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 generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi-Hamiltonian structures

For the orthosymplectic Lie superalgebra $\text{◂⋅▸}OSP(2,2)$, we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi-Hamiltonian structures.     

The single-polarization filter composed of gold-coated photonic crystal fiber

A single-polarization filter comprising a gold-coated photonic crystal fiber based on surface plasmon resonance is designed and investigated. The pattern matching and coupled polarization characteristics analyzed by the full-vector finite element method (FEM) and losses at 1,540 nm are achieved to 1,016.01739 dB/cm (x-pol core mode) and 33.81917 dB/cm (y-pol core mode). The crosstalk (CT) value of the 1,540 nm band is ?853.12653 dB for fiber length $L=1,000\mathrm{\mu }\text{m}$ and the bandwidth is 850 nm. The working wavelength of the filter ranges from 1,280 nm to 1,540 nm by varying the diameter of outer air holes ($d_1$), the diameter of inner air holes ($d_4$), the metal film thickness (t), as well as the liquid refractive index (n).     

S—R分解定理的唯一性,存在性和客观性

对于连续体的一切物理可能的变形场，其变形梯度张量F可被分解为一个对称张量S和一个正交张量R的直和，这便是S-R分解定理．本文通过矩阵方法和张量方法证明了S-R分解定理的唯一性、存在性和客观性．     

微孔和介孔材料中的热化学

这篇论文综述了美国加州大学戴维斯分校科学院院士Navrotsky课题组多年来在多孔材料上取得的一系列热化学研究结果。讨论了热化学对微孔、介孔材料的结构稳定性和合成过程的影响。借助多种测热手段对影响骨架结构的热焓、热熵和自由能进行了系统的测量和计算。研究数据表明一系列纯硅分子筛、介孔材料和磷酸铝多孔材料同相应的石英相和块磷铝矿相相比能量上最多只高出15 kJ·mol-1。一系列纯硅分子筛的熵值比石英相高出3.2—4.2 J·K-1·mol-1；在0—12.6 J·K-1·mol-1范围内相对应的自由能几乎没有差别。因此，对不同微孔、介孔材料，其骨架结构在能量上是几乎没有区别的。另外，本文通过介绍一种新型测热方法——原位测热，揭示了分子筛合成过程中的动力学和成核/结晶机理。     

铝(Ⅲ)-桑色素-十二烷基硫酸钠体系在滤纸基质的荧光反应研究及其分析应用

本文探讨了在滤纸基质上表面活性剂SLS对铝-桑色素荧光反应的增敏作用及最佳实验条件。铝含量在0～6.0ng范围的工作曲线呈良好的线性关系(r=0.9998),方法的检测限为0.001ng。含铝4.0ng的溶液11次测定结果的相对标准偏差为4.3％。方法用于含铝水样的测定获得满意结果,回收率为96.0～100.5％。     

Determination of manganese in grain by potential titration with catalytic end-point indication

A method of catalytic potential titration for the determination of trace manganese by using crystal violet ion selective electrode is reported in this paper. It is based on the catalytic effect of manganese (II) on the oxidation of crystal violet by potassium periodate in the presence of nitrilotriacetic acid (NTA). The experiments indicated that this technique showed high sensitivity and high accuracy. The results of determination of trace manganese in grain could be compared with the results obtained by means of atomic absorption spectrometry.     

Preparation and characterization of sulfated zirconia (SO4/ZrO2)/Nafion composite membranes for PEMFC operation at high temperature/low humidity

Fine particle superacidic sulfated zirconia (SO₄²⁻/ZrO₂, S-ZrO₂) was synthesized by ameliorated method, and composite membranes with different S-ZrO₂ contents were prepared by a recasting procedure from a suspension of S-ZrO₂ powder and Nafion solution. The physico-chemical properties of the membranes were studied by ion exchange capacity (IEC) and liquid water uptake measurements, scanning electron microscopy (SEM) and X-ray diffraction (XRD) analysis, thermogravimetry–mass spectrometry (TG–MS) and Fourier transform infrared (FT-IR) spectroscopy. The results showed that the IEC of composite membrane increased with the content of S-ZrO₂, S-ZrO₂ was compatible with the Nafion matrix, the incorporation of the S-ZrO₂ could increase the crystallinity and also improve the initial degradation temperature of the composite membrane. The performance of single cell was the best when the S-ZrO₂ content was 15 wt.%, and achieved 1.35 W/cm² at 80 °C and 0.99 W/cm² at 120 °C based on H₂/O₂ and at a pressure of 2 atm, the performance of the single cell with optimized S-ZrO₂ was far more than that of the Nafion at the same condition (e.g. 1.28 W/cm² at 80 °C, 0.75 W/cm² at 120 °C). The 15 wt.% S-ZrO₂/Nafion composite membrane showed lower fuel cell internal resistance than Nafion membranes at high temperature and low relative humidity (RH).