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.     

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     

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     

Magnetic-Field-Dependent Electronic Relaxation of Gd3+ in Aqueous Solutions of the Complexes [Gd(H2O)8]3+, [Gd(propane-1,3-diamine-N,N,N′,N′-tetraacetate)(H2O)2]−, and [Gd(N,N′-bis[(N-methylcarbamoyl)methyl]-3-azapentane-1,5-diamine-3,N,N′-triacetate)(H2O)] of interest in magnetic-resonance imaging

EPR Spectra have been measured for aqueous solutions of a series of Gd³⁺ complexes at variable temperature and a range of magnetic fields; S-band (0.14 T), X-band (0.34 T), Q-band (1.2 T), and 2-mm-band (5.0 T). The major contribution to the observed line widths is magnetic-field-dependent and is interpreted as being due to the modulation of the zero-field splitting produced by distortion of the complexes from perfect symmetry. The transverse and longitudinal relaxation matrices for an ⁸S ion with such an interaction have been calculated using Redfield theory with vector-coupling methods, and diagonalised numerically to obtain relaxation rates and intensities for the degenerate transitions which contribute to the multiplet. The observed line width, which is inversely proportional to the magnetic field at low temperatures, is best described by the intensity-weighted mean transverse relaxation time for the four transitions with non-zero intensity. A least-squares fit of the data yields the square of the zero-field splitting tensor, Δ², and a correlation time, τ_v, with activation energy, E_v. The physical significance of these parameters and the extent of validity of the theoretical approach are considered. The parameters are used to predict the magnetic-field dependence of the longitudinal and transverse electronic relaxation times, which are discussed in the context of their relevance to ¹H-NMR relaxivity.     

Algebraically flat or projective algebras

We define and study algebraically flat algebras in order to have a better understanding of algebraically projective algebras of finite type (the projective algebras of literature). A close examination of the differential properties of these algebras leads to our main structure theorem. As a corollary, we get that an algebraically projective algebra of finite type over a field is either a polynomial ring or the affine algebra of a complete intersection.     

RECURSIVE BAIRE CLASSIFICATION AND SPEEDABLE FUNCTIONS

Using recursive variants of Baire notions of nowhere dense and meagre sets we study the topological size of speedable and infinitely often speedable functions in a machine-independent framework. We show that the set of speedable functions is not “small” whereas the set of infinitely often speedable functions is “large”. In this way we offer partial answers to a question in [4].     

Microchemical Investigation of Archaeological Copper Based Artefacts Disclosing an Ancient Witness of the Transition from the Value of the Substance to the Value of the Appearance

In many Italian archaeological sites dated between the sixth and third centuries BC, unworked lumps of Cu-based materials are sometimes found, the so called Aes Rude, which according to archaeological considerations were appreciated as currency, as a medium of exchange and as a form of saving. The microchemical investigation of these ancient artefacts discloses their nature as apparently not usable for any functional applications or possible use. Indeed, Aes Rude resemble ordinary copper material, but microchemical results indicate that they are constituted by highly ferruginous leaded copper, making them useless for producing other metal objects by means of casting or hot and cold working. Notwithstanding this intrinsic negative feature, the production of these intractable Cu-based alloys was deliberately carried out to maximise the process yield in terms of produced metal from an impure and unselected metal ore by tailoring the smelting process parameters. With these considerations in mind, the microchemical investigation of these ancient iron–copper alloys gives evidence of the passage from the acceptance of an artefact value based on its true nature or potential use to the acceptance of the value based only on its appearance or form irrespective of its present or future use. This information could contribute to a better understanding of the evolution of human thought and economic and social interactions.