Article
Automation & Control Systems
Tatsuya Miyano, Justin Romberg, Magnus Egerstedt
Summary: This paper addresses the problem of collectively transporting multiple objects using air-ground multirobot teams. The objective is to minimize the energy of the overall system by finding the optimal matching between the objects and aerial/ground robots. The authors propose a combination of branch and bound algorithm with a negative-cycle canceling algorithm (NCCA) that proves to be an efficient solution for this combinatorial problem, providing the globally optimal solution. Numerical experiments demonstrate the practical performance of the proposed algorithm.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY
(2023)
Article
Mathematics
Tatsuya Miura, Felix Otto
Summary: This paper develops a boundary epsilon-regularity theory for optimal transport maps between bounded open sets with C-1,C-alpha-boundary, asserting sharp C-1,C-alpha-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions. The main quantitative assumptions are small local nondimensionalized transport cost and locally almost flat boundaries. The method is completely variational and builds on the recently developed interior regularity theory.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Maria Colombo, Max Fathi
Summary: The article focuses on strictly log-concave measures and Gaussian distributions, discussing the optimal transport map between them and proving some properties of this mapping.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Statistics & Probability
Jan-Christian Huetter, Philippe Rigollet
Summary: The Brenier's theorem establishes the existence of an optimal transport map T between two probability distributions under certain regularity conditions. This work aims to establish minimax estimation rates for such a transport map from data sampled from the distributions under smoothness assumptions on T. By developing an estimator based on empirical versions of the semidual optimal transport problem and providing numerical experiments supporting the theoretical findings, the study demonstrates the practical benefits of smoothness regularization in achieving near minimax optimality for transport maps in general dimension.
ANNALS OF STATISTICS
(2021)
Article
Mathematics
Takahiro Inayama
Summary: We prove the optimal L-2-extension theorem of Ohsawa-Takegoshi type on a tube domain and provide a simple proof of Prekopa's theorem.
JOURNAL OF GEOMETRIC ANALYSIS
(2022)
Article
Computer Science, Artificial Intelligence
Alireza Bahraini, Saeed Sadeghi
Summary: This article applies the concepts of Ricci curvature by Lott-Sturm-Villani and the optimal transport theorem by McCann-Brenier to investigate the properties and applications of mean-field variational inference (MFVI) in both continuous and continuous-discrete mixture cases.
INTERNATIONAL JOURNAL OF APPROXIMATE REASONING
(2023)
Article
Mathematics
Tianling Jin, Jingang Xiong
Summary: We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains, answering a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof utilizes a geometric type structure of the fast diffusion equations, where an important component is an evolution equation for a curvature-like quantity.
AMERICAN JOURNAL OF MATHEMATICS
(2023)
Article
Geography, Physical
Kristen L. Underwood, Donna M. Rizzo, Mandar M. Dewoolkar, Michael Kline
Summary: Given limited resources for managing erosion hazards and water quality along rivers, stakeholders in water resource management could benefit from tools to identify river reaches prone to sediment loading. The Self-Organizing Map (SOM) is a useful tool for clustering multivariate observations and analyzing complex, nonlinear river systems. Through multiple stages of SOM application, we identified seven sediment regimes in river reaches based on stream geomorphic assessment data.
Article
Mathematics
Qingshan Zhou, Liulan Li, Antti Rasila
Summary: A uniformly continuous identification between the inner boundary of Omega and the Gromov boundary with a visual metric is shown, leading to boundary continuity for both quasiconformal homeomorphisms and rough quasi-isometries between domains equipped with quasihyperbolic metrics.
MATHEMATICA SCANDINAVICA
(2021)
Article
Mathematics, Applied
Yang Liu
Summary: Global well-posedness for the half-wave map with S-2 target for small initial data and equation with H-2 target for small initial data has been proven. This result is significant for the understanding and study of the behavior of these equations with given initial conditions.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2023)
Article
Physics, Multidisciplinary
Giacomo De Palma, Dario Trevisan
Summary: We propose a method called specific quantum W-1 distance, which extends the Wasserstein distance of order 1 to quantum spin systems on the lattice Z(d). This method is based on the W-1 distance for qudits and recovers Ornstein's (d)-distance for certain quantum states. We also generalize the Lipschitz constant to quantum interactions on Z(d) and prove its duality with the specific quantum W-1 distance. Furthermore, we establish continuity bounds for the von Neumann entropy and the specific von Neumann entropy in terms of the specific quantum W-1 distance, and prove the uniqueness of Gibbs states for local quantum commuting interactions above a critical temperature.
ANNALES HENRI POINCARE
(2023)
Article
Materials Science, Multidisciplinary
Adebowale S. Borokinni
Summary: This work presents a mechanical isotropic rate-independent theory that couples plastically deformed materials with species transport. Natural ingredients such as mass and virtual power balances are used to obtain local balance laws, while the second law of thermodynamics is applied to derive thermodynamically consistent constitutive relations. The theory also includes a variational inequality formulation for the coupled system.
MATHEMATICS AND MECHANICS OF SOLIDS
(2023)
Article
Mathematics, Applied
Ping Chen
Summary: This paper focuses on the monotonicity of maps and the associated problems with optimal transport maps. By solving secondary variational problems, it is shown that ray increasing and decreasing optimal transport maps exist under certain conditions, and a classification for cost functions is provided.
