Operator Theory Arising from Systems Engineering

源于系统工程的算子理论

基本信息

  • 批准号:
    1500835
  • 负责人:
  • 金额:
    $ 32.44万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Continuing Grant
  • 财政年份:
    2015
  • 资助国家:
    美国
  • 起止时间:
    2015-07-01 至 2020-06-30
  • 项目状态:
    已结题

项目摘要

Optimization is one of the areas most critical to modern technology, since designers always try to minimize cost or maximize performance, safety, output, etc. It can be thought of in two parts: convex optimization and nonconvex optimization. The concept of a convex function is illustrated by a cup; it has unique lowest point (minimum), while for nonconvex problems one would think of a mountain range with many valleys, hence many lowest points (local minima). Computer algorithms are good at finding one (or even a few) of local minima, but a major open problem is this: out of all the local minima, find the lowest (global) one. For convex problems all local minima are global, which means that computer runs do not report a false minimum. Of course, in technology the number of variables is huge, so all that is available to the designer are algebraic formulas (not pictures); thus the cup and mountain metaphors are misleadingly simple. There are two major classes of convex optimization problems solvable on a computer: classical linear programing and (within the last twenty years) the more widely applicable linear matrix inequalities (LMIs). This project concerns many aspects of LMIs, including the scope of LMI techniques: problems treatable with LMIs are convex, but conversely, which convex problems are treatable with LMIs? With collaborators the PI has sketched out a roadmap for this problem and pursues its confirmation. What one sees in linear systems engineering and control are problems with matrix unknowns. Simplifying physical problems and converting them to convex ones is currently done by ad hoc algebraic tricks. A major goal in this project is to develop a theory that will help systematize this. A particular concern is changes of variables to convert nonconvex problems to LMIs. Another is approximating a set with a convex set. In addition, the PI's group is the main provider to the public of software (called NCAlgebra) for performing general noncommuting algebra calculations in Mathematica. NCAlgebra is developed in the course of doing experiments for the proposed research.Classical real algebraic geometry develops a theory of (commutative) polynomials and much of it concerns inequalities based on evaluating them on tuples of real numbers. A good part of this project concerns noncommutative polynomials and their properties when evaluated on tuples of matrices (of all sizes). This new (freely) noncommutative real algebraic geometry often behaves much more rigidly than classical real algebraic geometry. While seeing how classical structure transports to free real algebraic geometry is part of the pursuit, engineering motivation and the highly rigid structure opens up new classes of problems. For example, free convexity, change of variables to achieve free convexity, free convex hulls, and free dilation theory are mathematically rich areas involving mixtures of functional analysis, optimization theory, algebra, and several complex variables. Also studied in this project are interactions with other subjects such as free probability as well as commutative topics related mostly to so-called linear matrix inequalities.
优化是现代技术中最关键的领域之一,因为设计师总是试图最大限度地降低成本或最大限度地提高性能,安全性,产量等。凸函数的概念可以用一个杯子来说明;它有唯一的最低点(最小值),而对于非凸问题,人们会认为山脉有许多山谷,因此有许多最低点(局部最小值)。计算机算法擅长于找到一个(甚至几个)局部极小值,但一个主要的开放问题是:在所有的局部极小值中,找到最低的(全局)极小值。对于凸问题,所有的局部极小值都是全局的,这意味着计算机运行不会报告错误的极小值。当然,在技术中,变量的数量是巨大的,所以设计者所能得到的只是代数公式(而不是图片);因此,杯子和山的比喻简单得令人误解。有两大类凸优化问题可以在计算机上解决:经典线性规划和(在过去的二十年中)更广泛应用的线性矩阵不等式(LMI)。这个项目涉及到LMI的许多方面,包括LMI技术的范围:可以用LMI处理的问题是凸的,但反过来,哪些凸问题可以用LMI处理?PI与合作者一起为这个问题勾画了一个路线图,并寻求确认。我们在线性系统工程和控制中看到的是矩阵未知数的问题。简化物理问题并将其转化为凸问题目前是通过特别的代数技巧来完成的。这个项目的一个主要目标是发展一个理论,将有助于系统化。一个特别关注的是变量的变化,将非凸问题转化为LMI。另一个是用凸集来近似一个集合。此外,PI的小组是向公众提供软件(称为NCAlgebra)的主要供应商,用于在Mathematica中执行一般非交换代数计算。经典的真实的代数几何发展了一种(可交换的)多项式的理论,其中大部分涉及基于对真实的数的元组求值的不等式。这个项目的一个很好的部分涉及非交换多项式和它们的性质时,评估元组的矩阵(所有大小)。这种新的(自由的)非交换的真实的代数几何往往比经典的真实的代数几何表现得更加严格。虽然看到经典结构如何运输到自由的真实的代数几何是追求的一部分,工程动机和高度刚性的结构开辟了新的问题类别。例如,自由凸性,变量的变化以实现自由凸性,自由凸包和自由膨胀理论是数学上丰富的领域,涉及泛函分析,优化理论,代数和几个复变量的混合。该项目还研究了与其他学科的相互作用,如自由概率以及与所谓的线性矩阵不等式相关的交换主题。

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

J. William Helton其他文献

NonlinearH ∞ control theory for stable plants
Optimization over analytic functions whose founrier coefficients are constrained
  • DOI:
    10.1007/bf01203384
  • 发表时间:
    1995-12-01
  • 期刊:
  • 影响因子:
    0.900
  • 作者:
    J. William Helton;Orlando Merino;Trent E. Walker
  • 通讯作者:
    Trent E. Walker
The Hessian of a noncommutative polynomial has numerous negative eigenvalues
  • DOI:
    10.1007/s11854-007-0016-y
  • 发表时间:
    2007-08-01
  • 期刊:
  • 影响因子:
    0.900
  • 作者:
    Harry Dym;J. William Helton;Scott Mccullough
  • 通讯作者:
    Scott Mccullough
Factorization results related to shifts in an indefinite metric
Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form
  • DOI:
    10.1007/s11854-009-0017-0
  • 发表时间:
    2009-09-11
  • 期刊:
  • 影响因子:
    0.900
  • 作者:
    Harry Dym;Jeremy M. Greene;J. William Helton;Scott A. McCullough
  • 通讯作者:
    Scott A. McCullough

