On Generalized Douglas–Weyl(α, β)-Metrics

In this paper, we study generalized Douglas–Weyl(α, β)-metrics. Suppose that a regular(α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas–Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas–Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.     

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.     

平面热传导方程Douglas交替方向隐格式的稳定性与收敛性

1 引言 解抛物方程的交替方向差分方法产生于本世纪五十年代,由于其稳定性好、易编制计算程序,几十年来在科学与工程计算中得到了广泛的应用,最早的交替方向差分格式由Peace- man,D.W.和Rachford,H.H.以及Douglas,J,Jr.和Rachford,H.H.给出,Douglas,J.Jr.于1962年提出了后人称之为Douglas格式的一类交替方向隐格式,其特点是有二阶截断误差而且适用于三维情形.下面简要给出其格式。 考虑平面方形区域上热传导方程 (1.1) (1.2) (1.3)其中Ω=(0,1)×(0,1),为其边界,t为时间变量,以及u_o为已知函数. 令x与y方向网格步长均为h,x_i=ih,y_j=jh,N=1/h为正整数;时间步长为△t,t~n=n△t,记w_(ij)~n=w(x_i,y_j,t~n)以及     

A simplified proof of weak convergence in Douglas–Rachford method

Douglas–Rachford method is a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Weak convergence in this method to a solution of the underlying monotone inclusion problem in the general case remained an open problem for 30 years and was proved by the author 7 years ago. That proof was cluttered with technicalities because we considered the inexact version with summable errors. In this short communication we present a streamlined proof of this result.     

Triterpenoids fromAbies species

Four new compounds were identified in a mixture of tetracyclic methyl ketones obtained by alkaline cleavage of acids of Siberian fir needles. Two of them correspond to natural acids of the 17,13-fridolanostane series of yet unknown structure.For Part 19, see Ref. 1.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2997–2999, December, 1996.     

具有可反测地线的$(\alpha,\beta)$-度量

本文我们得到了$(\alpha,\beta)$-度量的测地系数$G^{i}(x,y)$和其逆$G^{i}(x,-y)$有相同Douglas曲率的充分必要条件.这个充分必要条件恰好是$(\alpha,\beta)$-度量具有可反测地线的充分必要条件.     

On the finite termination of the Douglas-Rachford method for the convex feasibility problem

In this paper we apply the Douglas–Rachford (DR) method to solve the problem of finding a point in the intersection of the interior of a closed convex cone and a closed convex set in an infinite-dimensional Hilbert space. For this purpose, we propose two variants of the DR method which can find a point in the intersection in a finite number of iterations. In order to analyse the finite termination of the methods, we use some properties of the metric projection and a result regarding the rate of convergence of fixed point iterations. As applications of the results, we propose the methods for solving the conic and semidefinite feasibility problems, which terminate at a solution in a finite number of iterations.     

平面光波导器件时域有限差分束传输法分析中边界和初始条件的处理

从慢变包络（SVEA）时域光传输方程出发,对比了综合道格拉斯（GD）和传统的中心差分（CN）两种离散化格式,指出GD法比CN法精度有了很大提高.重点讨论了GD法的TD-BPM边界处理问题,给出了吸收边界和透明边界离散的显式和隐式表达式,分析并确定了交替方向隐式法（ADIM）的边界处理方式;同时,对激励源加入的连接边界进行了仔细讨论.结果表明对于隐式GD-TD-BPM法边界处理应采用隐式,中间激励更能反映光波导反射场的情况.     

Relativistic and electron‐correlation effects on magnetizabilities investigated by the Douglas‐Kroll‐Hess method and the second‐order Møller‐Plesset perturbation theory

Isotropic and anisotropic magnetizabilities for noble gas atoms and a series of singlet and triplet molecules were calculated using the second‐order Douglas‐Kroll‐Hess (DKH2) Hamiltonian containing the vector potential A and in part using second‐order generalized unrestricted Møller‐Plesset (GUMP2) theory. The DKH2 Hamiltonian was resolved into three parts (spin‐free terms, spin‐dependent terms, and magnetic perturbation terms), and the magnetizabilities were decomposed into diamagnetic and paramagnetic terms to investigate the relativistic and electron‐correlation effects in detail. For Ne, Kr, and Xe, the calculated magnetizabilities approached the experimental values, once relativistic and electron‐correlation effects were included. For the IF molecule, the magnetizability was strongly affected by the spin‐orbit interaction, and the total relativistic contribution amounted to 22%. For group 17, 16, 15, and 14 hydrides, the calculated relativistic effects were small (less than 3%), and trends were observed in relativistic and electron‐correlation effects across groups and periods. The magnetizability anisotropies of triplet molecules were generally larger than those of similar singlet molecules. The so‐called relativistic‐correlation interference for the magnetizabilities computed using the relativistic GUMP2 method can be neglected for the molecules evaluated, with exception of triplet SbH. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009