Smoothing schemes for reaction-diffusion systems with nonsmooth data

Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002) 430–455] developed a class of Exponential Time Differencing Runge–Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [A.K. Kassam, Ll. N. Trefethen, Fourth-order time stepping for stiff pdes, SIAM J. Sci. Comput. 26 (2005) 1214–1233] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth-order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth-order Cox–Matthews, Kassam–Trefethen ETDRK4 scheme designed to eliminate the remaining computational difficulties. This new scheme utilizes an exponential time differencing Runge–Kutta ETDRK scheme using a diagonal Padé approximation of matrix exponential functions, while to deal with the problem of nonsmooth data we use several steps of an ETDRK scheme using a sub-diagonal Padé formula. The new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices, having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel. In this approach it is only required that several backward Euler linear problems be solved, in serial or parallel. Numerical experiments are described to support the new scheme.     

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.     

Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers

The Douglas–Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas–Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre–Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas–Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments.     

Erratum

Peculiar transitions in the optical spectra of Mn atoms isolated in argon matrices confirm the existence of Mn₂ molecule. The temperature variation of the absorption and MCD intensity in the range 13-27 K demonstrate that the Mn₂ molecule is antiferromagnetic and allows a precise determination of its ground-state exchange energy: J = ?10.3 ± 0.6 cm^?1.     

Novel results on structural investigations of natural minerals of clathrate hydrates

High-resolution powder X-ray diffraction measurements were carried out on gas hydrates in the angle-dispersive mode using a synchrotron radiation source. In contrast to the structural studies of laboratory-grown gas hydrates, this study has been performed on naturally grown clathrate hydrates obtained from the sea floor at different geographic locations. While the hydrate samples of the Cascadia Margin exhibit a preponderance of structure I, those from the Gulf of Mexico consist of mixed structures, namely structure II, structure I and structure H. Ice in structure Ih is inherently present in all the clathrate hydrate samples. PACS 61.10.Nz; 61.50.Ah; 61.66.Fn; 91.50.-r     

Local velocity measurements in the shear-thickening transition of dilute micellar solutions of surfactants

Local velocimetry and rheometric measurements are performed on three dilute micellar solutions which undergo the shear-thickening transition. The three surfactants, namely, alkyltrimethyl ammonium bromide (C(n)TAB), all belong to the same family and only differ by the length of the aliphatic chain. Simultaneous ultrasonic velocimetry and rheometry recordings provide convincing evidence for a heterogeneous flow in the shear-thickening domain. A detailed analysis allows us to demonstrate surprisingly similar evolutions of the wall slip magnitude and of the apparent viscosity as well as subtle differences between the three systems. Together with the velocimetry results, the direct observation of the flow in the vorticity-velocity plane reveals that the shear-thickening transition is associated with the emergence of a three-dimensional unstable flow.     

On the Synthesis of 3-Acetyl-1-aryl-1,4,5,6-tetrahydro-1 H-1,2,4-triazepin-7-ones by Reaction of Nitrilimines with 3-Aminopropanoic Acid

Nitrilimines were prepared from N-arylhydrazono chlorides and reacted with -alanine yielding the corresponding amidrazones, which were treated with 1,1-carbonyldiimidazole in THF affording the hitherto unknown 3-acetyl-1-aryl-1,4,5,6-tetrahydro-1,2,4-triazepin-7-ones.     

Electrosynthesis of well-organized nanoporous poly(3,4-ethylenedioxythiophene) by nanosphere lithography

The electrodeposition of poly(3,4-ethylenedioxythiophene) (PEDOT) films from aqueous surfactant solution through a two-dimensional poly(styrene) (PS) template onto indium tin oxide (ITO) substrate has been investigated. The polymer grows in the interstitial spaces of the self-assembled PS spheres which were subsequently removed by dissolution in tetrahydrofuran (THF). Surface characterization by scanning electron microscopy (SEM) and atomic force microscopy (AFM) reveals that two-dimensional nanoporous honeycomb PEDOT structures can easily be obtained by using PS spheres of different sizes. Gold electrodeposition onto the nanostructured PEDOT electrode was investigated and SEM images show preferential formation of nanoparticles (NP) on the wall and the rim of the PEDOT film but metal clusters inside the pores are also observed.     

Efficient One‐pot Synthesis of Alkyl 3‐(alkyl)‐4‐methyl‐2‐thioxo‐2,3‐dihydrothiazole‐5‐carboxylates in a Multi Component Reaction

A simple and efficient one‐pot synthesis of alkyl 2‐(alkyl)‐4‐methyl‐2‐thioxo‐2,3‐dihydrothiazole‐5‐carboxylates from the reaction of primary alkylamines and carbon disulfide in the presence of 2‐chloro‐1,3‐dicarbonyl compounds is described. This new protocol has several advantages such as lack of necessity of the catalyst, good yields, mild conditions and short times for reaction.     

An alternative synthesis of 2-(N-arylhydrazono)-1-benzothiophen-3-ones

In the presence of triethylamine, 2-mercaptobenzoic acid readily adds onto acylhydrazonoyl chlorides (1a-c) (precursors of the reactive nitrile imine 1,3-dipolar species) to afford good yields of the corresponding 2-[(2-oxo-1-arylhydrazonopropan-1-yl)mercapto]benzoic acids (2a-c). The latter acyclic adducts, in THF in the presence of 1,1'-carbonyldiimidazole, undergo intramolecular cyclization involving the activated carboxy and the enol functionality to deliver the respective 2-(N-arylhydrazono)-3-oxobenzothiophenes (3a-c). In the solid state, the latter compounds adopt the (Z)-geometry around the C=N double bond as evidenced by single crystal X-ray structure determination for 3b.