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.     

双层桥梁耦合系统的强迫振动

本文研究作为双层桥模型的梁方程耦合系统，利用Ｌｅｒａｙ－Ｓｃｈａｕｄｅｒ不动点定理，得到了一个关于这种系统的解的存在性定理，它类似于ＭｃＫｅｎｎａ和Ｗａｌｔｅｒ文２中关于吊桥方程的一个定理。     

激光等离子体探针及应用研究

本文介绍用阴影、纹影相干涉方法确定等离子体折射系数变化的原理。分析了等离子体中的非线性效应和对光探测系统的基本要求。并简要叙述了实验上采用的后向喇曼探测束,给出了等离子体干涉、纹影和阴影的测量结果。     

Some Notes about Tanaka''''s Equation

ＳｏｍｅＮｏｔｅｓａｂｏｕｔＴａｎａｋａ＇ｓＥｑｕａｔｉｏｎＹａｎＺｈｉｂｉｎ（严质彬）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨａｒｂｉｎＩｎｓｔｉｔｕｔｅｏｆＴｅｃｈｎｏｌｏｇｙ，Ｈａｒｂｉｎ，１５０００１）ＡｂｓｔｒａｃｔＬｅｔ｛Ｗｔ...     

Jensen’s Inequality for Backward Stochastic Differential Equations

Abstract Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen’s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) ≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen’s inequality for g- expectation in [4, 7–9]. *Project supported by the National Natural Science Foundation of China (No.10325101) and the Science Foundation of China University of Mining and Technology.     

New isothermal equation of state of solids applied to high pressures

In the present paper, a new two-parameter inverted equation of state (EOS) is developed which is found to be working very well in the high-pressure region. To check its success and validity, this EOS has been applied in a number of solids. The computed volume compression is found to be in very good agreement with the experimental data in the whole range of pressure in all the solids. The minimum and the maximum pressure range used in the present study is 0–320 kbar and 0–3000 kbar, respectively.     

Fourier regularization for a backward heat equation

In this paper a simple and convenient new regularization method for solving backward heat equation—Fourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.     

A Degenerate Stefan Problem with Two Free Boundaries

ＡＤｅｇｅｎｅｒａｔｅＳｔｅｆａｎＰｒｏｂｌｅｍｗｉｔｈＴｗｏＦｒｅｅＢｏｕｎｄａｒｉｅｓ￥（李辉来）ＬｉＨｕｉｌａｉ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＪｉｌｉｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈａｎｇｃｈｕｎ，１３００２３）Ａｂｓｔｒａｃｔ...     

一类混合单调算子的不动点定理及其应用

本文对一类重要的混合单调算子证明了不动点的存在、唯一与逼近定理，并应用于研究动态规划中泛函方程的正解问题．