J. William Helton的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

{{ truncateString('J. William Helton', 18)}}的其他基金

Operator Theory Arising from Systems Engineering
源于系统工程的算子理论
  • 批准号:
    1201498
  • 财政年份:
    2012
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
FRG: Collaborative Research: Semidefinite optimization and convex algebraic geometry
FRG:协作研究:半定优化和凸代数几何
  • 批准号:
    0757212
  • 财政年份:
    2008
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Standard Grant
Operator Theory Arising from Systems Engineering
源于系统工程的算子理论
  • 批准号:
    0700758
  • 财政年份:
    2007
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Operator Theory Arising from Systems Engineering
源于系统工程的算子理论
  • 批准号:
    0400794
  • 财政年份:
    2004
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Operatory Theory and Systems Engineering
操作理论与系统工程
  • 批准号:
    0100576
  • 财政年份:
    2001
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Operator Theory and Systems Engineering
算子理论与系统工程
  • 批准号:
    9732891
  • 财政年份:
    1998
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Operator Theory and Systems Engineering
数学科学:算子理论与系统工程
  • 批准号:
    9501064
  • 财政年份:
    1995
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Operator Theory and Systems Engineering
数学科学:算子理论与系统工程
  • 批准号:
    9207740
  • 财政年份:
    1992
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Operator Theory and Communications Engineering
数学科学:算子理论与通信工程
  • 批准号:
    8902098
  • 财政年份:
    1989
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Conference on Functional Analysis andApplications
数学科学:泛函分析与应用会议
  • 批准号:
    8703163
  • 财政年份:
    1987
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Standard Grant

相似国自然基金

Research on Quantum Field Theory without a Lagrangian Description
  • 批准号:
    24ZR1403900
  • 批准年份:
    2024
  • 资助金额:
    0.0 万元
  • 项目类别:
    省市级项目
基于isomorph theory研究尘埃等离子体物理量的微观动力学机制
  • 批准号:
    12247163
  • 批准年份:
    2022
  • 资助金额:
    18.00 万元
  • 项目类别:
    专项项目
Toward a general theory of intermittent aeolian and fluvial nonsuspended sediment transport
  • 批准号:
  • 批准年份:
    2022
  • 资助金额:
    55 万元
  • 项目类别:
英文专著《FRACTIONAL INTEGRALS AND DERIVATIVES: Theory and Applications》的翻译
  • 批准号:
    12126512
  • 批准年份:
    2021
  • 资助金额:
    12.0 万元
  • 项目类别:
    数学天元基金项目
基于Restriction-Centered Theory的自然语言模糊语义理论研究及应用
  • 批准号:
    61671064
  • 批准年份:
    2016
  • 资助金额:
    65.0 万元
  • 项目类别:
    面上项目

相似海外基金

Inverse Problems Arising from Kinetic Theory and Applications
动力学理论及其应用产生的反问题
  • 批准号:
    2306221
  • 财政年份:
    2023
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Continuing Grant
Modular properties of algebraic structures arising from conformal field theory
共形场论产生的代数结构的模性质
  • 批准号:
    2554038
  • 财政年份:
    2020
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Studentship
Novel Modeling of Multi-phase Flows using Methodologies Arising from Kinetic Theory
使用动力学理论产生的方法对多相流进行新颖的建模
  • 批准号:
    503009-2017
  • 财政年份:
    2019
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Postdoctoral Fellowships
Qualitative study on model equations arising in mathematical biology from a viewpoint of the bifurcation theory
分岔理论视角下数学生物学模型方程的定性研究
  • 批准号:
    17KK0086
  • 财政年份:
    2018
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Fund for the Promotion of Joint International Research (Fostering Joint International Research)
Novel Modeling of Multi-phase Flows using Methodologies Arising from Kinetic Theory
使用动力学理论产生的方法对多相流进行新颖的建模
  • 批准号:
    503009-2017
  • 财政年份:
    2018
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Postdoctoral Fellowships
Novel Modeling of Multi-phase Flows using Methodologies Arising from Kinetic Theory
使用动力学理论产生的方法对多相流进行新颖的建模
  • 批准号:
    503009-2017
  • 财政年份:
    2017
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Postdoctoral Fellowships
Dynamical theory of switching predation based on chemical substances arising learning behaviors in the parasitic wasp
基于化学物质引起寄生蜂学习行为的切换捕食动力学理论
  • 批准号:
    17H03731
  • 财政年份:
    2017
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
Solutions for boundary value problems arising in chemical reactor theory
化学反应器理论中边值问题的求解
  • 批准号:
    510295-2017
  • 财政年份:
    2017
  • 资助金额:
    $ 32.44万
  • 项目类别:
    University Undergraduate Student Research Awards
Study of parabolic systems with discontinuous nonlinearities arising in game theory
博弈论中具有不连续非线性的抛物线系统的研究
  • 批准号:
    16K05226
  • 财政年份:
    2016
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Study of independence of periods arising in number theory
数论中周期独立性的研究
  • 批准号:
    15K17525
  • 财政年份:
    2015
  • 资助金额:
    $ 32.44万
  • 项目类别:
    Grant-in-Aid for Young Scientists (B)
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了