A numerical algorithm for L2L_2 semi-discrete optimal transport in 3D

This paper introduces a numerical algorithm to compute the $L_2$ optimal transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a density $\rho$ defined as a piecewise linear function (supported by a tetrahedral mesh), and where $\nu$ is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on Computational Geometry conf. proc.}, ACM (1992)] showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure $\mu$. The numerical algorithm is experimented and evaluated on several datasets, with up to hundred thousands tetrahedra and one million Dirac masses.Comment: 23 pages, 14 figure

Notions of optimal transport theory and how to implement them on a computer

This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, with active research communities, both on its theoretical aspects and on more applicative considerations, such as geometry processing and machine learning. This article aims at explaining the main principles behind the theory of optimal transport, introduce the different involved notions, and more importantly, how they relate, to let the reader grasp an intuition of the elegant theory that structures them. Then we will consider a specific setting, called semi-discrete, where a continuous function is transported to a discrete sum of Dirac masses. Studying this specific setting naturally leads to an efficient computational algorithm, that uses classical notions of computational geometry, such as a generalization of Voronoi diagrams called Laguerre diagrams.Comment: 32 pages, 17 figure

Global and local aspects of spectral actions

The principal object in noncommutatve geometry is the spectral triple consisting of an algebra A, a Hilbert space H, and a Dirac operator D. Field theories are incorporated in this approach by the spectral action principle, that sets the field theory action to Tr f(D^2/\Lambda^2), where f is a real function such that the trace exists, and \Lambda is a cutoff scale. In the low-energy (weak-field) limit the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up the second order in gauge connection and consider limiting cases.Comment: 22 pages, dedicated to Stuart Dowker on his 75th birthda

Tadpoles and commutative spectral triples

25 pages, 1 figureInternational audienceUsing the Chamseddine--Connes approach of the noncommutative action on spectral triples, we show that there are no tadpoles of any order for compact spin manifolds without boundary, and also consider a case of a chiral boundary condition. Using pseudodifferential techniques, we track zero terms in spectral actions

Cardiovascular and metabolic responses to catecholamine and sepsis prognosis: a ubiquitous phenomenon?

Many parameters have been associated with sepsis prognosis. In the present issue of Critical Care, Kumar and colleagues demonstrate that a preserved cardiac answer to dobutamine evaluated by radionucleotide measurements was associated with a better prognosis during septic shock. In this context, it is interesting to note that not only is the cardiac response to catecholamine stimulation associated with prognosis, but also the vascular and metabolic responses are associated. The ability of exogenous catecholamine to increase the arterial pressure (dopamine test) or to increase the lactate level is also related to prognosis. According to the ubiquitous character of catecholamine sensitivity, therefore, we should think in terms of cellular ability to respond to catecholamines in defining the concept of physiological reserve

Where does the lactate come from? A rare cause of reversible inhibition of mitochondrial respiration

Biguanide poisoning is associated with lactic acidosis. The exact mechanism of biguanide-induced lactic acidosis is not well understood. In the previous issue of Critical Care, Protti and colleagues demonstrated that biguanide-induced lactic acidosis may be due in part to a reversible inhibition of mitochondrial respiration. Thus, in the absence of an antidote, increased drug elimination through dialysis is logical