of clarifying the interplay between dynamics, topology, and geometry.

This research program aims at realizing one of the most relevant scientific activities in the mathematical theory of optimal transport for the next years in Europe.

Initial Joint Publications

L. Ambrosio, M. Fornasier, M. Morandotti, and S. Savaré. Spatially inhomogeneous evolutionary games, May 2018, arXiv:1805.04027.

M. Di Francesco, M. Fornasier, J.-C. Hütter, and D. Matthes, Asymptotic behavior of gradient flows driven by nonlocal power repulsion and attraction potentials in one dimension, SIAM J. Math. Anal., 46 (2014), pp. 3814-3837.

M. Fornasier, S. Lisini, C. Orrieri and G. Savaré. Mean-field optimal control as Gamma-limit of finite agent controls, March 2018, arXiv:1803.04689.

D. Matthes, R. J. McCann, and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Di_erential Equations, 34 (2009), pp. 1352-1397.

Publications by the Focus Group

2025

  • Matthes, Daniel; Rott, Eva-Maria; Savaré, Giuseppe; Schlichting, André: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport. Numerische Mathematik 157 (4), 2025, 1347-1395 more…

2024

  • Dal Maso, Gianni; Rossi, Riccarda; Savaré, Giuseppe; Toader, Rodica: Visco-energetic solutions for a model of crack growth in brittle materials. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2024, 241-289 more…
  • Fornasier, Massimo; Klock, Timo; Riedl, Konstantin: Consensus-Based Optimization Methods Converge Globally. SIAM Journal on Optimization 34 (3), 2024, 2973-3004 more…
  • Rossi, Riccarda; ; Savaré, Giuseppe;: Abstract action spaces and their topological and dynamic properties. Discrete and Continuous Dynamical Systems - S 17 (1), 2024, 395-420 more…
  • Savaré, Giuseppe; Sodini, Giacomo Enrico: A relaxation viewpoint to Unbalanced Optimal Transport: duality, optimality and Monge formulation. , 2024 more…

2023

  • Bonnet, Benoît; Cipriani, Cristina; Fornasier, Massimo; Huang, Hui: A measure theoretical approach to the mean-field maximum principle for training NeurODEs. Nonlinear Analysis 227, 2023, 113161 more…
  • Cavagnari, Giulia; Savaré, Giuseppe; Sodini, Giacomo Enrico: Extension of monotone operators and Lipschitz maps invariant for a group of isometries. Canadian Journal of Mathematics, 2023, 1-38 more…
  • Cavagnari, Giulia; Savaré, Giuseppe; Sodini, Giacomo Enrico: Extension of monotone operators and Lipschitz maps invariant for a group of isometries. Canadian Journal of Mathematics 77 (1), 2023, 149-186 more…
  • Cavagnari, Giulia; Savaré, Giuseppe; Sodini, Giacomo Enrico: A Lagrangian approach to totally dissipative evolutions in Wasserstein spaces. , 2023 more…
  • Fiedler, Christian; Fornasier, Massimo; Klock, Timo; Rauchensteiner, Michael: Stable recovery of entangled weights: Towards robust identification of deep neural networks from minimal samples. Applied and Computational Harmonic Analysis 62, 2023, 123-172 more…
  • Fornasier, Massimo; Heid, Pascal; Sodini, Giacomo Enrico: Approximation Theory, Computing, and Deep Learning on the Wasserstein Space. , 2023 more…
  • Liero, Matthias; Mielke, Alexander; Savaré, Giuseppe: Fine Properties of Geodesics and Geodesic $$\lambda $$-Convexity for the Hellinger–Kantorovich Distance. Archive for Rational Mechanics and Analysis 247 (6), 2023 more…
  • Matthes, Daniel; Rott, Eva-Maria; Savaré, Giuseppe; Schlichting, André: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport. , 2023 more…
  • Mazzoleni, Dario; ; Savaré, Giuseppe;: $ L^2 $-Gradient flows of spectral functionals. Discrete and Continuous Dynamical Systems 43 (3&4), 2023, 1560-1594 more…

2022

  • BONAFINI, MAURO; FORNASIER, MASSIMO; SCHMITZER, BERNHARD: Data-driven entropic spatially inhomogeneous evolutionary games. European Journal of Applied Mathematics 34 (1), 2022, 106-159 more…
  • Cavagnari, Giulia; Lisini, Stefano; Orrieri, Carlo; Savaré, Giuseppe: Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence. Journal of Differential Equations 322, 2022, 268-364 more…
  • Cavagnari, Giulia; Savaré, Giuseppe; Sodini, Giacomo Enrico: Dissipative probability vector fields and generation of evolution semigroups in Wasserstein spaces. Probability Theory and Related Fields, 2022 more…
  • Fornasier, Massimo; Savaré, Giuseppe; Sodini, Giacomo: Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces. , 2022 more…
  • Naldi, Emanuele; Savaré, Giuseppe: Weak topology and Opial property in Wasserstein spaces, with applications to gradient flows and proximal point algorithms of geodesically convex functionals. Rendiconti Lincei - Matematica e Applicazioni 32 (4), 2022, 725-750 more…
  • Peletier, Mark A.; Rossi, Riccarda; Savaré, Giuseppe; Tse, Oliver: Jump processes as generalized gradient flows. Calculus of Variations and Partial Differential Equations 61 (1), 2022 more…
  • Sodini, Giacomo Enrico: The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson's inequalities. , 2022 more…
  • Sodini, Giacomo; Savaré, Giuseppe: A Simple Relaxation Approach to Duality for Optimal Transport Problems in Completely Regular Spaces. Journal of Convex Analysis 29, 2022, 1-12 more…

2021

  • Ambrosio, Luigi; ; Savaré, Giuseppe; ; ; ; ;: Duality properties of metric Sobolev spaces and capacity. Mathematics in Engineering 3 (1), 2021, 1-31 more…
  • Ambrosio, Luigi; Fornasier, Massimo; Morandotti, Marco; Savaré, Giuseppe: Spatially Inhomogeneous Evolutionary Games. Communications on Pure and Applied Mathematics 74 (7), 2021, 1353-1402 more…
  • Naldi, Emanuele; Savaré, Giuseppe: Weak topology and Opial property in Wasserstein spaces, with applications to Gradient Flows and Proximal Point Algorithms of geodesically convex functionals. 2021 more…
  • Savaré, Giuseppe: Sobolev Spaces in Extended Metric-Measure Spaces. In: Lecture Notes in Mathematics. Springer International Publishing, 2021 more…

2020

  • Cavagnari, Giulia; Lisini, Stefano; Orrieri, Carlo; Savaré, Giuseppe: Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence. 2020 more…

2019

  • Savaré, Giuseppe: Sobolev spaces in extended metric-measure spaces. 2019 more…

2018

  • Fornasier, Massimo; Lisini, Stefano; Orrieri, Carlo; Savaré, Giuseppe: Mean-field optimal control as Gamma-limit of finite agent controls. arXiv, 2018 more…
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