LUCA GRANIERI - Academia.edu (original) (raw)
Papers by LUCA GRANIERI
By disintegration of transport plans it is introduced the notion of transport class. This allows ... more By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case. Contents 19 4.1. Existence in some transport classes 20 4.2. Non-existence in some transport classes 20 References 20
Science & Philosophy, 2021
Modern science was historically built by avoiding a direct treatment of cause-effect relationshi... more Modern science was historically built by avoiding a direct treatment of cause-effect relationship questions. Recent developments in mathematical, probabilistic and statistical sciences make possible to pursue a more direct approach in cause-effect relationships leading to effective and systematic investigations in a wide range of scientific research fields.
Science & Philosophy, 2019
Mathematics plays a central role in modern science. However, it is very common an instrumental an... more Mathematics plays a central role in modern science. However, it is very common an instrumental and utilitarian view of math leading to the underestimation of its scientific nature. We propose some consideration to emphasize a different idea on the fundamental role of math in modern science.
We show how some problems coming from different fields of applied sciences such as physics, engin... more We show how some problems coming from different fields of applied sciences such as physics, engineering, biology, admit a common variational formulation characterized by the competition of two energetic terms. We discuss related problems and techniques studied by the authors and collaborators in the recent past as well open problems and further possible research directions in these topics.
Rendiconti del Seminario Matematico della Università di Padova, 2010
We investigate some geometric aspects of Wasserstein spaces through the continuity equation as wo... more We investigate some geometric aspects of Wasserstein spaces through the continuity equation as worked out in mass transportation theory. By defining a suitable homology on the flat torus T n , we prove that the space p (T n) has nontrivial homology in a metric sense. As a byproduct of the developed tools, we show that every parametrization of a Mather's minimal measure on T n corresponds to a mass minimizing metric current on p (T n) in its homology class. 1. Introduction. 1.1 ± The Monge-Kantorovich problem. Optimal transport problems, also known as Monge-Kantorovich problems, have been very intensively studied in the last 10 years and, due to the numerous and important applications to PDE, shape optimization and Calculus of Variations, we witnessed a spectacular development of the field. For the details of this theory, the interested reader may look at the book and lecture notes [1, 7, 31, 32], the paper [17] and for some of the applications [6, 22]. Our description will be restricted to the setting of a compact Riemannian manifold M without boundary, however many of the concepts of this section could be formulated in general metric spaces. Let c : M Â M 3 R be a Borel function. The Monge problem is formulated as follows: given two probability measures m; n find a map t : M 3 M such that t ] m n (] denotes the push-forward of measures) and such that t minimizes M c(x; t(x))dm
Analysis and Geometry in Metric Spaces, 2014
Optimization and Engineering, 2011
The quality assessment of manufacturing operations performed to obtain given flat surfaces is alw... more The quality assessment of manufacturing operations performed to obtain given flat surfaces is always a problem of comparing the substitute model (approximating the features of the true manufactured part) to the nominal specifications, at any stage of the manufacturing cycle. A novel methodology, based on applications of classical tools of Calculus of Variations, is here presented with the aim of assessing the output quality of manufactured flat surfaces based on the information available on transformation imposed by technological processes. By assuming that any manufacturing process operates under equilibrium states, the proposed variational methodology allows to account for the traces left by different stages of manufacturing processes. A simple twodimensional case is here discussed, to give the flavor of the methodology and its future potential developments.
Nonlinear Differential Equations and Applications NoDEA, 2007
In recent years different authors ([4, 16, 17]) have noticed and investigated some analogy betwee... more In recent years different authors ([4, 16, 17]) have noticed and investigated some analogy between Mather's theory of minimal measures in Lagrangian dynamic and the mass transportation (or Monge-Kantorovich) problem. We replace the closure and homological constraints of Mather's problem by boundary terms and we investigate the equivalence with the mass transportation problem. An Hamiltonian duality formula for the mass transportation and the equivalence with Brenier's formulation are also established.
Mathematical Methods in the Applied Sciences, 2010
We deal with a variational model of the Stock-Recruitment (S-R) relationship in fish dynamic. In ... more We deal with a variational model of the Stock-Recruitment (S-R) relationship in fish dynamic. In this model the S-R relationship is characterized as a minimizer of a suitable integral energy functional. By basic tools of Calculus of Variations a necessary condition is derived. As application, the derived condition is used to test the equilibrium of the recruitment level. An exploratory numerical procedure is also discussed.
Journal of Optimization Theory and Applications, 2010
Indagationes Mathematicae, 2009
The Monge-Kantorovich problem is equivalent to the problem of finding 1-currents with fixed bound... more The Monge-Kantorovich problem is equivalent to the problem of finding 1-currents with fixed boundary and minimal mass. We address the question of the stability for the mass minimizing currents. In particular, we state a-convergence result. We provide proofs relying just on basic properties of currents and on the notion of flat norm.
ESAIM: Control, Optimisation and Calculus of Variations, 2013
ESAIM: Control, Optimisation and Calculus of Variations, 2009
We provide an approximation of Mather variational problem by finite dimensional minimization prob... more We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [9] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.
Calculus of Variations and Partial Differential Equations, 2006
Applied Mathematics Letters, 2009
Acta Applicandae Mathematicae, 2013
IMA Journal of Applied Mathematics, 2009
We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this w... more We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this work, this problem is related to the minimization of a surface energy, depending on the stress tensor in the original equilibrium configuration. Special cases of mechanical interest are also analysed.
