求解稀疏相位恢复问题的随机交替方向法(英文)

近年来稀疏相位恢复问题受到了越来越多的关注.本文提出了一种随机交替方法方法求解稀疏相位恢复问题,该算法采用硬阈值追踪算法求解带稀疏约束的最小二乘子问题.大量的数值实验表明,该算法可以通过O(s log n)次测量(理论上最少测量值)稳定的恢复n维s稀疏向量,并且在随机初值下可以获得全局收敛性.     

VRPTW的扰动恢复及其TABU SEARCH算法

本文对带时间窗的车辆路线安排扰动恢复问题进行了讨论,分析了各种可能的扰动:增加减少客户,时间窗、客户需求及路线可行性的扰动,构造了扰动模型.利用禁忌搜索算法对问题进行求解,同时通过对模型参数重新设置,得到了多个满足要求的不同的解,这样使解更具有实际可行性和有效性.     

基于基底合并的分片稀疏恢复

如我们所知,诸如视频和图像等信号可以在某些框架下被表示为稀疏信号,因此稀疏恢复(或稀疏表示)是信号处理、图像处理、计算机视觉、机器学习等领域中被广泛研究的问题之一.通常大多数在稀疏恢复中的有效快速算法都是基于求解$l^0$或者$l^1$优化问题.但是,对于求解$l^0$或者$l^1$优化问题以及相关算法所得到的理论充分性条件对信号的稀疏性要求过严.考虑到在很多实际应用中,信号是具有一定结构的,也即,信号的非零元素具有一定的分布特点.在本文中,我们研究分片稀疏恢复的唯一性条件和可行性条件.分片稀疏性是指一个稀疏信号由多个稀疏的子信号合并所得.相应的采样矩阵是由多个基底合并组成.考虑到采样矩阵的分块结构,我们引入了子矩阵的互相干性,由此可以得到相应$l^0$或者$l^1$优化问题可精确恢复解的稀疏度的新上界.本文结果表明.通过引入采样矩阵的分块结构信息.可以改进分片稀疏恢复的充分性条件.以及相应$l^0$或者$l^1$优化问题整体稀疏解的可靠性条件.     

动态Gabor矩阵测量的相位恢复

相位恢复是一类由无相位采样值恢复待估信号的问题.本文讨论的采样是由动态Gabor系统得到的.我们证明了关于动态Gabor测量矩阵可相位恢复的充分条件,并给出了C2和R3中的例子.     

构建复杂系统的解景观

很多交叉科学的实际问题在数学上都可以被归为求解具有多个变量的非线性函数或泛函的极小值问题,如何有效地寻找其能量景观的全局极小和如何找到不同极小之间的关系是计算数学领域两个长久以来尚未解决的重要科学问题.本文着重介绍近年来提出的“解景观”概念和方法.我们将回顾解景观的概念、构建解景观的鞍点动力学方法、以及解景观在液晶和准晶方面的应用.     

增强解题优化意识

<正>同学们知道,一个数学问题,由于思维视角的不同,常有不同的解法.而且不同解法的繁简程度迥异不同.因此,在求解一个数学问题时,应当增强解题的优化意识.本文拟通过探究几个典型的解析几何问题的求解过程,介绍几种优化解题过程的常见途径,供同学们参考.一、返璞归真所谓返璞归真,是指在解题中,强化基本概念、基本性质与基本方法意识,通过比较,从中找到简洁求解问题的方法,从而达到优化解     

高可扩展区域分解算法及其在流体模拟和优化问题中的应用

本文介绍一套求解复杂流体模拟和优化控制问题的高可扩展并行算法.该算法基于非结构化网格,结合了加稳定化项的有限元空间离散方法、全隐的时间离散格式、多物理场全耦合的求解算法、区域分解算法及求解非线性系统的Newton-Krylov-Schwarz算法等多套先进算法.利用该算法,本文对多个实际工程应用中流体模拟和优化设计问题进行了测试,数值结果显示,该算法对本文研究的几类问题,具有很好的收敛性和并行可扩展性,当使用8192个处理器核求解规模超过两千万个网格单元的问题时,仍然具有超过40%的并行效率.     

