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"Verdier, Olivier"
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Orwell : old Etonian, copper, prole, dandy, militiaman, journalist, rebel, novelist, eccentric, socialist, patriot, gardener, hermit, visionary
George Orwell's most celebrated work, '1984', and the prescient vision it contains of a society governed by Big Brother predates the constant monitoring of people and data we are familiar with today by almost 70 years. But his life was every bit as fascinating and forward-looking as his books. Orwell studied at Eton, joined the police in Burma, fought in the Spanish Civil War, fiercely opposed Stalinism, and lived in London's slums while working as a journalist. With extra illustrations by a team of artists including Annie Goetzinger, Juanjo Guarnido, Enki Bilal, Manu Larcenet, Blutch, and Andre Juillard, Pierre Christin and Sebastian Verdier's Orwell offers readers an intimate yet definitive portrait of our greatest political writer.
BOOK
Statistical Model and ML-EM Algorithm for Emission Tomography with Known Movement
In positron emission tomography, movement leads to blurry reconstructions when not accounted for. Whether known a priori or estimated jointly to reconstruction, motion models are increasingly defined in continuum rather that in discrete, for example by means of diffeomorphisms. The present work provides both a statistical and functional analytic framework suitable for handling such models. It is based on time-space Poisson point processes as well as regarding images as measures, and allows to compute the maximum likelihood problem for line-of-response data with a known movement model. Solving the resulting optimisation problem, we derive an maximum likelihood expectation maximisation (ML-EM)-type algorithm which recovers the classical ML-EM algorithm as a particular case for a static phantom. The algorithm is proved to be monotone and convergent in the low-noise regime. Simulations confirm that it correctly removes the blur that would have occurred if movement were neglected.
Journal Article
The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge–Kutta method cannot preserve volume.
Journal Article
Spatio-Temporal Positron Emission Tomography Reconstruction with Attenuation and Motion Correction
The detection of cancer lesions of a comparable size to that of the typical system resolution of modern scanners is a long-standing problem in Positron Emission Tomography. In this paper, the effect of composing an image-registering convolutional neural network with the modeling of the static data acquisition (i.e., the forward model) is investigated. Two algorithms for Positron Emission Tomography reconstruction with motion and attenuation correction are proposed and their performance is evaluated in the detectability of small pulmonary lesions. The evaluation is performed on synthetic data with respect to chosen figures of merit, visual inspection, and an ideal observer. The commonly used figures of merit—Peak Signal-to-Noise Ratio, Recovery Coefficient, and Signal Difference-to-Noise Ration—give inconclusive responses, whereas visual inspection and the Channelised Hotelling Observer suggest that the proposed algorithms outperform current clinical practice.
Journal Article
Reductions of operator pencils
We study problems associated with an operator pencil, i.e., a pair of operators on Banach spaces. Two natural problems to consider are linear constrained differential equations and the description of the generalized spectrum. The main tool to tackle either of those problems is the reduction of the pencil. There are two kinds of natural reduction operations associated to a pencil, which are conjugate to each other. Our main result is that those two kinds of reductions commute, under some mild assumptions that we investigate thoroughly. Each reduction exhibits moreover a pivot operator. The invertibility of all the pivot operators of all possible successive reductions corresponds to the notion of regular pencil in the finite dimensional case, and to the inf-sup condition for saddle point problems on Hilbert spaces. Finally, we show how to use the reduction and the pivot operators to describe the generalized spectrum of the pencil.
Journal Article
A minimal-variable symplectic integrator on spheres
We construct a symplectic, globally defined, minimal-variable, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau–Lifshitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic.
Journal Article
A NUMERICAL ALGORITHM FOR C²-SPLINES ON SYMMETRIC SPACES
Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesies are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized Bézier curves. To construct C²-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C²-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the -sphere, complex projective space, and the real Grassmannian.
Journal Article
A Numerical Algorithm for$C^2$ -Splines on Symmetric Spaces
Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.
Journal Article
Simplified a priori estimate for the time periodic Burgers’ equation; pp. 34–41
We present here a version of the existence and uniqueness result of time periodic solutions to the viscous Burgersâ equation with irregular forcing terms (with Sobolev regularity â1 in space). The key result here is an a priori estimate which is simpler than the previously treated case of forcing terms with regularity â½ in time.
Journal Article
High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations
We propose a class of semi-Lagrangian methods of high approximation order in space and time, based on spectral element space discretizations and exponential integrators of Runge–Kutta type. The methods were presented in Celledoni and Kometa (J Sci Comput 41(1):139–164,
2009
) for simpler convection–diffusion equations. We discuss the extension of these methods to the Navier–Stokes equations, and their implementation using projections. Semi-Lagrangian methods up to order three are implemented and tested on various examples. The good performance of the methods for convection-dominated problems is demonstrated with numerical experiments.
Journal Article