ADVANCES IN CALCULUS OF VARIATIONS
(2022)
Article
Computer Science, Theory & Methods
Kristian Bredies, Marcello Carioni, Silvio Fanzon, Francisco Romero
Summary: We propose a dynamic generalized conditional gradient method for dynamic inverse problems with optimal transport regularization. The method effectively reconstructs heavily undersampled dynamic data and demonstrates convergence and optimization performance.
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics
Ping Chen, Hai-Rong Liu, Xiao-Ping Yang
Summary: This study proves the existence of solutions to the Monge problem with an absolutely continuous initial measure by solving a secondary variational problem with any strictly convex function. It shows that under certain conditions, the same optimal transport map can be obtained even with different strictly convex functions.
PACIFIC JOURNAL OF MATHEMATICS
(2021)
Article
Mathematics
Alessio Figalli, Joaquim Serra
INVENTIONES MATHEMATICAE
(2019)
Article
Mathematics
Alessio Figalli, Joaquim Serra
INVENTIONES MATHEMATICAE
(2020)
Article
Mathematics, Applied
Matteo Bonforte, Alessio Figalli
Summary: The study focuses on the homogeneous Dirichlet problem for the fast diffusion equation in a smooth bounded domain, showing that bounded positive solutions extinguish in a finite time and approach a separate variable solution as they near extinction. Further investigation into the fine asymptotic behavior of the relative error and proof of sharp rates of convergence are conducted, based on an entropy method and the concept of almost-orthogonality.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Alessio Figalli, Federico Glaudo
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2020)
Article
Mathematics, Applied
L. Caffarelli, F. Cagnetti, A. Figalli
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2020)
Article
Mathematics, Applied
Alessio Figalli, Yi Ru-Ya Zhang
Summary: This paper investigates the relationship between the Wulff energy of a convex polyhedron and its approximations, proves partial results, and provides related stability inequalities and rigidity results.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2022)
Article
Mathematics
Giulio Ciraolo, Alessio Figalli, Alberto Roncoroni
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2020)
Article
Mathematics
Alessio Figalli, Xavier Ros-Oton, Joaquim Serra
PUBLICATIONS MATHEMATIQUES DE L IHES
(2020)
Article
Mathematics
A. Belotto da Silva, A. Figalli, A. Parusinski, L. Rifford
Summary: In this paper, we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. We investigate the size of the set of points that can be reached by singular horizontal paths starting from a given point and prove that it has Hausdorff dimension at most 1. Moreover, we demonstrate that such a set is a semianalytic curve under the condition that the lengths of the singular curves are bounded with respect to a given complete Riemannian metric. We also establish that sub-Riemannian geodesics in 3-dimensional analytic manifolds are always of class C-1 and analytic outside of a finite set of points, combining our techniques with recent developments on the regularity of sub-Riemannian minimizing geodesics.
INVENTIONES MATHEMATICAE
(2022)
Article
Mathematics, Applied
Ludovic Rifford
Summary: This article investigates the Lonely Runner Conjecture, proving its validity in certain cases and presenting an upper bound related to the number of rounds. It also explores a conjecture regarding a covering problem.
ACTA APPLICANDAE MATHEMATICAE
(2022)
Article
Mathematics, Applied
Alessio Figalli, Sunghan Kim, Henrik Shahgholian
Summary: In this paper, we investigate vector-valued solutions to a linear transmission problem and prove the transmission of Lipschitz regularity from one phase to the next. We present the conditions for a solution to an elliptic system and demonstrate its Lipschitz property in a specific space. Similar results are also derived for the parabolic counterpart of the problem.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics
Alessio Figalli, Yi ru-ya Zhang
Summary: This article proves a sharp quantitative version of the p-Sobolev inequality, with a control on the maximum possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for p < 2, while it depends on p for p > 2.
DUKE MATHEMATICAL JOURNAL
(2022)
Article
Mathematics
Alessio Figalli, David Jerison
Summary: This paper proves a sharp form of the analogous result in dimensions 2 and 3, which is related to the approximate structure of sets of integers in Freiman's theorem concerning real numbers estimation.
ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE
(2021)
Article
Mathematics, Interdisciplinary Applications
Xavier Fernandez-Real, Alessio Figalli
MATHEMATICS IN ENGINEERING
(2020)
Article
Mathematics
Xavier Cabre, Xavier Ros-Oton, Alessio Figalli, Joaquim Serra
Article
Mathematics, Applied
Davide Parise, Alessandro Pigati, Daniel Stern
Summary: This paper studies the self-dual Yang-Mills-Higgs energies on a closed Riemannian manifold and proves their convergence to minimal submanifolds. The author establishes a connection between the energies and the Euler class by introducing a suitable gauge invariant Jacobian, and shows the existence of a recovery sequence under certain conditions. Furthermore, a comparison between the min-max values obtained from the Almgren-Pitts theory and the Yang-Mills-Higgs framework is made, with the former always providing a lower bound for the latter.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Wenkui Du, Robert Haslhofer
Summary: This paper explores ancient noncollapsed mean curvature flows and provides insights into their behavior and properties through spectral analysis and precise asymptotic analysis in various cases.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2024)