Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

Remarks on exact controllability for Stokes and Navier–Stokes systems

This Note deals with the controllability of Stokes and Navier–Stokes systems with distributed controls with support in possibly small subdomains. We first present a new global Carleman inequality for the solutions to Stokes-like systems that leads to the null controllability at any time T>0. Then, we present a local result concerning exact controllability to trajectories of the Navier–Stokes system. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Synthesis of 2-pyrone-4-carboxaldehydes from acetylene dicarboxaldehyde monoacetal

Reaction of acetylene dicarboxaldehyde monoacetal with substituted Meldrum’s acid leads good yields in 2-pyranones-4-carboxaldehydes substituted in position 3.     

A mixed‐FEM formulation for nonlinear incompressible elasticity in the plane

This article deals with an expanded mixed finite element formulation, based on the Hu‐Washizu principle, for a nonlinear incompressible material in the plane. We follow our related previous works and introduce both the stress and the strain tensors as further unknowns, which yields a two‐fold saddle point operator equation as the corresponding variational formulation. A slight generalization of the classical Babu?ka‐Brezzi's theory is applied to prove unique solvability of the continuous and discrete formulations, and to derive the corresponding a priori error analysis. An extension of the well‐known PEERS space is used to define an stable associated Galerkin scheme. Finally, we provide an a posteriori error analysis based on the classical Bank‐Weiser approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 105–128, 2002     

High-frequency magnetoimpedance in nanocomposites

The transmission of millimeter-range electromagnetic waves (30–50 GHz) through a magnetic nanocomposite thin film exhibiting tunnel magnetoresistance (TMR) is calculated. The relative change of transmission coefficient in an applied magnetic field due to the magnetorefractive effect is approximately linear with TMR and strongly depends on nanocomposite resistivity and film thickness. The obtained results are in a good agreement with experiment.     

Approche non paramétrique du filtrage de système non linéaire à temps discret et à paramètres inconnus

Particle filters are presently among the most powerful tools for filtering discrete time non linear systems. However the presence of unknown parameters in the system equations makes their use more complex and can even impair their convergence properties. This Note shows how an on-line consistent estimation of these parameters can be obtained simultaneously to that of the state variables to be filtered. This approach relies upon a kernel-based non parametric estimation of conditional probability densities from successive Monte Carlo generations of system particles. To cite this article: V. Rossi, J.-P. Vila, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Viscoelastic properties of nanostructured natural rubber/polystyrene interpenetrating polymer networks

The effects of the blend ratio and initiating system on the viscoelastic properties of nanostructured natural rubber/polystyrene‐based interpenetrating polymer networks (IPNs) were investigated in the temperature range of ?80 to 150 °C. The studies were carried out at different frequencies (100, 50, 10, 1, and 0.1 Hz), and their effects on the damping and storage and loss moduli were analyzed. In all cases, tan δ and the storage and loss moduli showed two distinct transitions corresponding to natural rubber and polystyrene phases, which indicated that the system was not miscible on the molecular level. However, a slight inward shift was observed in the IPNs, with respect to the glass‐transition temperatures (T_g's) of the virgin polymers, showing a certain degree of miscibility or intermixing between the two phases. When the frequency increased from 0.1 to 100 Hz, the T_g values showed a positive shift in all cases. In a comparison of the three initiating systems (dicumyl peroxide, benzoyl peroxide, and azobisisobutyronitrile), the dicumyl peroxide system showed the highest modulus. The morphology of the IPNs was analyzed with transmission electron microscopy. The micrographs indicated that the system was nanostructured. An attempt was made to relate the viscoelastic behavior to the morphology of the IPNs. Various models, such as the series, parallel, Halpin–Tsai, Kerner, Coran, Takayanagi, and Davies models, were used to model the viscoelastic data. The area under the linear loss modulus curve was larger than that obtained by group contribution analysis; this showed that the damping was influenced by the phase morphology, dual‐phase continuity, and crosslinking of the phases. Finally, the homogeneity of the system was further evaluated with Cole–Cole analysis. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1680–1696, 2003     

On the Curvature of the Central Path of Linear Programming Theory

We prove a linear bound on the average total curvature of the central path of linear programming theory in terms of the number of independent variables of the primal problem, and independent of the number of constraints.     

Conversations on mind, matter, and mathematics

Conclusion  The debate between Jean-Pierre Changeux and Alain Connes is one of the most, interesting to take place in recent years. It re-frames in a very up-to-date context a whole series of traditional and difficult questions from the standpoint of the knowledge and experience of two of the leading protagonists of contemporary science. To the choice presented by the neurobiologist between a Platonist ontology and a neurocognitive psychology of mathematical activity, the mathematician replies with a conception that is objective (neither ontological nor psychological) of the thoroughly consistent universe of mathematical idealities. It is indeed in this three-sided arena that the major difficulties play themselves out. One of the great virtues of the book is to cast a spotlight on this confrontation.