基于非线性相位恢复的X射线相位衬度断层成像算法

对全息测量下的X射线相位衬度断层成像问题提出了一种新的重建算法．该算法的主要想法是利用牛顿迭代法求解非线性的相位恢复问题．我们证明了牛顿方向满足的线性方程是非适定的,并利用共轭梯度法得到方程的正则化解．最后利用模拟数据进行了数值实验,数值结果验证了算法的合理性以及对噪声数据的数值稳定性,同时通过与线性化相位恢复算法的数值结果比较说明了新算法对探测数据不要求限制在Fresnel区域的近场,适用范围更广．     

可相位恢复p-框架的稳定性

相位恢复问题在物理和工程中有着广泛的应用.设X是Banach空间,1＜p＜∞.设Φ={xn}n∈I是X上的p-框架.若对任意x*,y*∈X*,等式|x*(xn)|=|y*(xn)|对任意n∈I成立蕴涵存在|α|=1使得x*=αy*,则称Φ是可相位恢复的.本文证明在有限维Banach空间上,可相位恢复p-框架是稳定的,但...     

图像恢复的一种快速迭代正则化方法

本文研究了将图像恢复问题转化为大型的线性不适定问题的求解.利用由Landweber迭代正则化方法改进所得到的快速收敛的迭代正则化方法,处理具有可分离点扩散函数的图像恢复问题.图像恢复实验表明该方法可大大提高收敛速度,且在计算中只需要较少的存储量.     

The Minimal Measurement Number Problem in Phase Retrieval: A Review of Recent Developments

Phase retrieval is to recover the signals from phaseless measurements which is raised in many areas. A fundamental problem in phase retrieval is to determine the minimal measurement number $m$ so that one can recover $d$-dimensional signals from $m$ phaseless measurements. This problem attracts much attention of experts from different areas. In this paper, we review the recent development on the minimal measurement number and also raise many interesting open questions.     

压缩感知

压缩感知是近来国际上热门的研究方向. 其主要思想为: 利用信号稀疏性的特征, 通过尽量少的观测信息恢复信号. 压缩感知在多个应用领域, 如医学成像、图像处理、地质勘探等中具有很好的应用前景. 此外, 它与逼近论、最优化、随机矩阵及离散几何等领域密切相关, 由此产生了一些漂亮的数学结果. 本文综述压缩感知一些基本结果并介绍最新进展. 主要包括RIP 矩阵编码与l₁ 解码的性能、RIP (restricted isometry property) 矩阵的构造、Gelfand 宽度、个例最优性及OMP (orthogonalmatching pursuit) 解码等.     

Hopf monads

We introduce and study Hopf monads on autonomous categories (i.e., monoidal categories with duals). Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. In particular, any monoidal adjunction between autonomous categories gives rise to a Hopf monad. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke's criterium of semisimplicity, etc.) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again defined in a non-braided setting).     

问卷设计知识管理中的范例推理

将范例推理技术应用于问卷设计知识重用过程中,提出了基于范例推理的问卷设计知识重用系统的应用框架,研究了问卷设计CBR系统中的一些关键技术:范例的表示、范例组织、范例检索算法、范例修正优化和重用、范例学习等.范例检索采用KNN法与遗传禁忌算法相结合的混合检索机制,通过建立GA适应度函数模型以及选择、禁忌交叉、禁忌变异算子等自动确定权重.我们组织和建立了初步的范例知识库,进行了相关实验.结果表明,融合遗传算法的范例检索方法可较大程度上缩短设计时间、提高问卷设计效率和质量,并有效地支持问卷设计的知识重用和知识管理创新.     

Fuzzy Topological Representation Theory For D_(01)

In this paper, we shall introduce the concepts of fuzzy prime ideal, fuzzy Stone space, fuzzy fundamental set, etc. And the representations of a distributive lattice with 0, 1 (D_(01) denotes the equational class of such lattices) by fuzzy sets will be investigated and many useful results are obtained.     