Journal of Elasticity, 2011
By disintegration of transport plans it is introduced the notion of transport class. This allows ... more By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case. Contents 19 4.1. Existence in some transport classes 20 4.2. Non-existence in some transport classes 20 References 20
Science & Philosophy, 2021
Modern science was historically built by avoiding a direct treatment of cause-effect relationshi... more Modern science was historically built by avoiding a direct treatment of cause-effect relationship questions. Recent developments in mathematical, probabilistic and statistical sciences make possible to pursue a more direct approach in cause-effect relationships leading to effective and systematic investigations in a wide range of scientific research fields.
Science & Philosophy, 2019
Mathematics plays a central role in modern science. However, it is very common an instrumental an... more Mathematics plays a central role in modern science. However, it is very common an instrumental and utilitarian view of math leading to the underestimation of its scientific nature. We propose some consideration to emphasize a different idea on the fundamental role of math in modern science.
We show how some problems coming from different fields of applied sciences such as physics, engin... more We show how some problems coming from different fields of applied sciences such as physics, engineering, biology, admit a common variational formulation characterized by the competition of two energetic terms. We discuss related problems and techniques studied by the authors and collaborators in the recent past as well open problems and further possible research directions in these topics.
Rendiconti del Seminario Matematico della Università di Padova, 2010
We investigate some geometric aspects of Wasserstein spaces through the continuity equation as wo... more We investigate some geometric aspects of Wasserstein spaces through the continuity equation as worked out in mass transportation theory. By defining a suitable homology on the flat torus T n , we prove that the space p (T n) has nontrivial homology in a metric sense. As a byproduct of the developed tools, we show that every parametrization of a Mather's minimal measure on T n corresponds to a mass minimizing metric current on p (T n) in its homology class. 1. Introduction. 1.1 ± The Monge-Kantorovich problem. Optimal transport problems, also known as Monge-Kantorovich problems, have been very intensively studied in the last 10 years and, due to the numerous and important applications to PDE, shape optimization and Calculus of Variations, we witnessed a spectacular development of the field. For the details of this theory, the interested reader may look at the book and lecture notes [1, 7, 31, 32], the paper [17] and for some of the applications [6, 22]. Our description will be restricted to the setting of a compact Riemannian manifold M without boundary, however many of the concepts of this section could be formulated in general metric spaces. Let c : M Â M 3 R be a Borel function. The Monge problem is formulated as follows: given two probability measures m; n find a map t : M 3 M such that t ] m n (] denotes the push-forward of measures) and such that t minimizes M c(x; t(x))dm
Analysis and Geometry in Metric Spaces, 2014
Optimization and Engineering, 2011
The quality assessment of manufacturing operations performed to obtain given flat surfaces is alw... more The quality assessment of manufacturing operations performed to obtain given flat surfaces is always a problem of comparing the substitute model (approximating the features of the true manufactured part) to the nominal specifications, at any stage of the manufacturing cycle. A novel methodology, based on applications of classical tools of Calculus of Variations, is here presented with the aim of assessing the output quality of manufactured flat surfaces based on the information available on transformation imposed by technological processes. By assuming that any manufacturing process operates under equilibrium states, the proposed variational methodology allows to account for the traces left by different stages of manufacturing processes. A simple twodimensional case is here discussed, to give the flavor of the methodology and its future potential developments.
Nonlinear Differential Equations and Applications NoDEA, 2007
In recent years different authors ([4, 16, 17]) have noticed and investigated some analogy betwee... more In recent years different authors ([4, 16, 17]) have noticed and investigated some analogy between Mather's theory of minimal measures in Lagrangian dynamic and the mass transportation (or Monge-Kantorovich) problem. We replace the closure and homological constraints of Mather's problem by boundary terms and we investigate the equivalence with the mass transportation problem. An Hamiltonian duality formula for the mass transportation and the equivalence with Brenier's formulation are also established.
Mathematical Methods in the Applied Sciences, 2010
We deal with a variational model of the Stock-Recruitment (S-R) relationship in fish dynamic. In ... more We deal with a variational model of the Stock-Recruitment (S-R) relationship in fish dynamic. In this model the S-R relationship is characterized as a minimizer of a suitable integral energy functional. By basic tools of Calculus of Variations a necessary condition is derived. As application, the derived condition is used to test the equilibrium of the recruitment level. An exploratory numerical procedure is also discussed.
Journal of Optimization Theory and Applications, 2010
Indagationes Mathematicae, 2009
The Monge-Kantorovich problem is equivalent to the problem of finding 1-currents with fixed bound... more The Monge-Kantorovich problem is equivalent to the problem of finding 1-currents with fixed boundary and minimal mass. We address the question of the stability for the mass minimizing currents. In particular, we state a-convergence result. We provide proofs relying just on basic properties of currents and on the notion of flat norm.
ESAIM: Control, Optimisation and Calculus of Variations, 2013
ESAIM: Control, Optimisation and Calculus of Variations, 2009
We provide an approximation of Mather variational problem by finite dimensional minimization prob... more We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [9] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.
Calculus of Variations and Partial Differential Equations, 2006
Applied Mathematics Letters, 2009
Acta Applicandae Mathematicae, 2013
IMA Journal of Applied Mathematics, 2009
We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this w... more We consider the problem of mass reduction for elastic bodies by appearance of cavities. In this work, this problem is related to the minimization of a surface energy, depending on the stress tensor in the original equilibrium configuration. Special cases of mechanical interest are also analysed.
Journal of Elasticity, 2011