算子的拟相似与(Q)类算子

本文引进定义于L(H)上的集值函数β(S)和(Q)类算子(指本质谱含于其一切拟相似算子的本质谱的算子),用β(S)刻划 (Q)类算子的特征;证明(Q)类算子范围广泛,次可分解算子(包括次标量算子,M-亚正常算子,半亚正常算子等等)是其中的一部分;(Q)类算子在L(H)中稠密.     

On signal reconstruction from FROG measurements

Phase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. Unfortunately, phase retrieval is ill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the ill-posedness, it is common to use a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. The FROG trace results in a system of phaseless quartic Fourier measurements. In this paper, we prove that it suffices to consider only three translations of the signal to determine almost all bandlimited signals, up to trivial ambiguities. In practice, one usually also has access to the signal's Fourier magnitude. We show that in this case only two translations suffice. Our results significantly improve upon earlier work.     

Information dynamics algorithm for detecting communities in networks

The problem of community detection is relevant in many scientific disciplines, from social science to statistical physics. Given the impact of community detection in many areas, such as psychology and social sciences, we have addressed the issue of modifying existing well performing algorithms by incorporating elements of the domain application fields, i.e. domain-inspired. We have focused on a psychology and social network-inspired approach which may be useful for further strengthening the link between social network studies and mathematics of community detection. Here we introduce a community-detection algorithm derived from the van Dongen’s Markov Cluster algorithm (MCL) method [4] by considering networks’ nodes as agents capable to take decisions. In this framework we have introduced a memory factor to mimic a typical human behavior such as the oblivion effect. The method is based on information diffusion and it includes a non-linear processing phase. We test our method on two classical community benchmark and on computer generated networks with known community structure. Our approach has three important features: the capacity of detecting overlapping communities, the capability of identifying communities from an individual point of view and the fine tuning the community detectability with respect to prior knowledge of the data. Finally we discuss how to use a Shannon entropy measure for parameter estimation in complex networks.     

Mixed-integer programming models for nesting problems

Several industrial problems involve placing objects into a container without overlap, with the goal of minimizing a certain objective function. These problems arise in many industrial fields such as apparel manufacturing, sheet metal layout, shoe manufacturing, VLSI layout, furniture layout, etc., and are known by a variety of names: layout, packing, nesting, loading, placement, marker making, etc. When the 2-dimensional objects to be packed are non-rectangular the problem is known as the nesting problem. The nesting problem is strongly NP-hard. Furthermore, the geometrical aspects of this problem make it really hard to solve in practice. In this paper we describe a Mixed-Integer Programming (MIP) model for the nesting problem based on an earlier proposal of Daniels, Li and Milenkovic, and analyze it computationally. We also introduce a new MIP model for a subproblem arising in the construction of nesting solutions, called the multiple containment problem, and show its potentials in finding improved solutions.     

Local linear convergence of approximate projections onto regularized sets

The numerical properties of algorithms for finding the intersection of sets depend to some extent on the regularity of the sets, but even more importantly on the regularity of the intersection. The alternating projection algorithm of von Neumann has been shown to converge locally at a linear rate dependent on the regularity modulus of the intersection. In many applications, however, the sets in question come from inexact measurements that are matched to idealized models. It is unlikely that any such problems in applications will enjoy metrically regular intersection, let alone set intersection. We explore a regularization strategy that generates an intersection with the desired regularity properties. The regularization, however, can lead to a significant increase in computational complexity. In a further refinement, we investigate and prove linear convergence of an approximate alternating projection algorithm. The analysis provides a regularization strategy that fits naturally with many ill-posed inverse problems, and a mathematically sound stopping criterion for extrapolated, approximate algorithms. The theory is demonstrated on the phase retrieval problem with experimental data. The conventional early termination applied in practice to unregularized, consistent problems in diffraction imaging can be justified fully in the framework of this analysis providing, for the first time, proof of convergence of alternating approximate projections for finite dimensional, consistent phase retrieval problems.